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2025 SSMO Speed Round Problems/Problem 10

Problem

Let $p$ be a quadratic with a positive leading coefficient, and let $r$ be a real number satisfying $r < 1 < \tfrac{5}{2r} < 5$. Given that $p(p(x)) = x$ for $x \in \{ r,1, \tfrac{5}{2r} , 5\}$, find $p(12)$.

Solution

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