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 The 28th International Mathematical Olympiad Shortlist Problems 1987年第28届国际数学奥林匹克备选题 f is a real-valued function on the reals such that: (1) if x ≥ y and f(y) - y ≥ v ≥ f(x) - x, then f(z) = v + z for some z between x and y; (2) for some k, f(k) = 0 and if f(h) = 0, then h <= k; (3) f(0) = 1; (4) f(1987) ≤ 1988; (5) f(x) f(y) = f(x f(y) + y f(x) - xy) for all x, y. Find f(1987). (Australia 6) S = {a1, a2, ... , an, b1, b2, ... , bn}. There are subsets C1, C2, ... , Ck, such that (1) for no i, j do both ai and bi both belong to Cj, (2) for any pair of distinct elements of S, not of the form ai, bi, there is just one Cj containing both elements. Show that if n > 3, then k ≥ 2n. (USA 3) Does there exist a polynomial p(x, y) of degree 2 such that, for each non-negative integer n, we have n = p(a, b) for just one pair (a, b) of non-negative integers? (Finland 3) ABCDA'B'C'D' is any parallelepiped (with ABCD, A'B'C'D' faces and AA', BB', CC', DD' edges). Show that AC + AB'+ AD' <= AB + AD + AA' + AC' (the sum of the thre short diagonals from A is less than the sum of the three edges from A plus the long diagonal from A). (France 5) Find the smallest real c such that x11/2 + x21/2 + ... + xn1/2 ≤ c (x1 + x2 + ... + xn)1/2 for all n and all real sequences x1, x2, x3, ... which satisfy x1 + x2 + ... + xn ≤ xn+1. (United Kingdom 6) Show that an/(b+c) + bn/(c+a) + cn/(a+b) ≥ (2/3)n-2 sn-1 for all n ≥ 1, where a, b, c are the sides of a triangle and s = (a + b + c)/2. (Greece 4) Given any 5 real numbers u0, u1, u2, u3, u4, show that we can always find 5 real numbers v0, v1, v2, v3, v4 such that each ui - vi is integral and ∑i 1, there is an irrational number r such that [rm] = -1 mod k for every natural number m. (Yugoslavia 2) (USSR 4), has been used in the Olympiad. (Vietnam 4),has been used in the Olympiad. 点击此处查看相关视频讲解 在方框内输入单词或词组