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there exist numbers a_1,a_2,...,a_n

Source: Polish MO Round 3 2009,problem 4

November 3, 2009
pigeonhole principlevectorinequalities unsolvedinequalities

Problem Statement

Let x1,x2,..,xn x_1,x_2,..,x_n be non-negative numbers whose sum is 1 1 . Show that there exist numbers a1,a2,…,an a_1,a_2,\ldots ,a_n chosen from amongst 0,1,2,3,4 0,1,2,3,4 such that a1,a2,…,an a_1,a_2,\ldots ,a_n are different from 2,2,…,2 2,2,\ldots ,2 and 2\leq a_1x_1\plus{}a_2x_2\plus{}\ldots\plus{}a_nx_n\leq 2\plus{}\frac{2}{3^n\minus{}1}.
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