2000年IMO中国国家集训队选拔考试试题

利用高斯函数求某些难度大的数列项之和往往能起到常规方法无能为力的作用但有时还要用到一特殊的方法与技巧才能使问题快速获解
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中 等 数 学
2002 年 IMO 中国国家集训队选拔赛试题
(2002203231 8 :00～12 :30)
一、 设凸四边形 ABCD 的两组对边所在的直线分 别交于 E 、 F 两点 ,两对角线的交点为 P ,过 P 作 PO ⊥EF 于 O . 求证 : ∠BOC = ∠AOD . ( 冷岗松 命题) 肖振纲 1 1 2 (1 + an - 1 ) , n ≥ 二、 设 a1 = , an = 2. 求最小 4 4 实数 λ,使得对任意非负实数 x1 , x2 , …, x2 002 ,都有
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求函数 g ( r , s , t ) 的最小值 .
( 黄玉民 命题)
( 上接第 20 页) ∴ 数列{ an }的前 n 项之和为 2 n + n +2 Sn = . 4 注 : 运用 “多退少补” 的试算方法 , 把 ① 折 成 ②,即 S n = f ( n ) - f ( i , k ) 的形式 , 使不等 2 i - 3i - 4k +2 式0≤ < 1 成立而符合要求 ,从 4 而使问题获解 . 例9 设数列{ an } 的前十项为 1 ,1 ,2 ,3 , ). 4 ,4 ,5 ,6 ,7 ,8 , 且 an + 10 = a n + 8 ( n = 1 ,2 , … 求 S n ,并计算 S 2 002 之值 . 解 :设待定常数 i ∈ {1 ,2 ,3 , …,10} , m 为 分段顺序数 , k 为非负整数 , 使 n = 10 m + i ,

Romanian Olympiad 2000 IMO Team Selection Tests
1st Test - April 27, 2000
1. Let n ≥ 2 be a positive integer. Find the number of functions f : {1, 2, . . . , n} → {1, 2, 3, 4, 5} which have the following property
Work time: 4 hours. TeX (c) 2003 Valentin Vornicu - MathLinks.ro
n = ± a1 ± a2 ± a3 ± a4 ± a5 .
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Radu Ignat
4. Let P1P2 . . . Pn be a convex polygon in the plane. We assume that for any arbitrary choice of vertices Pi, Pj there exists a vertex in the polygon Pk, such that ∠PiPkPj = 60◦. Prove that n = 3.
Work time: 4 hours. TeX (c) 2003 Valentin Vornicu - MathLinks.ro
4th Test - May 21, 2000
1. Let n ≥ 3 be an odd integer and m ≥ n2 − n + 1 be an integer number. The sequence of polygons P1, P2, . . . , Pm is defined as follows (i) P1 is a regular polygon with n vertices; (ii) Pk+1 is a regular polygon whose vertices are the midpoints of the sides of Pk, for any k ∈ {1, 2, . . . , m − 1}. Find, with proof, the maximum number of colors which can be used to color all the vertices of these polygons such that, for each such coloring, one could find four vertices A, B, C, D which have the same color and the quadrilateral ABCD is an isosceles trapezoid. Radu Ignat

1.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 cos x的二次方程a cos2x +b cos x +c = 0,试用a,b,c作出一个关于 cos 2x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 cos x和cos 2x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。

1.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：tan α = 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'）。

41st IMO2000Problem1.AB is tangent to the circles CAMN and NMBD.M lies between C and D on the line CD,and CD is parallel to AB.The chords NA and CM meet at P;the chords NB and MD meet at Q.The rays CA and DB meet at E.Prove that P E=QE.Problem2.A,B,C are positive reals with product1.Prove that(A−1+1 B )(B−1+1C)(C−1+1A)≤1.Problem3.k is a positive real.N is an integer greater than1.N points are placed on a line,not all coincident.A move is carried out as follows. Pick any two points A and B which are not coincident.Suppose that A lies to the right of B.Replace B by another point B to the right of A such that AB =kBA.For what values of k can we move the points arbitrarily far to the right by repeated moves?Problem4.100cards are numbered1to100(each card different)and placed in3boxes(at least one card in each box).How many ways can this be done so that if two boxes are selected and a card is taken from each,then the knowledge of their sum alone is always sufficient to identify the third box?Problem5.Can wefind N divisible by just2000different primes,so that N divides2N+1?[N may be divisible by a prime power.]Problem6.A1A2A3is an acute-angled triangle.The foot of the altitude from A i is K i and the incircle touches the side opposite A i at L i.The line K1K2is reflected in the line L1L2.Similarly,the line K2K3is reflected in L2L3and K3K1is reflected in L3L1.Show that the three new lines form a triangle with vertices on the incircle.1。