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2024 Mongolian Mathematical Olympiad, 1
Let $P(x)$ and $Q(x)$ be polynomials with nonnegative coefficients. We denote by $P'(x)$ the derivative of $P(x)$. Suppose that $P(0)=Q(0)=0$ and $Q(1) \leq 1 \leq P'(0)$. $(1)$ Prove that $0 \leq Q(x) \leq x \leq P(x)$ for all $0 \leq x \leq 1$. $(2)$ Prove that $P(Q(x)) \leq Q(P(x))$ for all $0 \leq x \leq 1$. [i]Proposed by Otgonbayar Uuye.[/i]
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