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IMC 1994 D1 P3

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March 6, 2017
IMCirrational numbercollege contestsreal analysis

Problem Statement

Given a set SS of 2n−12n-1, n∈Nn\in \mathbb N, different irrational numbers. Prove that there are nn different elements x1,x2,…,xn∈Sx_1, x_2, \ldots, x_n\in S such that for all non-negative rational numbers a1,a2,…,ana_1, a_2, \ldots, a_n with a1+a2+…+an>0a_1+a_2+\ldots + a_n>0 we have that a1x1+a2x2+⋯+anxna_1x_1+a_2x_2+\cdots +a_nx_n is an irrational number.
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