This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND
OR
NO

Found problems: 2

2025 Alborz Mathematical Olympiad, P2
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
positive integer polynomial positive coefficients algebra number theory polynomials

2016 Singapore MO Open, 4
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.
coefficient polynomial positive coefficients algebra number theory polynomials