Problem

Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 11.5

Tags: algebra



You are given some $12$ non-zero, not necessarily distinct real numbers. Find all positive integers $k$ from $1$ to $12$, such that among these numbers you can always choose $k$ numbers whose sum has the same sign as their product, that is, either both the sum and the product are positive, or both are negative. Proposed by Anton Trygub