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xy is perfect square if yx^2+(y^2-z^2)x+y(y-z)^2=0 in integers x,y,z

Source: 2009 Belarus TST 1.4

November 8, 2020
number theoryPerfect SquarediophantineDiophantine equation

Problem Statement

Let x,y,zx,y,z be integer numbers satisfying the equality yx2+(y2−z2)x+y(y−z)2=0yx^2+(y^2-z^2)x+y(y-z)^2=0 a) Prove that number xyxy is a perfect square. b) Prove that there are infinitely many triples (x,y,z)(x,y,z) satisfying the equality.
I.Voronovich
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