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Arithmetic means of subsequences bounded by arithmetic mean

Source: Baltic Way 2004 Problem 4

November 19, 2004
inequalitiesinequalities proposed

Problem Statement

Let x1x_1, x2x_2, ..., xnx_n be real numbers with arithmetic mean XX. Prove that there is a positive integer KK such that for any integer ii satisfying 0≤i<K0\leq i<K, we have 1K−i∑j=i+1Kxj≤X\frac{1}{K-i}\sum_{j=i+1}^{K} x_j \leq X. (In other words, prove that there is a positive integer KK such that the arithmetic mean of each of the lists {x1,x2,...,xK}\left\{x_1,x_2,...,x_K\right\}, {x2,x3,...,xK}\left\{x_2,x_3,...,x_K\right\}, {x3,...,xK}\left\{x_3,...,x_K\right\}, ..., {xK−1,xK}\left\{x_{K-1},x_K\right\}, {xK}\left\{x_K\right\} is not greater than XX.)
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