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 The 6th Asian Pacific Mathematical Olympiad 1994年第6届亚太地区数学奥林匹克 Find all real-valued functions f on the reals such that (1) f(1) = 1, (2) f(-1) = -1, (3) f(x) ≤ f(0) for 0 < x < 1, (4) f(x + y) ≥ f(x) + f(y) for all x, y, (5) f(x + y) ≤ f(x) + f(y) + 1 for all x, y. Given a nondegenerate triangle ABC, with circumcentre O, orthocentre H, and circumradius R, prove that |OH|< 3R. Let n be an integer of the form a2 +b2, where a and b are relatively prime integers and such that if p is a prime, p ≤ √n, then p divides ab. Determine all such n. Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational? You are given three lists A, B, and C. List A contains the numbers of the form 10k in base 10, with k any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively: ``` A B C 10 1010 20 100 1100100 400 1000 1111101000 13000 ...``` Prove that for every integer n > 1, there is exactly one number in exactly one of the lists Bor C that has exactly n digits. 点击此处查看相关视频讲解 在方框内输入单词或词组