高中数学竞赛 历届IMO竞赛试题(1-46完整) 37页

AccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas第1届IMO1. 求证(21n+4)/(14n+3)对每个自然数n都是最简分数。2.  设√(x+√(2x-1))+√(x-√(2x-1))=A，试在以下3种情况下分别求出x的实数解： (a)A=√2；(b)A=1；(c)A=2。3. a、b、c都是实数，已知cosx的二次方程acos2x+bcosx+c=0,试用a,b,c作出一个关于cos2x的二次方程，使它的根与原来的方程一样。当a=4,b=2,c=-1时比较cosx和cos2x的方程式。4. 试作一直角三角形使其斜边为已知的c，斜边上的中线是两直角边的几何平均值。5. 在线段AB上任意选取一点M，在AB的同一侧分别以AM、MB为底作正方形AMCD、MBEF，这两个正方形的外接圆的圆心分别是P、Q，设这两个外接圆又交于M、N，   (a.)求证AF、BC相交于N点；  （b.)求证不论点M如何选取直线MN都通过一定点S；   (c.)当M在A与B之间变动时，求线断PQ的中点的轨迹。6. 两个平面P、Q交于一线p，A为p上给定一点，C为Q上给定一点，并且这两点都不在直线p上。试作一等腰梯形ABCD（AB平行于CD），使得它有一个内切圆，并且顶点B、D分别落在平面P和Q上。第2届IMO1. 找出所有具有下列性质的三位数N：N能被11整除且N/11等于N的各位数字的平方和。2. 寻找使下式成立的实数x：面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas4x2/(1-√(1+2x))2 < 2x+93. 直角三角形ABC的斜边BC的长为a，将它分成n等份（n为奇数），令a为从A点向中间的那一小段线段所张的锐角，从A到BC边的高长为h，求证：tana=4nh/(an2-a).4. 已知从A、B引出的高线长度以及从A引出的中线长，求作三角形ABC。5. 正方体ABCDA'B'C'D'（上底面ABCD，下底面A'B'C'D'）。X是对角线AC上任意一点，Y是B'D'上任意一点。a.求XY中点的轨迹；b.求(a)中轨迹上的、并且还满足ZY=2XZ的点Z的轨迹。6. 一个圆锥内有一内接球，又有一圆柱体外切于此圆球，其底面落在圆锥的底面上。令V1为圆锥的体积，V2为圆柱的体积。   (a). 求证：V1不等于V2；   (b). 求V1/V2的最小值；并在此情况下作出圆锥顶角的一般。7. 等腰梯形ABCD，AB平行于DC，BC=AD。令AB=a,CD=c，梯形的高为h。X点在对称轴上并使得角BXC、AXD都是直角。试作出所有这样的X点并计算X到两底的距离；再讨论在什么样的条件下这样的X点确实存在。第3届IMO1. 设a、b是常数，解方程组x+y+z=a;  x2+y2+z2=b2;  xy=z2并求出若使x、y、z是互不相同的正数，a、b应满足什么条件？2. 设a、b、c是某三角形的边，A是其面积，求证：a2+b2+c2>=4√3A.并求出等号何时成立。　面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas3. 解方程cosnx-sinnx=1,其中n是一个自然数。4. P是三角形ABC内部一点，PA交BC于D，PB交AC于E，PC交AB于F，求证AP/PD,BP/PE,CP/PF中至少有一个不大于2，也至少有一个不小于2。5. 作三角形ABC使得AC=b,AB=c，锐角AMB=a,其中M是线断BC的中点。求证这个三角形存在的充要条件是btan(a/2)<=c1/2.3. 正方体ABCDA'B'C'D'（ABCD、A'B'C'D'分别是上下底）。一点x沿着正方形ABCD的边界以方向ABCDA作匀速运动；一点Y以同样的速度沿着正方形B'C'CB的边界以方向B'C'CBB'运动。点X、Y在同一时刻分别从点A、B'开始运动。求线断XY的中点的轨迹。4. 解方程cos2x+cos22x+cos23x=1。5. 在圆K上有三个不同的点A、B、C。试在K上再作出一点D使得这四点所形成的四边形有一个内切圆。6. 一个等腰三角形，设R为其外接圆半径，内切圆半径为r，求证这两个圆的圆心的距离是√(R(R-2r))。面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas7. 求证：正四面体有5个不同的球，每个球都与这六条边或其延长线相切；反过来，如果一个四面体有5个这样的球，则它必然是正四面体。第5届IMO1. 找出下列方程的所有实数根（其中p是实参数）：　√(x2-p)+2√(x2-1)=x.2. 给定一点A及线断BC，设空间中一点P使得存在线段BC上有一点X满足角APX是直角，试求出所有这样的点P的轨迹。3. 在一个n边形中，所有内角都相等，边长依次是 a1>=a2>=...>=an，求证：所有边长都相等。4. 设y是一个参数，试找出方程组xi+xi+2=yxi+1(i=1,...,5)的所有解x1,...,x5。5. 求证cospi/7-cos2pi/7+cos3pi/7=1/2.6. 五个同学A、B、C、D、E参加竞赛，一种猜测说比赛结果的名次依然是ABCDE。但是实际上没有一位同学的名次被猜中，而且预测中名次相邻的同学也没有真的相邻（例如，C、D两位同学名次不是(1,2)、(2,3)、(3,4)、(4,5)中的任何一种）。还有一种猜测说结果会是DAECB的顺序。实际上是恰好有两个同学所得的名次与预测的一样；而且有两对同学（4个不同的同学）的名次像预测中的一样是相连。试讨论最后的名次如何？第6届IMO面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas1.  (a) 求所有正整数n使得 2n-1能被7整除；  (b)求证不存在正整数n使得2n+1能被7整除。2. 假设a、b、c是某三角形的三边长，求证：a2(b+c-a)+b2(c+a-b)+c2(a+b-c)<=3abc.3. 三角形ABC的三边长为别为a、b、c。分别平行于ABC的各边作三角形ABC内切圆的切线，每条切线都在ABC中又切出一个小三角形，再在每个这样的小三角形中作内切圆，求这四个内切圆的面积之和（用a,b,c表示）。4.十七个人互相通信，每一个人都和其他人写信。在他们的信上一共讨论有三个不同的话题，每两个人只讨论一个话题，求证：这些人当中至少有三个人他们所讨论的话题是一样的。5.平面上有五个点，任意两点的连线都不平行，也不垂直，现从每一个点向其他四点两两连接的直线作垂线，试求出所有这些垂线的交点的最大数目。6.四面体ABCD的中心是D0，分别过A、B、C作DD0的平行线，这些线分别交平面BCD、CAD、ABD于点A0、B0、C0，求证：ABCD的体积是A0B0C0D0的三分之一；再问如果D0为三角形ABC内的任意一点，结果是否仍然成立？第7届IMO1. 试找出所有位于区间[0,2pi]的x使其满足    2cosx≤|√(1+sin2x)-√(1-sin2x)|≤√2.2. 如下方程组的系数aij，        a11x1+a12x2+a13x3=0        a21x1+a22x2+a23x3=0        a31x1+a32x2+a33x3=0面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas满足：a. a11、a22、a33是正数，其余是负数；b.每个方程中的系数之和是正的。求证：该方程组的有唯一的解x1=x2=x3=0。3. 四面体ABCD被平行于AB、CD边的一个平面分割成两部分，并且该平面到AB边的距离是该平面到CD边距离的k倍。试求出这两部分的体积比。4. 四个实数，它们中的任何三个的乘积再加上第四个数都等于2，求出这四个数的所有可能值。5. 三角形OAB中的角O是锐角，M是边AB上任意一点，从M向OA、OB边引垂线，垂足分别为P、Q。设三角形OPQ的垂心为，求出当M在AB边上移动时点H的轨迹；若M在三角形OAB内部移动是H的轨迹又是什么？6. 平面上给定了n>2个点，任何两点之间都有线断相连，这些线断长度中的最大值被定义为这个点集的直径，求证：长度为直径的线断至多有n条。第8届IMO1. 在一次数学竞赛中共有A、B、C三道题，25名参赛者每人至少答对了一题。在所有没有答对A的学生中，答对B的人数是答对C的人数的两倍，只答对问题A的人数比既答对A又至少答对其他一题的人数多1。又已知在所有恰好答对一题的参赛者中，有一半没有答对A。请问有多少学生只答对B？2. 三角形ABC，如果，BC+AC=tanC/2(BCtanA+ACtanB).则该三角形为等腰三角形。3. 求证：从正四面体的内切圆圆心到各顶点距离之和小于从空间中任意其他点到各顶点距离之和。4. 对任何自然数n以及满足sin2nx不为0的实数x，求证：面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas1/sin2x+1/sin4x+...+1/sin2nx=cotx-cot2nx.5.  ai（i=1,2,3,4）是互不相同的实数，解方程组（i=1,2,3,4）|ai-a1|x1+|ai-a2|x2+|ai-a3|x3+|ai-a4|x4=1。6. 在三角形ABC的边BC、CA、AB上分别任选三内点K、L、M，求证三角形AML、BKM、CLK之中至少有一个的面积小于活等于三角形ABC的四分之一。第9届IMO1. 平行四边形ABCD，边长AB=a,AD=1, 角BAD=A,已知三角形ABD是一个锐角三角形，求证以A,B,C,D为圆心半径为1的四个圆能够覆盖此平行四边形的充要条件是 a≤cosA+√3sinA.2. 若四面体有且仅有一边大于1，求证其体积≤1/8.3. k,m,n是自然数且m+k+1是一个大于n+1的素数，令cs=s(s+1)，求证(cm+1-ck)(cm+2-ck)...(cm+n-ck)可被乘积c1c2...cn整除。4. 任意两个锐角三角形A0B0C0和A1B1C1。考虑所有与三角形A1B1C1相似且外接于三角形A0B0C0的所有三角形ABC（即BC边包含A0，CA边包含B0，AB边包含C0），试构造出满足此条件的面积最大的三角形ABC。5. a1,...,a8是不全为0的实数，令cn=a1n+a2n+...+a8n(n=1,2,3,...)，如果数列{cn}中有无穷多项等于0，试求出所有使cn＝0的自然数n。6. 在一次运动会中，连续n天内（n>1）一共颁发了m块奖牌。在第一天，颁发了一块奖牌以及剩下m-1个中的1/7；在第二天颁发了两块奖牌以及剩下的1/7；依此类推。在最后一天即第n天，剩下的n块奖牌全部颁发完毕。问该运动会共进行了几天，一共颁发了多少块奖牌？面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas第10届IMO1. 求证有且仅有一个三角形，它的边长为连续整数，有一个角是另外一个角的两倍。2. 试找出所有的正整数n，其各位数的乘积等于n2-10n-22。3. a,b,c是不全为0的实数。x1,x2,...,xn是满足下述方程组的未知数：    axi2+bxi+c=xi+1,对于i=1,2,...,n-1；    axn2+bxn+c=x1；若设M=(b-1)2-4ac，求证：a.若M<0，则方程组无解；b.若M=0，则方程组恰有一解；c.若M>0，则方程组不止有一个解。4. 求证任何四面体上都有一个顶点使得经过该顶点的三条边可构成一个三角形的三边。5. 令f是定义在所有实数并取值实数的函数，并且对于某个a>0及任何x>0有f(x+a)=1/2+√[f(x)-f(x)2]求证f是周期函数，并且当a=1时请给出一个非常值函数的例子。6. 对任何自然数n，试计算下式的值[(n+1)/2]+[(n+2)/4]+[(n+4)/8]+...+[(n+2k)/2k+1]+...其中[x]表示不超过x的最大整数。第11届IMO1. 对任意正整数n，求证有无穷多个正整数m使得n4+m不是质数。2. 令f(x)=cos(a1+x)+1/2cos(a2+x)+1/4cos(a3+x)+...+1/2n-1cos(an+x),其中ai是实数常量，x是实数变量。现已知f(x1)=f(x2)=0，求证x1-x2是面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhasπ的整数倍。3. 对每一个k=1,2,3,4,5，试找出a>0应满足的充要条件使得存在一个四面体，其中k个边长均为a，其余6-k个边的长度均为1。4. 以AB为直径的半圆弧，C是其上不同于A、B的一点，D是C向AB作垂线的垂足。K1是三角形ABC的内切圆，圆K2与CD、DA以及半圆都相切，圆K3与CD、DB及半圆相切。求证：圆K1、K2、K3除AB外还有一条公切线。5. 平面上已给定了n>4个点，无三点共线。求证至少有(n-3)(n-4)/2个凸四边形，其顶点都是已给点集中的点。6. 给定实数x1,x2,y1,y2,z1,z2,满足x1>0,x2>0,x1y1>z12,x2y2>z22，求证：8≤1+1(x1+x2)(y1+y2)-(z1+z2)2x1y1-z12x2y2-z22并给出等号成立的充分必要条件。第12届IMO1. M是三角形ABC的边AB上的任何一点，r、r1、r2分别是三角形ABC、AMC、BMC的内切圆的半径，q是AB外旁切圆的半径（即与AB边相切，与CA、CB的延长线上相切的圆），类似的，q1、q2分别是AC、BC外旁切圆的圆心。求证：r1r2q=rq1q2。2. 已知0≤xi0,xn-1>0。如果a>b，xnxn-1...x0是数A在a进制下的表示、也是B在b进制下的表示，则xn-1xn-2...x0表示了A'在a进制下的表示、B'在b进制下的表示。求证：A'Bc成立。4. 试找出所有的正整数n使得集合{n,n+1,n+2,n+3,n+4,n+5}可被分拆成两个子集合，每个子集合的元素的乘积相等。5. 四面体ABCD，角BDC是直角，D向平面ABC作垂线的垂足恰好是三角形ABC的垂心。求证：(AB+BC+CA)2≤6(AD2+BD2+CD2).并问何时等号成立？6. 平面上给定100个点，无三点共线，求证：这些点构成的三角形中至多70%是锐角三角形。第13届IMO1. 令En=(a1-a2)(a1-a3)...(a1-an)+(a2-a1)(a2-a3)...(a2-an)+...+(an-a1)(an-a2)...(an-an-1).求证 En>=0对于n=3或5成立，而对于其他自然数n>2不成立。2. 凸多边形P1的顶点是A1,A2,...,A9，若将顶点A1平移至Ai时则P1平移成了多边形Pi，求证P1,P2,...,P9之中至少有两个具有一共同内点。3. 求证能够找到一个由形式2n-3（n是正整数）的整数构成的集合并满足任何两个元素互质。4. 四面体ABCD的所有面都是锐角三角形，在线段AB上取一内点X，现在BC上取内点Y，CD上取内点Z，AD上内点T。求证：a.如果∠DAB+∠BCD≠∠CDA+∠ABC，则没有一条闭路径XYZTX具有最小值；b.如果∠DAB+∠BCD＝∠CDA+∠ABC，则有无穷多最短路径XYZTX，它们的长度是2ACsin(k/2)，其中k=∠BAC+∠CAD+∠DAB。5. 对任何自然数m，求证存在平面上一有限点集S，满足：对S中的每一个点A，存在S中的恰好m个点与A的距离为单位长。面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas6. 设A=(aij),其中i,j=1,2,...,n，是一个方阵，元素aij都是非负整数。若i、j使得aij=0，则第i行和第j列的元素之和大于或等于n。求证：该方阵中所有元素之和大于或等于n2/2。第14届IMO1.有十个互不相同的二位数，求证必可从中选出两个不相交的子集，使得这两个子集中的元素之和相等。2. 设n>4，求证每一个圆内接四边形都可以分割成n个圆内接四边形。3. m,n是任意非负整数，求证下式是一整数。(2m)!(2n)! m!n!(m+n)!4. 试找出下述方程组的所有正实数解：    (x12-x3x5)(x22-x3x5)<=0    (x22-x4x1)(x32-x4x1)<=0    (x32-x5x2)(x42-x5x2)<=0    (x42-x1x3)(x52-x1x3)<=0    (x52-x2x4)(x12-x2x4)<=05. f、g都是定义在实数上并取值实数的函数，并且满足方程f(x+y)+f(x-y)=2f(x)g(y)，又已知f不恒等于0且|f(x)|<=1。求证对所有x同样有|g(x)|<=1。6. 给定四个不相同的平行平面，求证存在一个正四面体，它的四个定点分别在这四个平面上。面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas第15届IMO1. OP1,OP2,...,OP2n+1是平面上的单位向量，其中点P1,P2,...,P2n+1都是位于通过点O的一条直线的同一侧，求证|OP1+...+OP2n+1|>=1.2. 问能否在空间中找到一个不共面的有限点集M使得，对M中的任何两点A、B，都可以再在M中寻找到两点C、D，而直线AB、CD是不相同的并且是互相平行的。3. 考虑所有这样的实数a、b使得方程x4+ax3+bx2+ax+1=0至少有一个实根。试找出a2+b2的最小值。4. 一个士兵需要在一个等边三角形的区域内探测有没有地雷，他的扫雷器的半径是三角形高的一半，士兵从三角形的一个定点出发，试问如果要完成任务且使行程最短他应该走什么样的路径？5. G是具有下述形式且非常值的函数的集合：f(x)=ax+b，其中a,b,x都是实数。并且已知G具有这些性质：·如果f,g都属于G，则fg(x)=f(g(x))也属于G；·如果f属于G，则f-1(x)=x/a-b/a也属于G；·对任何f属于G，存在一个实数xf使得f(xf)=xf成立。求证：存在实数M使得f(M)=M对所有G中的函数f都成立。6. a1,a2,...,an是正实数，实数q满足00的整系数多项式，n是P(X)=1或-1的不同整根的个数，则有 n<=d+2.第17届IMO1. 已知x1>=x2>=...>=xn,以及y1>=y2>=...>=yn都是实数，求证若面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhasz1,z2,...,zn是yi的任意排列则有∑(xi-yi)2  <= ∑(xi-zi)2上式中左右两边的求和都是i从1到n。2. 令a1=1，存在无穷多个an可以写成 an=rai+saj的形式，其中r，s是正实数且j>i。3. 任意三角形ABC的边上，向外作三角形ABR,BCP，CAQ，使角CBP、角CAQ都是45度，角BCP、角ACQ都是30度，角ABR、角BAR都是15度。求证角QRP是直角并且QR=RP。4. 令A是将写成十进制数字时的各位数字之和，令B时A的各位数字之和，求B的各位数字之和。5. 判定并证明能否在单位圆上找到1975个点使得任意两点间的距离为有理数。6. 找出所有两个变量的多项式P(x,y)使其满足：I. 对某一正整数n及所有实数t、x、y有P(tx,ty)=tnP(x,y)成立；II.对所有实数x、y、z有P(y+z,x)+P(z+x,y)+P(x+y,z)=0;III.P(1,0)=1。第18届IMO1. 平面上一凸四边形的面积是32，两对边与一对角线之和为16，求另外一个对角线的所有可能的长度。2. 令P1(x)=x2-2,Pi+1=P1(Pi(x)),i=1,2,3,...，求证对任何一个正整数n，方程式Pn(x)=x的所有根都是互不相同的实数。3. 一个长方形的箱子可以用单位正方体完全装满，如果用体积为面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas2的正方体来尽量装填，使得每个边都与箱子的边平行，则恰能装满箱子的40％，求所有这种箱子的可能尺寸（长、宽、高）。4. 试将1976分解成一些正整数之和，求这些正整数乘积的最大值，并加以证明。5. n是一个正整数，m=2n，aij=0、1或-1（1<=i<=n,1<=j<=m）。还有m个未知数x1,x2,...,xm满足下面n个方程：ai1x1+ai2x2+...+aimxm=0,其中i=1,2,...,n。求证这n个方程有一组不全为0的整数解（x1,x2,...,xm）使得|xi|<=m。6. 一个序列u0,u1,u2,...定义为：u0=2,u1=5/2,un+1=un(un-12-2)-u1，n=1,2,...求证[un]=2(2n-(-1)n)/3,其中[x]表示不大于x的最大整数。第19届IMO1. 在正方形ABCD中作等边三角形ABK、BCL、CDM、DAN，证明线段KL、LM、MN、NK的四个中点以及线段AK、BK、BL、CL、CM、DM、DN、AN的八个中点构成一个正十二边形的定点。2. 在一个有限项的实数序列中，任意的相连七项之和为负，任意的相连十一项之和为正。求出这种序列最多有几项。3. n>2是一给定整数，Vn是所有1+kn形式的整数构成的集合，其中k是正整数，对于Vn中的一个数m，如果不存在Vn中的两个数p、q使得m=pq，则称m是不可分解的。求证：Vn中存在一数r，它可有多于一种的方式表示为Vn中不可分解数的乘积。（乘积中若仅仅是因数的顺序不同则视为是同一种分解。）4. 定义f(x)=1-acosx-bsinx-Acos2x-Bsin2x，其中a,b,A,B都是实数常量。如果f(x)>=0对所有实数x都成立，求证面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhasa2+b2<=2且A2+B2<=1.5. a,b是正整数，设a2+b2除以a+b得到商为q，余数是r。试求出所有的正整数对（a,b）使得q2+r=1977。6. f是定义在所有正整数上且取值也是正整数的函数，求证如果f(n+1)>f(f(n))对所有正整数n都成立，则f(n)=n对每个n都成立。第20界IMO　1. m、n都是正整数且n>m。如果1978m和1978n的十进制表示法的末三位数字相同，试求满足此条件并使m+n达到最小的m与n。2. P是某已知球内部一点，A、B、C是球面上三点，且有PA、PB、PC相互垂直，由PA、PB、PC决定的平行六面体与P点对角相向的顶点为Q，试求出Q点的轨迹。3. 两不交集合{f(1),f(2),f(3),...}和{g(1),g(2),g(3),...}的并集是全部的正整数，其中f(1)=　∑1/k；上式中两边的求和都是k从1到n。6. 某国际组织共有来自六个国家的共1978名会员，会员编号分别是1,2,...,1978。求证至少有某一会员的编号，恰为与他同国家的另外两位会员编号的和，或者是他同国家的两外一名会员编号的两倍。面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas第21界IMO　1. m,n是满足下述条件的正整数:m/n=1-1/2+1/3-1/4+...-1/1318+1/1319.求证：m可被1979整除。2. 一个棱柱的上底和下底分别是正五边形A1A2A3A4A5、B1B2B3B4B5。这两个正五边形的每条边以及每个AiBj边都被染上红色或蓝色。又已知每个边都被着色的三角形（其顶点即这个棱柱的顶点）必有两边着不同色，求证：上、下底的十条边都被染上了同一种颜色。3. 平面上的两个圆相交，A是其中一个交点。现有两质点同时从A出发各自以恒定的速度，同以顺时针方向或同以逆时针方向绕各自的圆移动，在绕过一周之后这两点又同时回到了A点。求证：在这个平面上一定存在某个固定的点P使得在任意时刻P点都与这两动点的距离相等。4. 给定一平面k，在这个平面上有一点P，平面外有一点Q，试找出平面k上的所有的点R使得(QP+PR)/QR为最大值。5. 试求出所有的实数a，使得存在非负实数x1,x2,x3,x4,x5满足下列关系式：x1+2x2+3x3+4x4+5x5=a；x1+23x2+33x3+43x4+53x5=a2；x1+25x2+35x3+45x4+55x5=a3。6. 令A、E是一个正八边形的两相对顶点，一只青蛙从A点开始跳动，除了E点外，从八边形中的其他每一个顶点都可以跳至与它相邻两顶点中的任何一个。当它跳到E点时就停止运动。设an为恰好经过n步跳动以后到达E点的所有可能线路的个数，求证：   a2n-1=0   a2n=(2+√2)n-1/√2-(2-√2)n-1/√2。第22界IMO面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas　1. P是三角形ABC内部一点，D、E、F分别是从P点向边BC、CA、AB所引垂线的垂足。试找出BC/PD+CA/PE+AB/PF式达到最小值的所有P点。2. 取r满足1<=r<=n，并考虑集合{1,2,...,n}的所有r元子集，每个子集都有一个最小元素。设F(n,r)是所有这些最小元素的算术平均值。求证：F(n,r)=(n+1)/(r+1)。3. 设m、n是属于{1,2,...,1981}的整数并且满足(n2-mn-m2)2=1。试计算m2+n2的最大值。4.设n>2，问a.n为何值时，存在一个由n个连续的正整数构成的集合使得其中的最大元是其它n-1个元素最小公倍数的因子？b.n为何值时，恰好值存在一个满足条件的集合？5. 三个都通过点O的等半径的圆位于一个给定三角形的内部，并且每个圆都相切于这个三角形的两条边。求证：这个三角形的内心、外心、O点三点共线。6. 函数f(x,y)，对于任何非负整数x,y都满足f(0,y)=y+1,f(x+1,0)=f(x,1),f(x+1,y+1)=f(x,f(x+1,y))。试计算f(4,1981)的值。第23届IMO　1. f(n)是定义在正整数上且取值为非负整数的函数，f(2)=0,f(3)>0,f(9999)=3333，并对所有m,n有f(m+n)-f(m)-f(n)=0或1。试求出f(1982)。2. A1A2A3是不等腰三角形，其三边为a1,a2,a3，其中ai是角Ai的对边，设Mi是边ai的中点，Ti是三角形的内切圆在边ai上的切点，记Si为点Ti关于内角Ai的角平分线的对称点，求证线M1S1,M2S2和M3S3共点。3. 考虑无限正实数序列{xn}满足x0=1及x0>=x1>=x2>=...，面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhasa.求证对每个这样的序列都有存在一个n>=1使得x02/x1+x12/x2+...+xn-12/xn>=3.999.b.试寻找一个这样的序列使其满足 x02/x1+x12/x2+...+xn-12/xn<4 对所有n成立。4. n使正整数，求证如果方程x3-3xy2+y3=n有关于整数x,y的一个解，则其至少有三个解；当n=2891时再证明这个方程无整数解。5. 正六边形ABCDEF的对角线AC、CE上分别有分点M、N并且AM/AC=CN/CE=r，如果B、M、N共线，试求r的值。6. 设S是边长为100的正方形，L是在S内部不自交的系列线段A0A1,A1A2,A2A3,...,An-1An并且A0与An不重合。已知对于每一个在S边界上的点P，L中存在一个点与P之间的距离不大于1/2。求证：L中存在两点X、Y，X与Y的距离不大于1，并且L上位于X和Y之间的部分不少于198。第24届IMO　1. 试找出所有定义在正实数并取值正实数的函数f，使其满足f(x(f(y))=yf(x)对所有x,y成立，并且当x趋向于无穷大时f(x)趋向于0.2. 圆C1、C2的圆心分别是O1、O2，它们相交于两个不同的点，设A是其中一个交点。这两个圆的一条公切线切C1、C2分别于点P1、P2，另外一条公切线分别切C1、C2于点Q1、Q2，再设M1、M2分别是P1Q1和P2Q2的中点，求证：角O1AO2=角M1AM2。3. a,b,c是正整数，并且它们中的任何两个都没有大于1的公约数。求证2abc-ab-bc-ca是不能表示成形式xbc+yca+zab的最大整数，其中x,y,z是非负整数。4. 等边三角形ABC，设集合E是该三角形的所有边界点（即边AB,BC,CA），任意将E分拆成两个不相交的子集合（它们的并集是E），试证明这两个集合中的至少一个包含有三点构成一直角三角形。面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas5. 问是否可能存在小于或等于105的1983个不同的正整数，任何三个都不构成一等茶数列。6. 设a,b,c是一个三角形的三边长，求证a2b(a-b)+b2c(b-c)+c2a(c-a)>=0.并判断何时等号成立。第25届IMO　1. 求证0<=yz+zx+xy-2xyz<=7/27,其中x,y，z是非负实数并满足x+y+z=1.2. 试找出所有的正整数对（a,b）满足ab(a+b)不能被7整除,但(a+b)7-a7-b7可被77整除。3. 给定平面上的点O、A。平面上的每个点都被染色成有限种颜色中的一个。设X是平面上一给定点，以O为圆心的圆C(X)的半径是OX+(∠AOX)/OX，其中角∠AOX是用弧度衡量（即范围是[0,2л)），求证能够找到不在OA上的一点X使得它的颜色出现在圆C(X)的圆周上。4. 凸四边形ABCD的边CD与以AB为直径的圆相切，求证：AB与以CD为直径的圆相且当且仅当BC和AD是平行的。5. 设d是平面上一凸n边形（n>3）的所有对角线的长度之和，p是它的周长。求证: n-3<2d/p<[n/2][(n+1)/2]-2,其中[x]表示不超过x的最大整数。6. 0=o(Qi1).4. 集合M由1985个不同的正整数组成，且每个数都有一个大于23的素因子，求证M中存在4个元素的积是某个整数的4次方。5. 圆心为O的一个圆经过三角形ABC的顶点A和C，并与AB,BC分别交于不同的两点K、N，三角形ABC的外接圆和三角形KBN的外接圆相交于两个不同的点B、M，求证角OMB是直角。6. 对于任何一个实数x1，可通过递推式xn+1=xn(xn+1/n) 构造序列x1,x2,...，求证存在唯一的一个x1满足对所有的n都有0=4时定义As=As-3，现使用以下的方法构造一系列点P1,P2,P3,...：Pk+1是Pk绕Ak+1顺时针旋转120度得到的点（k=0,1,2,...）。如果P1986=P0，求证A1A2A3是等边三角形。3. 给正五边形的每个顶点赋值一个整数，使这5个整数之和是正的。对于任何三个连续的顶点设它们所赋予的数分别是x,y,z，如果y<0则执行下述操作：将x,y,z分别替换为x+y,-y,z+y。重复执行这样的操作直到这5个顶点数中至少有一个是负值。试问能否经过有限步之后操作结束。4. O是正n(n>=5)边形的中心，设A,B是一对相邻的顶点。设开始的时候三角形XYZ与三角形OAB重合，现用如下的方式移动三角形XYZ：保持Y、Z始终在多边形的边界上、X在多边形的内部。试求出当Y、Z都走遍多边形的边界时X点所形成的轨迹。5. 试找出所有定义在非负实数并取值也是非负实数的函数f，使其满足f(2)=0；当0<=x<2时f(x)不等于0；对所有x,y都有f(xf(y))f(y)=f(x+y)。6. 给定平面上的一个有限点集，每个点的坐标都是整数，问有没有一种将这些点涂成红色或白色的染色方法使得在任何一条平行于坐标轴（两个坐标轴中的任何一个）的直线L上的红点和白点的个数之差不大于1？第28届IMO　1. 设pn(k)是集合{1,2,3,...,n}上具有k个固定点的排列的个数，求证k从0到n对(kpn(k))的求和是n!。[一个集合S的一个排列是从S到它自身的一一映射。元素i称为是f固定点如果f(i)=i。]2. 锐交三角形ABC的内角A的角平分线交BC于L，交ABC的外接圆于N，从L点向AB,AC做垂线，垂足分别是K、M，求证四边形AKNM的面积与三角形ABC的面积相等。3. x1,x2,...,xn是实数并且满足x12+x22+...+xn2=1，求证对每个正整数k>=2存在不全为0的整数a1,a2,...,an，使得对每个i有|ai|<=k-1及|a1x1+a2x2+...+anxn|<=(k-1)√n/(kn-1)。4. 求证不存在从非负整数到非负整数的函数f满足对所有n有f(f(n))=n+1987面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas成立。 5. n是大于或等于3的整数，求证存在一个由平面上n个点构成的集合满足任何两点的距离都是无理数并且任何三点构成一个面积为有理数的非退化的三角形。6. n是大于或等于2的整数，如果对所有0<=k<=√n/3都有k2+k+n是素数，则当0<=k<=n-2时，k2+k+n都是素数。第29届IMO1. 考虑平面上同一圆心的两个半径分别为R>r的圆。P点是小圆上一个固定的点，B使大圆上的动点，BP交大圆于C，过P点作BP的垂线交小圆于A点（如果相切则A=P），a.试确定AB2+BC2+CA2的所有可能值；b.试确定BC中点的轨迹。2. n是正整数，A1,A2,...,A2n+1都是集合B的子集，假设i.每个Ai都恰有2n个元素；ii.任何两个不同的Ai恰有一个公共元素；iii.B中的每个元素至少属于两个Ai。试问对于什么样的n值有办法将B中的元素都标上0或1使得每个Ai都恰好包含n个标0的元素。3. 函数f定义在正整数集上：f(1)=1;f(3)=3;且对每个正整数n有f(2n)=f(n),f(4n+1)=2f(2n+1)-f(n)。试确定小于或等于1988并满足f(n)=n的正整数n的个数。4. 试证明满足1/(x-1)+2/(x-2)+3/(x-3)+...+70/(x-70)>=5/4.的所有实数x的集合是一些互不相交的区间的并集，并且这些区间的长度之和是1988。5. 三角形△ABC,角∠A是直角，D是BC边上的高的垂足。三角形△ABD、三角形△ACD的内心的连线分别交边AB,面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".HistoryhasAC于K,L。求证：三角形ABC的面积是三角形AKL的面积的至少两倍。6. a,b都是正整数，且ab+1整除a2+b2.求证(a2+b2)/(ab+1)是完全平方数。第30界IMO试题1. 试证明集合{1,2,...,1989}可以分拆成117个子集合A1,A2,...,A117（即这些子集合互不相交且并集为整个集合），满足每个Ai包含17个元素，并且每个Ai中元素之和都相等。2. 锐角△ABC，内角∠A的角平分线交△ABC的外界圆于A_1，类似定义B1,C1点。设AA1与∠B,∠C的外交平分线交于A0点，类似定义B0,C0点。 求证：△A0B0C0的面积是六边形AC1BA1CB1的两倍也是△ABC面积的至少4倍。3.设n,k是正整数，S是由平面上n个点构成的集合并且无三线共点，对任何S中的点P至少存在S中的k个点与P等距离。求证：k<1/2+√2n。4.凸四边形ABCD的边AB,AD,BC满足AB=AD+BC，四边形内部有一与直线CD距离为h的点P，并且AP=h+AD，BP=h+BC，求证：1/√h<=1/√AD+1/√BC。5.试证明对每个正整数n，存在n个连续的正整数使得其中无素数或素数的幂。6.设{x1,x2,...,xm}是{1,2,...,2n}的一个排列，其中n是一个正整数。如果|xi-xi+1|=n对至少{1,2,...,2n-1}中的一个i成立就说这个排列{x1,x2,...,xm}具有性质P。试证明对于任意的n，具有性质P的排列都比不具有的多。第31界IMO试题1.面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas弦AB,CD相交于圆内一点E，M是线段EB上的一点，过E点与△DEM外接圆的切线分别交BC,AC于F,G。设t=AM/AB，试用t表示EF/EG。2.设n>=3，考虑一个圆上由2n-1个不同点构成的集合E。现给E中恰好k个点染上黑色，如果至少有一对黑点使得这两个黑点之间的弧上（两段弧中的某一个）包含恰好E中的n个点，就成这样的染色方法是“好的”。试找出对于集合E能保证任意一种染色方法都是“好的”的最小的k值。3.试找出所有大于1的正整数n满足(2n+1)/n2也是整数。4.试构造一个从正有理数集到正有理数集的函数f使  f(xf(y))=f(x)/y对任何x,y都成立。5.给定一个初始整数n0>1，两个玩家A,B根据下述规则交替的选择整数n1,n2,n3,...：a.设B已选择n2k，则A选择n2k+1满足    n2k<=n2k+1<=n2k2； b.设A已选择n2k+1，则B选择n2k+2满足    n2k+1/n2k+2=pr 对某个p及r>=1成立。若A选到了数1990就获胜；若B选到了1就获胜。分别求除满足下述条件之一的n0： (1)A有必胜策略； (2)B有必胜策略； (3)A,B都没有必胜策略。6.求证存在一个凸1990边形使得所有角都相等并且边长是12,22,...,19902（顺序不定）。第32界IMO试题1.设I是△ABC的内心，∠A,∠B,∠C的交平分线分别交对边于A',B',C'点，求证: 16是一个整数，a1,a2,...,ak都是小于n的正整数并且与n互素。如果a2-a1=a3-a2=...=ak-ak-1>,求证，n是质数或者是2的幂次方。3.试找出最小的整数n使得每一个S的n元子集都包含5个两两互素的数。4.设G是一个有k条边的连通图，试证明可是对这些边编号1,2,...,k使得对于每个属于两条或两条以上的边的顶点，从这个顶点出发的所有边的标号的最大公约数是1。　注：一个图是由一组顶点和一些连接这些顶点的线段（称为边）组成。每对顶点之间最多有1条边。如果对图中的任何两个不同的顶点x,y都有一些顶点x=v0,v1,...,vm=y使得vi,vi+1(0<=i1，试构造一个有界的无限序列x0,x1,x2,... 使得对任何x≠y都有|xi-xj||i-j|a>=1。 　注：一个无限实数序列x0,x1,x2,...是有界的如果存在一个常数C使得|xi|=4有S(n)<=n2-14；·b.试找出一个整数n使得S(n)=n2-14；·c.试证明有无穷多个整数n使得S(n)=n2-14。　第34届IMO试题1.设f(x)=xn+5xn-1+3，其中n>1是一个整数。求证f(x)不能表示成两个非常数的整系数得多项式的乘积。2.设D是锐角三角形ABC内部一点且∠ADB=∠ACB+90o，AC·BD=AD·BC，oa.计算(AB·CD)/(AC·BD)；ob.求证△ACD,△BCD的外界圆在C处的切线互相垂直。　3.在一个无限大的棋盘上以如下方式做游戏。开始时棋盘中的一个n×n的框上整齐的摆放着n2个棋子（每个小方格上放着一个棋子），游戏的每一步都是在水平或者竖直方向上跨越一个棋子而跳到一个空格子上去，并同时取走所跨越过的棋子。试找出所有的n值使得游戏以只留一个棋子在棋盘上而结束。面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas4.对平面上的三个点P,Q,R，定义m(PQR)为△PQR的最短高的长度(如果P,Q,R共线当然有m(PQR)=0)。求证对任何点A,B,C,X有m(ABC)≤m(ABX)+m(AXC)+m(XBC)。5.问是否存在一个从正整数到正整数的函数f使得f(1)=2,f(f(n))=f(n)+n对所有n，并且f(n1盏灯L0,L1，...,Ln-1绕成一圈，为方便Ln+k也表示Lk。一盏灯只有开或关两个状态，初始时刻它们全是开着的，依次执行步骤s0,s1,...,：在步骤si，如果Li-1点燃，就关掉Li，否则什么都不做。试证明：oa.存在一个正整数M(n)使得在第M(n)步之后所有的灯都开着；ob.若n=2k，则可使M(n)=n2-1；oc.若n=2k+1，则可使M(n)=n2-n+1。　第35届IMO试题1.m和n都是正整数，a1,a2,...,am是{1,2,...,n}中不同的数，只要有ai+aj≤n（i,j可能相同）那么就有某个k使ai+aj=ak，求证(a1+...+am)/m≥(n+1)/2。2.△ABC是等腰三角形，AB=AC，M是BC的中点，O是线AM上的点且OB⊥AB，Q为线段BC上不同于B,C的任意一点，E,F分别在AB,AC上使得E,Q,F不同并共线。求证：OQ⊥EF当且仅当QE=QF。3.对任何正整数k，定义f(k)为集合{k+1,k+2,...,2k}中的用二进制表示后恰有3个1的元素的个数，求证对于每个正整数m，存在至少一个k使f(k)=m；并求出使得恰有一个k的所有m值。4.试求出所有的正整数对(m,n)使得(n3+1)/(mn-1)是整数。5.S是所有大于-1的实数集，试找出所有的从S到S的函数f满足对所有x,y,f(x+f(y)+xf(y))=y+f(x)+yf(x),并且对于-13，使得在平面上存在n个点A1,A2，...,An（无三点共线）及n个实数r1,r2,...,rn满足△AiAjAk的面积是ri+rj+rk，其中是对每个三元组1≤ip+q，令x0,x1,xn都是整数，x0=xn=0且对每个1≤i≤n，xi-xi-1=p或q。 求证存在下标iL>M>N是正整数且KM+LN=(K+L-M+N)(-K+L+M+N)。求证KL+MN是合数。第43界IMO试题1.设n是给定的正整数，T是一个集合，其元素是平面上满足x,y是非负整数且x+y=3.t_1,t_2,...,t_n>0满足n^2+1>(t_1+t_2+...+t_n)(1/t_1+1/t_2+...+1/t_n)证明t_1,t_2,...,t_n中随便取3个数都能构成一个三角5.凸四边形ABCD的对角线BD不平分∠ABC和∠CDA.ABCD内一点P满足∠PBC=∠DBA和∠PDC=∠BDA.求证：ABCD是圆的内接四边形当且仅当AP=CP.6.称一个正整数为“交替的”，如果它的十进表示的任两个连续数位的奇偶性不同.求所有的正整数n，n的某个倍数是交替的.面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome\nAccordingtoOfficePartyofmasslineeducationpracticeactivitiesledgroupofunifiedarrangements,unitsmainledtoforcadresworkersShangatimespartylecture,duetohimselflevellimited,onlyputthisstageconcentratedlearningofexperiencetodayandeveryonecommonwithlearningexchange,purposeisletwefurtherdeepunderstandingmasslineofconnotation,insistedpartyofmassline,domasseswork,followingmainfromfouraaspectstold:a,andfullawarenesspartyofmasslineeducationpracticeactivitiesofmajormeaningParty's18majordistinctproposedtofocusontheparty'sadvancednatureandpurity,intheparty,with"pragmaticandcleanforthepeople"asthemaincontentoftheparty'smasslineandeducationalpractice.　　ThisiscomradeXIJinpingaspartyGeneralSecretaryandadheretostrictlyadministeringtheparty,strengthentheparty'smajordecisionisinlinewiththepublicexpectationsandstrengtheningtheconstructionoflearning-orientedserviceinnovationofMarxistparty'smajordeployment,istheimportantmeasuretoadvancethegreatcauseofsocialismwithChinesecharacteristics.First,carryouttheparty'smasslineineducationpractice,wascarrying18ofthepartyspirit,summonapowerfulforceforrealizingthegreatrejuvenationoftheChinesenationtheChinesedreamofurgentneeds.Party's18topromotestrategicdeploymenttobuildingsocialismwithChinesecharacteristics,presentstwo100-yeargoal,namely,by2020China'sgrossdomesticproductandpercapitaincomeofurbanandruralresidentsin2010onthebasisofdoublecompletethebuildingofawell-offsociety;themiddleofthiscenturyintoaprosperous,democratic,civilizedandharmoniousSocialistmoderncountry,andonthisbasis,theChinesedreamofrealizingthegreatrejuvenationoftheChinesenation.18Congressoftheparty,ComradeXIJinpingrepeatedlyexpoundedtheChinesedream.ThebasicconnotationoftheChinesedream,isthenationalprosperity,nationalrevitalization,thehappinessofthepeople,ChinaimportanttofollowistokeepChinaRoad,carryforwardthespiritandcohesionstrengthofChina,dreamdreamsofthepeopleofChina,butdeterminedconfidence,enhanceself-awareness,achieveself-reliance,tobuildastrongerChina,civilization,harmony,beauty,China,China,China.　　StrategicconceptionofChinawas18thespiritoftheenrichmentanddevelopmentoftheparty,conscientiouslystudytheChinesedream,leadingpeopletohardtorealizetheChinesedream,masseducationactivitiesindepththefundamentalstartingpointandendingpoint.Strengthentheeducationoftheparty'smassline,helpstotrainthecollectiveconsciousnessoftheunityofthepartyandthepeople,playingtheChinesedreamideologicalbases.ComradeMaoZedongstressedthatpartiesshouldhave"commonlanguage"socialistcountriesmusthave"unityofwill".Historyhas2005InternationalMathematicalOlympiad第一天(4.5小时)1.等边三角形ABC各边上的六个点A1,A2(∈BC),B1,B2(∈CA),C1,C2(∈AB)构成六边长相等的凸六边形A1A2B1B2C1C2.求证:三条直线A1B2,B1C2,C1A2交于一点.2.整数数列a1,a2,……中有无穷多个正项及无穷多个负项.已知,对每个正整数n,数a1,a2,…,an除以n所得到的余数互不相同.证明:每个整数在数列a1,a2,……中都出现且只出现一次.3.x,y,z为正数且xyz≥1.求证:(x5-x2)/(x5+y2+z2)+(y5-y2)/(y5+z2+x2)+(z5-z2)/(z5+x2+y2)≥0.第二天(4.5小时)4.试求与无穷数列an=2n+3n+6n－1(n≥1)的一切项均互素的所有正整数.5.取定凸四边形ABCD,其中BC=DA,BC与DA不平行.动点E,F分别在线段BC,DA上且满足BE=DF.直线AC与BD交于P,BD与EF交于Q,EF与AC交于R.求证:当E,F变动时,所有三角形PQR的外接圆周除了P外还有一个公共点.6.一次数学竞赛共给出6道题.已知,每两题均被多于2/5的选手同时解出,但无一人解出所有6道题.证明:至少有两人各解出5道题.面雕美容术是根据脸型以及五官的比例等黄金比例数据，精确设计出最适合您且只属于您的精致脸型。弧度优美，精雕细琢，使得面雕后无论从正面、侧面看都，脸部弧度都自然流畅，立体完美。provedthatapeopleandanation,ifitdoesnothaveitsownspiritualpillar,thereisnounityofspiritualsupport,itmeansthatnosoul,willlosecohesionandvitality.Figurativelyspeaking,asackofpotatoes,quantitycannotbetooateam,formajointforce;onlylikegranite,socloselyUnitedtoforminvincible,nodifficultmeansofstrength.NationalrejuvenationofChinesedream,notonlyembodiesthecommoninterestsandpursuits,andcoversavarietyofgroupsandclasses,hasawideapplicabilityandinclusive,withastrongintegratingforceandleadtheforce.SheembodiestheaspirationsofseveralgenerationsofChinese,reflectingtoday...Intosexandpurity,andtheinevitablerequirementofconstantlyimprovingtheparty'sgoverningcapacity.Strengtheningtheconstructionoftheadvancednatureandpurity,thecoreproblemisalwaysmaintainingtheparty'sflesh-and-bloodtieswiththemasses,sothatourpartywillalwaysbethemostwholeheartedsupportofthemasses.　　Musttakethefundamentalinterestsoftheoverwhelmingmajorityofthepeopleasthepartyallthestartingpointandendingpoint,thepartyhasalwaysbeenacommonfatewiththemasses,thisisthemagicweaponforourpartytoalwaysmaintaintheadvancednatureandpurity.Strengtheningtheconstructionofparty'sadvancednatureandpurityandenhancingtheparty'srulingabilitywilleventuallybeimplementedtorealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Alltheworkmeritsthisisameasureoftheparty'sfundamentalstandards,isameasureoftheparty'sadvancednatureandpuritystandards.Believewho,dependingonwho,who,standingonthepositionoftheoverwhelmingmajorityofthepeoplearealways,alwaysrealize,safeguardanddevelopthefundamentalinterestsoftheoverwhelmingmajorityofthepeople,itisatouchstoneoftheMarxistpartyofjudgement,isalsosignofMarxistpartyisdifferentfromotherpoliticalparties.Marxistpoliticalpartyhasaclearpoliticalpositionandhistoricalmission:fortheinterestsofthevastmajorityofpeople,committedtothefundamentalinterestsoftheoverwhelmingmajorityofthepeople.FromthedayoftheestablishmentoftheChineseCommunistParty,alwaysadheretobasicprinciplesofhistoricalmaterialism,fortheinterestsofthemostpeopleastheirgoal,alwaysservingasitsfundamentalpurposeandvalues.Fullytrustthemasses,closelyrelyonthemassesmaintainflesh-and-bloodtieswiththemassesisalwaysthesourceofourpartyfullofvigorandvitality,isalwaysthemostdeterminedinthedevelopmentofourpartyandourcountry.　　Conscientiouslyacceptsupervisionbythemasses,payattentiontolistentothemasses'assessment,whichitselfwastheparty'sadvancednatureandpurityintopracticalactiontorealizethefundamentalinterestsoftheoverwhelmingmajorityofthepeople.Atpresent,thepartycommitteesandleadingcadresandtheGovernmentgenerallyvaluethecloserelationswiththemasses,doalotofproductivework,buttherearestillsome