2019 IMO Shortlist A6

A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities $$P(x, y, z)=P(x, y, x y-z)=P(x, z x-y, z)=P(y z-x, y, z) .$$ Prove that there exists a polynomial $F(t)$ in one variable such that $$P(x, y, z)=F\left(x^{2}+y^{2}+z^{2}-x y z\right) .$$

具有实系数的三个变量的多项式 $P(x, y, z)$ 满足恒等式 $$P(x, y, z)=P(x, y, x y-z)=P(x, z x-y, z)=P(y z-x, y, z) 。$$ 证明一个变量中存在多项式 $F(t)$ ，使得 $$P(x, y, z)=F\left(x^{2}+y^{2}+z^{2}-x y z\right) 。$$