IMC 1994 · Problem 3

Given a set $S$ of $2n - 1$, $n \in \mathbb{N}$, different irrational numbers. Prove that there are $n$ different elements $x_1, x_2, \dots, x_n \in S$ such that for all non-negative rational numbers $a_1, a_2, \dots, a_n$ with $a_1 + a_2 + \dots + a_n > 0$ we have that $a_1 x_1 + a_2 x_2 + \dots + a_n x_n$ is an irrational number.