2022 SSMO Accuracy Round Problems/Problem 10

Problem

Let $S=\sum^{\infty}_{k=5}2^{-10k}\dbinom{2k}{10}.$ Then the value of $S$ can be expressed as $\frac mn$ , where $m$ and $n$ are relatively prime positive integers. Find the largest positive integer $a$ such that $2^a\mid mn$ .

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2022 SSMO Accuracy Round Problems/Problem 10

Problem

Solution