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x^2 + y^2 + z^2 − xy − yz − zx = n , x^2 + y^2 − xy = n, integer solution

Source: Tournament of Towns, Junior O-Level Paper, Spring 2020 , p4

June 3, 2020
Diophantine equationdiophantinenumber theory

Problem Statement

For some integer n the equation x2+y2+z2−xy−yz−zx=nx^2 + y^2 + z^2 -xy -yz - zx = n has an integer solution x,y,zx, y, z. Prove that the equationx2+y2−xy=n x^2 + y^2 - xy = n also has an integer solution x,yx, y.
Alexandr Yuran
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