MathDB
Every Integer is a divisor

Source: IMO Shortlist 2023 N5

July 17, 2024
IMO Shortlistnumber theoryAZE IMO TST

Problem Statement

Let a1<a2<a3<…a_1<a_2<a_3<\dots be positive integers such that ak+1a_{k+1} divides 2(a1+a2+⋯+ak)2(a_1+a_2+\dots+a_k) for every k⩾1k\geqslant 1. Suppose that for infinitely many primes pp, there exists kk such that pp divides aka_k. Prove that for every positive integer nn, there exists kk such that nn divides aka_k.
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