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is (1+a^2)(1+b^2)(1+c^2) perfect square when ab + bc + ca = 1 ??

Source: 2011 Grand Duchy of Lithuania, Mathematical Contest p1 (Baltic Way TST)

October 3, 2020
number theoryPerfect Square

Problem Statement

Integers a,ba, b and cc satisfy the condition ab+bc+ca=1ab + bc + ca = 1. Is it true that the number (1+a2)(1+b2)(1+c2)(1+a^2)(1+b^2)(1+c^2) is a perfect square? Why?
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