MathDB
Existence of indices

Source: Baltic Way 2021, Problem 6

November 15, 2021
combinatoricscombinatorics proposed

Problem Statement

Let nn be a positive integer and tt be a non-zero real number. Let a1,a2,…,a2n−1a_1, a_2, \ldots, a_{2n-1} be real numbers (not necessarily distinct). Prove that there exist distinct indices i1,i2,…,ini_1, i_2, \ldots, i_n such that, for all 1≤k,l≤n1 \le k, l \le n, we have aik−ail≠ta_{i_k} - a_{i_l} \neq t.
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