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2023 SSMO Tiebreaker Round Problems/Problem 3

Revision as of 22:30, 15 December 2023 by Pinkpig (talk | contribs) (Created page with "==Problem== For <math>n\geq4,</math> let <math>a_n</math> be the maximum possible value of <math>P(n+1)</math> given that <math>P(x)</math> is a <math>n</math> degree monic po...")
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Problem

For $n\geq4,$ let $a_n$ be the maximum possible value of $P(n+1)$ given that $P(x)$ is a $n$ degree monic polynomial that satisfies $P(i)\in\{1,2,3\dots,n\}$ for $1\leq i \leq n.$ If $\frac{m}{n} = \sum_{n=4}^{\infty}\frac{a_n-n!}{3^n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

Solution

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