A bit of casework but imo pleasant
Source: Ukrainian Mathematical Olympiad 2024. Day 2, Problem 11.5
March 20, 2024
algebra
Problem Statement
You are given some non-zero, not necessarily distinct real numbers. Find all positive integers from to , such that among these numbers you can always choose 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