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Found problems: 2

2016 Singapore MO Open, 4
Tags: positive coefficients , coefficient , algebra , polynomial
Let $b$ be a number with $-2 < b < 0$. Prove that there exists a positive integer $n$ such that all the coefficients of the polynomial $(x + 1)^n(x^2 + bx + 1)$ are positive.

2025 Alborz Mathematical Olympiad, P2
Tags: polynomial , positive integer , positive coefficients , algebra
Suppose that for polynomials \( P, Q, R \) with positive integer coefficients, the following two conditions hold: \(\bullet\) The constant terms of \( P, Q, R \) are equal. \(\bullet\) For all real numbers \( x \), the following relations hold: \[ P(Q(R(x))) = Q(R(P(x))) = R(P(Q(x))) = P(R(Q(x))) = Q(P(R(x))) = R(Q(P(x))). \] Prove that for every real number \( x \), \( P(x) = Q(x) = R(x) \). Proposed by Soroush Behroozifar & Ali Nazarboland