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Powers of Two and Sums

Source: Ukrainian Mathematical Olympiad 2023. Day 2, Problem 9.7

April 5, 2023
number theory

Problem Statement

You are given n2n \ge 2 distinct positive integers. Let's call a pair of these integers elegant if their sum is an integer power of 22. For every nn find the largest possible number of elegant pairs.
Proposed by Oleksiy Masalitin