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Perfect square under odd conditions

Source: Czech and Slovak Olympiad 2018, National Round, Problem 4

April 5, 2018
number theorynational olympiad

Problem Statement

Let a,b,ca,b,c be integers which are lengths of sides of a triangle, gcd⁡(a,b,c)=1\gcd(a,b,c)=1 and all the values \frac{a^2+b^2-c^2}{a+b-c}, \frac{b^2+c^2-a^2}{b+c-a}, \frac{c^2+a^2-b^2}{c+a-b} are integers as well. Show that (a+b−c)(b+c−a)(c+a−b)(a+b-c)(b+c-a)(c+a-b) or 2(a+b−c)(b+c−a)(c+a−b)2(a+b-c)(b+c-a)(c+a-b) is a perfect square.
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