olymmath

ISL 2019 A6

    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, x y-z)=P(x, z x-y, z)=P(y z-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\left(x^{2}+y^{2}+z^{2}-x y z\right) .

    Dataset text NuminaMath olympiads. Quoted with source.

    Official text: IMO 2019 Shortlist booklet, page 7.

    Solutions

    Official solution Solution 1 by the IMO Problem Selection Committee

    The terms of the source let olymmath link to this solution but not copy it. Read the solution (IMO official site).

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