Let a,b,c be integers which are lengths of sides of a triangle, 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) or 2(a+b−c)(b+c−a)(c+a−b) is a perfect square. number theorynational olympiad