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 The 14th Asian Pacific Mathematical Olympiad 2002年第14届亚太地区数学奥林匹克 Let a1, a2, a3, ... , an be a sequence of non-negative integers, where n is a positive integer. Let An = (a1 + a2 + a3 + ... + an)/n Prove that a1!a2!a3! ... an! ≥ ([An]!)n where [An] is the greatest integer less than or equal to Anand a! = 1 × 2 × ... × a for a ≥ 1 (and 0! = 1) When does equality hold ? Find all positive integers a and b such that (a2 + b)/(b2-a) and (b2 + a)/(a2-b) are both integers. Let ABC be an equilateral triangle. Let P be a point on the side AC and Q be a point on the side AB so that both triangles ABP and ACQ are acute. Let R be the orthocentre of triangle ABP and S be the orthocentre of triangle ACQ. Let T be the point common to the segments BP and CQ. Find all possible values of ∠CBP and ∠BCQ such that triangle TRS is equilateral. Let x, y, z be positive numbers such that 1/x + 1/y +1/z =1 Show that √(x + yz) + √(y + zx) + √(z + xy) ≥ √(xyz) + √x + √y + √z . Let R denote the set of all real numbers. Find all functions f from R to R satisfying: (i) there are only finitely many s in R such that f(s) = 0, and (ii) f(x4 + y) = x3f(x) + f(f(y)) for all x, y in R. 点击此处查看相关视频讲解 在方框内输入单词或词组