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Nice problem on polynomials, x^2+y^2+z^2+2xyz=1

Source: SAMO, 2022, R3, P6

July 28, 2022
algebrapolynomial

Problem Statement

Show that there are infinitely many polynomials P with real coefficients such that if x, y, and z are real numbers such that x2+y2+z2+2xyz=1x^2+y^2+z^2+2xyz=1, then P(x)2+P(y)2+P(z)2+2P(x)P(y)P(z)=1P\left(x\right)^2+P\left(y\right)^2+P\left(z\right)^2+2P\left(x\right)P\left(y\right)P\left(z\right) = 1
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