Problem

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Tags: algebra, polynomial, algebra unsolved



Find all polynomials $P(x)$ with real coefficients such that for all pairwise distinct positive integers $x, y, z, t$ with $x^2 + y^2 + z^2 = 2t^2$ and $\gcd(x, y, z, t ) = 1,$ the following equality holds \[2P^2(t ) + 2P(xy + yz + zx) = P^2(x + y + z) .\] Note. $P^2(k)=\left( P(k) \right)^2.$