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2022 Combo Div 1 P6

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February 28, 2022
combinatorics

Problem Statement

Barry has a standard die containing the numbers 1-6 on its faces.
He rolls the die continuously, keeping track of the sum of the numbers he has rolled so far, starting from 0. Let EnE_n be the expected number of time he needs to until his recorded sum is at least nn.
It turns out that there exist positive reals a,ba, b such that lim⁡n→∞En−(an+b)=0\lim_{n \rightarrow \infty} E_n - (an + b) = 0
Find (a,b)(a,b).
Proposed by Dilhan Salgado
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