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Coprime integers with product being perfect squares

Source: 2022 Taiwan Mathematics Olympiad

February 19, 2022
number theoryTaiwan

Problem Statement

Let x,y,zx,y,z be three positive integers with gcd⁡(x,y,z)=1\gcd(x,y,z)=1. If x∣yz(x+y+z),x\mid yz(x+y+z), y∣xz(x+y+z),y\mid xz(x+y+z), z∣xy(x+y+z),z\mid xy(x+y+z), and x+y+z∣xyz,x+y+z\mid xyz, show that xyz(x+y+z)xyz(x+y+z) is a perfect square.
Proposed by usjl
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