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2022 SSMO Accuracy Round Problems/Problem 10

Revision as of 12:04, 14 December 2023 by Pinkpig (talk | contribs) (Created page with "==Problem== Let<cmath>S=\sum^{\infty}_{k=5}2^{-10k}\dbinom{2k}{10}.</cmath>Then the value of <math>S</math> can be expressed as <math>\frac mn</math>, where <math>m</math> and...")
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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 $2^a\mid mn$.

Solution

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