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Three-variable polynomial

Source: IMO Shortlist 2019 A6

September 22, 2020
IMO ShortlistalgebrapolynomialIMO Shortlist 2019

Problem Statement

A polynomial P(x,y,z)P(x, y, z) in three variables with real coefficients satisfies the identities
P(x,y,z)=P(x,y,xy−z)=P(x,zx−y,z)=P(yz−x,y,z).P(x, y, z)=P(x, y, xy-z)=P(x, zx-y, z)=P(yz-x, y, z).
Prove that there exists a polynomial F(t)F(t) in one variable such that
P(x,y,z)=F(x2+y2+z2−xyz).P(x,y,z)=F(x^2+y^2+z^2-xyz).
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