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Let polynomials $P(x)$, $Q(x)$, $R(x)$, and $S(x)$ satisfy: $$P(x^5) + xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x)$$ Prove: $(x-1) | P(x)$

Determine all polynomials such that $P(0) = 0$ and $P(x^2 + 1) = P(x)^2 + 1$.


Prove there cannot exist a $998$-degree polynomial with real number coefficients $P(x)$ such that $$[P(x)]^2-1=P(x^2+1)$$ holds for any $x\in\mathbb{C}$.