第 届IMO中国国家队选拔考试试题及部分试题答案

历届IMO试题（1-44届）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：4x2/(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α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：tanα=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''）。

1988Day 11Suppose real numbers A,B,C such that for all real numbers x,y,z the following inequality holds:A (x −y )(x −z )+B (y −z )(y −x )+C (z −x )(z −y )≥0.Find the necessary and sufficient condition A,B,C must satisfy (expressed by means of an equality or an inequality).2Find all functions f :Q →C satisfying(i)For any x 1,x 2,...,x 1988∈Q ,f (x 1+x 2+...+x 1988)=f (x 1)f (x 2)...f (x 1988).(ii)f (1988)f (x )=f (1988)f (x )for all x ∈Q .3In triangle ABC ,∠C =30◦,O and I are the circumcenter and incenter respectively,Points D ∈AC and E ∈BC ,such that AD =BE =AB .Prove that OI =DE and OI ⊥DE .4Let k ∈N ,S k ={(a,b )|a,b =1,2,...,k }.Any two elements (a,b ),(c,d )∈S k are called ”undistinguishing”in S k if a −c ≡0or ±1(mod k )and b −d ≡0or ±1(mod k );otherwise,we call them ”distinguishing”.For example,(1,1)and (2,5)are undistinguishing in S 5.Considering the subset A of S k such that the elements of A are pairwise distinguishing.Let r k be the maximum possible number of elements of A .(i)Find r 5.(ii)Find r 7.(iii)Find r k for k ∈N ./This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 1http://www.mathlinks.ro/1988Day 21Let f (x )=3x +2.Prove that there exists m ∈N such that f 100(m )is divisible by 1988.2Let ABCD be a trapezium AB//CD,M and N are fixed points on AB,P is a variable point on CD .E =DN ∩AP ,F =DN ∩MC ,G =MC ∩P B ,DP =λ·CD .Find the value of λfor which the area of quadrilateral P EF G is maximum.3A polygon is given in the OXY plane and its area exceeds n.Prove that there exist n +1points P 1(x 1,y 1),P 2(x 2,y 2),...,P n +1(x n +1,y n +1)in such that ∀i,j ∈{1,2,...,n +1},x j −x i and y j −y i are all integers.4There is a broken computer such that only three primitive data c ,1and −1are reserved.Only allowed operation may take u and v and output u ·v +v.At the beginning,u,v ∈{c,1,−1}.After then,it can also take the value of the previous step (only one step back)besides {c,1,−1}.Prove that for any polynomial P n (x )=a 0·x n +a 1·x n −1+...+a n with integer coefficients,the value of P n (c )can be computed using this computer after only finite operation./This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 2http://www.mathlinks.ro/。

1987Day 111a.)For all positive integer k find the smallest positive integer f (k )such that 5sets s 1,s 2,...,s 5exist satisfying:I.each has k elements;II.s i and s i +1are disjoint for i =1,2,...,5(s 6=s 1)III.the union of the 5sets has exactly f (k )elements.b.)Generalisation:Consider n ≥3sets instead of 5.Corrected due to the courtesy of[url=http://www.mathlinks.ro/Forum/profile.php?mode=viewprofileu=2616]zhaoli.[/url]2A closed recticular polygon with 100sides (may be concave)is given such that it’s vertices have integer coordinates,it’s sides are parallel to the axis and all it’s sides have odd length.Prove that it’s area is odd.Corrected due to the courtesy of[url=http://www.mathlinks.ro/Forum/profile.php?mode=viewprofileu=2616]zhaoli.[/url]3Let r 1=2and r n =n −1 k =1r i +1,n ≥2.Prove that among all sets of positive integers such that nk =11a i<1,the partial sequences r 1,r 2,...,r n are the one that gets nearer to 1./This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 1http://www.mathlinks.ro/1987Day 21Given a convex figure in the Cartesian plane that is symmetric with respect of both axis,we construct a rectangle A inside it with maximum area (over all posible rectangles).Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure.Find the smallest lamda such that it works for all convex figures.2Find all positive integer n such that the equation x 3+y 3+z 3=n ·x 2·y 2·z 2has positive integer solutions.3Let G be a simple graph with 2·n vertices and n 2+1edges,then there is a K 4-one edge,that is two triangles with a common edge./This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 2http://www.mathlinks.ro/。

2008IMO 中国国家队训练题及解答2008年IMO 中国队培训的主要阶段于6月15日至7月5日在上海中学进行,后期在清华附中调整.在培训期间,单墫、陈永高、冷岗松、余红兵、李伟固、熊斌等教授以及叶中豪、冯志刚先生为国家队队员作了讲座.我们从培训题中精选了一部分,配以个别队员们的解答,推荐给各位读者.1. 设G 为△ABC 内的一点,AG 、BG 、CG 分别交对边于点D 、E 、F.设△AEB和△AFC 的外接圆的公共弦所在的直线为l a ,类似定义l b ,l c .证明:直线l a ,l b ,l c 三线共点.证明:设∆AEB 的外接圆和∆AFC 的外接圆交于1,A A ,则a l 即1AA ,易知1A 在角BAC ∠内,1BAEA 共圆,1CAFA 共圆,类似地定义11,B C . 因为BAEA 1共圆,111FBA ABA A EC ∠=∠=∠故(1.1), 111A A C A A E A B E ∠=∠=∠(1.2), 11A AB A EB ∠=∠(1.3)因为CAFA 1共圆,故∠BFA 1=∠ACA 1=∠ECA 1 (1.4) 由(1.1)、(1.4)得:∆BFA 1~∆ECA 1,1BA BFA E CE=1故(1.5) 对∆BA 1E 用正弦定理并结合(1.2)、(1.3)得111111sin sin sin sin BA A EB A ABA E A BE A AC∠∠==∠∠(1.6) 1111,,b c B ABC C ACB l BB l CC ∠∠同理，在内，在内，即即11sin ,sin C CA AE C CB BD ∠=∠且有11sin sin B BC CDB BA AF∠=∠.故BDCDCE AE AF BF AF CD BD AE CE BF BA BC B CB CA C AC AB A ⋅⋅=⋅⋅=∠∠⋅∠∠⋅∠∠111111B sin sin C sin sin A sin sin 111AC BC CC BC CA AB D E F 而由、、分别交对边、、于、、及塞瓦定理得BD CDCE AE AF BF ⋅⋅=1. BA BC B CB CA C AC AB A 111111B sin sin C sin sin A sin sin ∠∠⋅∠∠⋅∠∠=1,这样利用角元形式的塞瓦定理可知直线AA 1,BB 1,CC 1三线共点.,,a b c l l l 即共点，命题得证。

1996Day 11Let side BC of ABC be the diameter of a semicircle which cuts AB and AC at D and E respectively.F and G are the feet of the perpendiculars from D and E to BC respectively.DG and EF intersect at M .Prove that AM ⊥BC .2S is the set of functions f :N →R that satisfy the following conditions:I.f (1)=2II.f (n +1)≥f (n )≥n n +1f (2n )for n =1,2,...Find the smallest M ∈N such that for any f ∈S and any n ∈N ,f (n )<M .3Let M ={2,3,4,...1000}.Find the smallest n ∈N such that any n -element subset of M contains 3pairwise disjoint 4-element subsets S,T,U such thatI.For any 2elements in S ,the larger number is a multiple of the smaller number.The same applies for T and U .II.For any s ∈S and t ∈T ,(s,t )=1.1.For any s ∈S and u ∈U ,(s,u )>1./This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 1http://www.mathlinks.ro/1996Day 213countries A,B,C participate in a competition where each country has 9representatives.The rules are as follows:every round of competition is between 1competitor each from 2countries.The winner plays in the next round,while the loser is knocked out.The remaining country will then send a representative to take on the winner of the previous round.The competition begins with A and B sending a competitor each.If all competitors from one country have been knocked out,the competition continues between the remaining 2countries until another country is knocked out.The remaining team is the champion.I.At least how many games does the champion team win?II.If the champion team won 11matches,at least how many matches were played?2Let α1,α2,...,αn ,β1,β2,...,βn (n ≥4)be 2sets of real numbers such that n i =1α2i <1,ni =1β2i < 1.Let A 2=1−n i =1α2i ,B 2=1−n i =1β2i ,W =12(1−n i =1αi βi )2.Find allreal numbers λsuch that x n +λ(x n −1+···+x 3+W x 2+ABx +1)=0only has real roots.Corrected due to the courtesy of[url=http://www.mathlinks.ro/Forum/profile.php?mode=viewprofileu=2616]zhaoli.[/url]3Does there exist non-zero complex numbers a,b,c and natural number h such that if integers k,l,m satisfy |k |+|l |+|m |≥1996,then |ka +lb +mc |>1his true?/This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 2http://www.mathlinks.ro/。

2003Day 11ABC is an acute-angled triangle.Let D be the point on BC such that AD is the bisector of ∠A .Let E,F be the feet of perpendiculars from D to AC,AB respectively.Suppose the lines BE and CF meet at H .The circumcircle of triangle AF H meets BE at G (apart from H ).Prove that the triangle constructed from BG ,GE and BF is right-angled.2Suppose A ⊆{0,1,...,29}.It satisfies that for any integer k and any two members a,b ∈A (a,b is allowed to be same),a +b +30k is always not the product of two consecutive integers.Please find A with largest possible cardinality.3Suppose A ⊂{(a 1,a 2,...,a n )|a i ∈R ,i =1,2...,n }.For any α=(a 1,a 2,...,a n )∈A and β=(b 1,b 2,...,b n )∈A ,we defineγ(α,β)=(|a 1−b 1|,|a 2−b 2|,...,|a n −b n |),D (A )={γ(α,β)|α,β∈A }.Please show that |D (A )|≥|A |./This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 1http://www.mathlinks.ro/2003Day 21Find all functions f :Z +→R ,which satisfies f (n +1)≥f (n )for all n ≥1and f (mn )=f (m )f (n )for all (m,n )=1.2Suppose A ={1,2,...,2002}and M ={1001,2003,3005}.B is an non-empty subset ofA .B is called a M -free set if the sum of any two numbers in B does not belong to M .If A =A 1∪A 2,A 1∩A 2=∅and A 1,A 2are M -free sets,we call the ordered pair (A 1,A 2)a M -partition of A .Find the number of M -partitions of A .3x n is a real sequence satisfying x 0=0,x 2=3√2x 1,x 3is a positive integers and x n +1=13√4x n +3√4x n −1+12x n −2for n ≥2.How many integers at least belong to this sequence?/This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 2http://www.mathlinks.ro/。