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Estonian Math Competitions 2005/2006

Source: Seniors Problem 10

July 30, 2008
inequalitiesalgebra unsolvedalgebra

Problem Statement

Let n≥2 n \ge 2 be a fixed integer and let ai,j(1≤i<j≤n) a_{i,j} (1 \le i < j \le n) be some positive integers. For a sequence x1,...,xn x_1, ... , x_n of reals, let K(x1,....,xn) K(x_1, .... , x_n) be the product of all expressions (x_i \minus{} x_j)^{a_{i,j}} where 1≤i<j≤n 1 \le i < j \le n. Prove that if the inequality K(x1,....,xn)≥0 K(x_1, .... , x_n) \ge 0 holds independently of the choice of the sequence x1,...,xn x_1, ... , x_n then all integers ai,j a_{i,j} are even.
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