Problem

Source: Iran 2024 3rd round p6

Tags: algebra



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