2006 USAMO Problems/Problem 3

Problem

For integral $\displaystyle m$ , let $\displaystyle p(m)$ be the greatest prime divisor of $\displaystyle m$ . By convention, we set $p(\pm 1)=1$ and $p(0)=\infty$ . Find all polynomials $\displaystyle f$ with integer coefficients such that the sequence $\{ p(f(n^2))-2n) \} _{n\ge 0}$ is bounded above. (In particular, this requires $f(n^2)\neq 0$ for $n\ge 0$ .)

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