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A number that is sum of two squares

Source: BMO SL 2023 N1

May 3, 2024
number theoryBMO Shortlist

Problem Statement

For positive integers a,b,ca, b, c (not necessarily distinct), suppose that a+bc,b+ac,c+aba+bc, b+ac, c+ab are all perfect squares. Show that a2(b+c)+b2(a+c)+c2(a+b)+2abca^2(b+c)+b^2(a+c)+c^2(a+b)+2abc can be written as sum of two squares.
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