Question 30
Show that there are infinitely many polynomials $P$ with real coefficients such that if $x$, $y$, and $z$ are real numbers such that $x^2 + y^2 + z^2 + 2xyz = 1$, then
$$
P(x)^2 + P(y)^2 + P(z)^2 + 2P(x)P(y)P(z) = 1.
$$
$$
P(x)^2 + P(y)^2 + P(z)^2 + 2P(x)P(y)P(z) = 1.
$$