2002年IMO中国国家队选拔考试试题

China Team Selection Test 2002 Day 1 1 Let E and F be the intersections of opposite sides of a convex quadrilateral ABCD. The two diagonals meet at P . Let O be the foot of the perpendicular from O to EF . Show that ∠BOC = ∠AOD. 2 Suppose a1 = 1 4 , an = 1 4 (1 + an−1)2, n ≥ 2. Find the minimum real λ such that for any non-negative reals x1, x2, . . . , x2002, it holds 2002∑ k=1 Ak ≤ λa2002, where Ak = xk − k (xk + · · ·+ x2002 + k(k−1)2 + 1)2 , k ≥ 1. 3 Seventeen football fans were planning to go to Korea to watch the World Cup football match. They selected 17 matches. The conditions of the admission tickets they booked were such that - One person should book at most one admission ticket for one match; - At most one match was same in the tickets booked by every two persons; - There was one person who booked six tickets. How many tickets did those football fans book at most? 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 2002 Day 2 1 Find all natural numbers n(n ≥ 2) such that there exists reals a1, a2, . . . , an which satisfy {|ai − aj | | 1 ≤ i < j ≤ n} = { 1, 2, . . . , n(n− 1) 2 } . Let A = {1, 2, 3, 4, 5, 6}, B = {7, 8, 9, . . . , n}. Ai(i = 1, 2, . . . , 20) contains eight numbers, three of which are chosen from A and the other five numbers from B. |Ai ∩Aj | ≤ 2, 1 ≤ i < j ≤ 20. Find the minimum possible value of n. 2 Given an integer k. f(n) is defined on negative integer set and its values are integers. f(n) satisfies f(n)f(n+ 1) = (f(n) + n− k)2, for n = −2,−3, · · · . Find an expression of f(n). 3 Let f(x1, x2, x3) = −2 · (x31+x32+x33)+3 · (x21(x2+x3)+x22 · (x1+x3)+x23 · (x1+x2)− 12x1x2x3. For any reals r, s, t, we denote g(r, s, t) = max t≤x3≤t+2 |f(r, r + 2, x3) + s|. Find the minimum value of g(r, s, t). http://www.artofproblemsolving.com/ This file was downloaded from the AoPS − MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/