HMMT 2023 T8

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Find, with proof, all nonconstant polynomials $P(x)$ with real coefficients such that, for all nonzero real numbers $z$ with $P(z) \neq 0$ and $P\left(\frac{1}{z}\right) \neq 0$ , we have

$\frac{1}{P(z)}+\frac{1}{P\left(\frac{1}{z}\right)}=z+\frac{1}{z}$

Dataset text NuminaMath olympiads. Quoted with source.

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P(x)=\frac{x\left(x^{4 k+2}+1\right)}{x^{2}+1} \text{ or } P(x)=\frac{x\left(1-x^{4 k}\right)}{x^{2}+1}

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