1986-2005年IMO中国国家队选拔考试试题

China Team Selection Test 1986 Day 1 1 Let ABCD be a cyclic quadrilateral. Prove that the incenters of the triangles ABC, BCD, CDA and DAB form a rectangle. 2 Let ai , 1 ≤ i ≤ n , and ai , 1 ≤ i ≤ n be 2 · n real numbers. Prove that for xi , 1 ≤ i ≤ n satisfying x1 ≤ x2 ≤ xn the following statemenst are equivalent: i) n∑ k=1 ak · xk ≤ n∑ k=1 bk · xk, ii.) s∑ k=1 ak ≤ s∑ k=1 bk for s = 1, 2, . . . , n− 1 and n∑ k=1 ak = n∑ k=1 bk. 3 Given a positive integer A written in decimal expansion: (an, an−1, . . . , a0) and let f(A) denote n∑ k=0 2n−k · ak. Define A1 = f(A), A2 = f(A1). Prove that: I. There exists positive integer k for which Ak+1 = Ak. II. Find such Ak for 1986. Corrected due to the courtesy of [url=http://www.mathlinks.ro/Forum/profile.php?mode=viewprofileu=2616]zhaoli.[/url] 4 Given a triangle ABC for which C = 90 degrees, prove that given n points inside it, we can name them P1, P2, . . . , Pn in some way such that: n−1∑ k=1 (PKPk+1) 2 ≤ AB2 (the sum is over the consecutive square of the segments from 1 up to n− 1). Edited by orl. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/ China Team Selection Test 1986 Day 2 1 Given a square ABCD whose side length is 1, P and Q are points on the sides AB and AD. If the perimeter of APQ is 2 find the angle PCQ. 2 Given a tetrahedron ABCD, E, F , G, are on the respectively on the segments AB, AC and AD. Prove that: i) area EFG ≤ maxarea ABC,area ABD,area ACD,area BCD. ii) The same as above replacing ”area” for ”perimeter”. 3 Let xi, 1 ≤ i ≤ n be real numbers with n ≥ 3. Let p and q be their symmetric sum of degree 1 and 2 respectively. Prove that: i) p2 · n− 1 n − 2q ≥ 0 ii) ∣∣∣xi − p n ∣∣∣ ≤√p2 − 2nq n− 1 · n− 1 n for every meaningful i. 4 Mark 4 · k points in a circle and number them arbitrarily with numbers from 1 to 4 · k. The chords cannot share common endpoints, also, the endpoints of these chords should be among the 4 · k points. I. Prove that 2 · k pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most 3 · k − 1. II. Prove that the 3 · k − 1 cannot be improved. Corrected due to the courtesy of [url=http://www.mathlinks.ro/Forum/profile.php?mode=viewprofileu=2616]zhaoli.[/url] http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/ China Team Selection Test 1987 Day 1 1 1 a.) For all positive integer k find the smallest positive integer f(k) such that 5 sets s1, s2, . . . , s5 exist satisfying: I. each has k elements; II. si and si+1 are disjoint for i = 1, 2, ..., 5 (s6 = s1) III. the union of the 5 sets has exactly f(k) elements. b.) Generalisation: Consider n ≥ 3 sets instead of 5. Corrected due to the courtesy of [url=http://www.mathlinks.ro/Forum/profile.php?mode=viewprofileu=2616]zhaoli.[/url] 2 A closed recticular polygon with 100 sides (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] 3 Let r1 = 2 and rn = n−1∏ k=1 ri+1, n ≥ 2. Prove that among all sets of positive integers such that n∑ k=1 1 ai < 1, the partial sequences r1, r2, ..., rn are the one that gets nearer to 1. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/ China Team Selection Test 1987 Day 2 1 Given 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. 2 Find all positive integer n such that the equation x3 + y3 + z3 = n · x2 · y2 · z2 has positive integer solutions. 3 Let G be a simple graph with 2 · n vertices and n2 + 1 edges, then there is a K4-one edge, that is two triangles with a common edge. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/ China Team Selection Test 1988 Day 1 1 Suppose 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). 2 Find all functions f : Q 7→ C satisfying (i) For any x1, x2, . . . , x1988 ∈ Q, f(x1 + x2 + . . .+ x1988) = f(x1)f(x2) . . . f(x1988). (ii) f(1988)f(x) = f(1988)f(x) for all x ∈ Q. 3 In 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. 4 Let k ∈ N, Sk = {(a, b)|a, b = 1, 2, . . . , k}. Any two elements (a, b), (c, d) ∈ Sk are called ”undistinguishing” in Sk if a− c ≡ 0 or ±1 (mod k) and b−d ≡ 0 or ±1 (mod k); otherwise, we call them ”distinguishing”. For example, (1, 1) and (2, 5) are undistinguishing in S5. Considering the subset A of Sk such that the elements of A are pairwise distinguishing. Let rk be the maximum possible number of elements of A. (i) Find r5. (ii) Find r7. (iii) Find rk for k ∈ N. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/ China Team Selection Test 1988 Day 2 1 Let f(x) = 3x+ 2. Prove that there exists m ∈ N such that f100(m) is divisible by 1988. 2 Let 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 ∩ PB, DP = λ ·CD. Find the value of λ for which the area of quadrilateral PEFG is maximum. 3 A polygon ∏ is given in the OXY plane and its area exceeds n. Prove that there exist n+1 points P1(x1, y1), P2(x2, y2), . . . , Pn+1(xn+1, yn+1) in ∏ such that ∀i, j ∈ {1, 2, . . . , n + 1}, xj − xi and yj − yi are all integers. 4 There is a broken computer such that only three primitive data c, 1 and −1 are 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 Pn(x) = a0 · xn + a1 · xn−1 + . . .+ an with integer coefficients, the value of Pn(c) can be computed using this computer after only finite operation. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/ China Team Selection Test 1989 Day 1 1 A triangle of sides 3 2 , √ 5 2 , √ 2 is folded along a variable line perpendicular to the side of 3 2 . Find the maximum value of the coincident area. 2 Let v0 = 0, v1 = 1 and vn+1 = 8 · vn − vn−1, n = 1, 2, .... Prove that in the sequence {vn} there aren’t terms of the form 3α · 5β with α, β ∈ N. 3 Find the greatest n such that (z + 1)n = zn + 1 has all its non-zero roots in the unitary circumference, e.g. (α+ 1)n = αn + 1, α 6= 0 implies |α| = 1. 4 Given triangleABC, squaresABEF,BCGH,CAIJ are constructed externally on sideAB,BC,CA, respectively. Let AH ∩ BJ = P1, BJ ∩ CF = Q1, CF ∩ AH = R1, AG ∩ CE = P2, BI ∩AG = Q2, CE∩BI = R2. Prove that triangle P1Q1R1 is congruent to triangle P2Q2R2. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/ China Team Selection Test 1989 Day 2 1 Let N = {1, 2, . . .}. Does there exists a function f : N 7→ N such that ∀n ∈ N, f1989(n) = 2 · n ? 2 AD is the altitude on side BC of triangle ABC. If BC +AD−AB−AC = 0, find the range of ∠BAC. 3 1989 equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently. 4 ∀n ∈ N, P (n) denotes the number of the partition of n as the sum of positive integers (disregarding the order of the parts), e.g. since 4 = 1+1+1+1 = 1+1+2 = 1+3 = 2+2 = 4, so P (4) = 5. ”Dispersion” of a partition denotes the number of different parts in that partitation. And denote q(n) is the sum of all the dispersions, e.g. q(4) = 1 + 2 + 2 + 1 + 1 = 7. n ≥ 1. Prove that (1) q(n) = 1 + n−1∑ i=1 P (i). (2) 1 + n−1∑ i=1 P (i) ≤ √ 2 · n · P (n). http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/ China Team Selection Test 1990 Day 1 1 In a wagon, every m ≥ 3 people have exactly one common friend. (When A is B’s friend, B is also A’s friend. No one was considered as his own friend.) Find the number of friends of the person who has the most friends. 2 Finitely many polygons are placed in the plane. If for any two polygons of them, there exists a line through origin O that cuts them both, then these polygons are called ”properly placed”. Find the least m ∈ N, such that for any group of properly placed polygons, m lines can drawn through O and every polygon is cut by at least one of these m lines. 3 In set S, there is an operation ′′◦′′ such that ∀a, b ∈ S, a unique a ◦ b ∈ S exists. And (i) ∀a, b, c ∈ S, (a ◦ b) ◦ c = a ◦ (b ◦ c). (ii) a ◦ b 6= b ◦ a when a 6= b. Prove that: a.) ∀a, b, c ∈ S, (a ◦ b) ◦ c = a ◦ c. b.) If S = {1, 2, . . . , 1990}, try to define an operation ′′◦′′ in S with the above properties. 4 Number a is such that ∀a1, a2, a3, a4 ∈ R, there are integers k1, k2, k3, k4 such that ∑ 1≤i 0, i = 1, 2, . . . , n, we have n∑ k=1 a · k + a24 Sk < T 2 · n∑ k=1 1 ak , where Sk = k∑ i=1 ai. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/ China Team Selection Test 1993 Day 1 1 For all primes p ≥ 3, define F (p) = p−1 2∑ k=1 k120 and f(p) = 1 2 − { F (p) p } , where {x} = x − [x], find the value of f(p). 2 Let n ≥ 2, n ∈ N, a, b, c, d ∈ N, a b + c d < 1 and a+ c ≤ n, find the maximum value of a b + c d for fixed n. 3 A graph G = (V,E) is given. If at least n colors are required to paints its vertices so that between any two same colored vertices no edge is connected, then call this graph ”n−colored”. Prove that for any n ∈ N, there is a n−colored graph without triangles. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/ China Team Selection Test 1993 Day 2 1 Find all integer solutions to 2 · x4 + 1 = y2. 2 Let S = {(x, y)|x = 1, 2, . . . , 1993, y = 1, 2, 3, 4}. If T ⊂ S and there aren’t any squares in T. Find the maximum possible value of |T |. The squares in T use points in S as vertices. 3 Let ABC be a triangle and its bisector at A cuts its circumcircle at D. Let I be the incenter of triangle ABC, M be the midpoint of BC, P is the symmetric to I with respect to M (Assuming P is in the circumcircle). Extend DP until it cuts the circumcircle again at N. Prove that among segments AN,BN,CN , there is a segment that is the sum of the other two. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/ China Team Selection Test 1994 Day 1 1 Find all sets comprising of 4 natural numbers such that the product of any 3 numbers in the set leaves a remainder of 1 when divided by the remaining number. 2 An n by n grid, where every square contains a number, is called an n-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an n-code to obtain the numbers in the entire grid, call these squares a key. a.) Find the smallest s ∈ N such that any s squares in an n−code (n ≥ 4) form a key. b.) Find the smallest t ∈ N such that any t squares along the diagonals of an n-code (n ≥ 4) form a key. 3 Find the smallest n ∈ N such that if any 5 vertices of a regular n-gon are colored red, there exists a line of symmetry l of the n-gon such that every red point is reflected across l to a non-red point. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/ China Team Selection Test 1994 Day 2 1 Given 5n real numbers ri, si, ti, ui, vi ≥ 1(1 ≤ i ≤ n), let R = 1 n n∑ i=1 ri, S = 1 n n∑ i=1 si, T = 1 n n∑ i=1 ti, U = 1 n n∑ i=1 ui, V = 1 n n∑ i=1 vi. Prove that n∏ i=1 risitiuivi + 1 risitiuivi − 1 ≥ ( RSTUV + 1 RSTUV − 1 )n . 2 Given distinct prime numbers p and q and a natural number n ≥ 3, find all a ∈ Z such that the polynomial f(x) = xn + axn−1 + pq can be factored into 2 integral polynomials of degree at least 1. 3 For any 2 convex polygons S and T , if all the vertices of S are vertices of T , call S a sub- polygon of T . I. Prove that for an odd number n ≥ 5, there exists m sub-polygons of a convex n-gon such that they do not share any edges, and every edge and diagonal of the n-gon are edges of the m sub-polygons. II. Find the smallest possible value of m. http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/ China Team Selection Test 1995 Day 1 1 Find the smallest prime number p that cannot be represented in the form |3a − 2b|, where a and b are non-negative integers. 2 Given a fixed acute angle θ and a pair of internally tangent circles, let the line l which passes through the point of tangency, A, cut the larger circle again at B (l does not pass through the centers of the circles). Let M be a point on the major arc AB of the larger circle, N the point where AM intersects the smaller circle, and P the point on ray MB such that ∠MPN = θ. Find the locus of P as M moves on major arc AB of the larger circle. Corrected due to the courtesy of [url=http://www.mathlinks.ro/Forum/profile.php?mode=viewprofileu=2616]zhaoli.[/url] 3 21 people take a test with 15 true or false questions. It is known that every 2 people have at least 1 correct answer in common. What is the minimum number of people that could have correctly answered the question which the most people were correct on? http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/ China Team Selection Test 1995 Day 2 1 Let S = {A = (a1, . . . , as) | ai = 0 or 1, i = 1, . . . , 8}. For any 2 elements of S, A = {a1, . . . , a8} and B = {b1, . . . , b8}. Let d(A,B) = ∑ i=1 8|ai − bi|. Call d(A,B) the distance between A and B. At most how many elements can S have such that the distance between any 2 sets is a