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x^{2n}+y^{2n}+z^{2n}-xy-yz-zx divides (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}

Source: Balkan MO Shortlist 2013 A5 BMO

March 9, 2020
dividespolynomialalgebraInteger Polynomial

Problem Statement

Determine all positive integersn n such that fn(x,y,z)=x2n+y2n+z2n−xy−yz−zxf_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx divides gn(x,y,z)=(x−y)5n+(y−z)5n+(z−x)5ng_n(x,y, z) = (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}, as polynomials in x,y,zx, y, z with integer coefficients.
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