Problem

Source: 2022 China TST, Test 1, P2 (posting for better LaTeX)

Tags: combinatorics, number theory, prime numbers



Let $p$ be a prime, $A$ is an infinite set of integers. Prove that there is a subset $B$ of $A$ with $2p-2$ elements, such that the arithmetic mean of any pairwise distinct $p$ elements in $B$ does not belong to $A$.