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Polynomial and sequence

Source: Iran 2024 3rd round p6

August 25, 2024
algebra

Problem Statement

Sequence of positive integers {xk}k≥1\{x_k\}_{k\geq 1} is given such that x1=1x_1=1 and for all n≥1n\geq 1 we have xn+12+P(n)=xnxn+2x_{n+1}^2+P(n)=x_n x_{n+2} where P(x)P(x) is a polynomial with non-negative integer coefficients. Prove that P(x)P(x) is the constant polynomial.
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