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BMT 2021 Guts Round p23

Source:

October 7, 2022
combinatorics

Problem Statement

Alireza is currently standing at the point (0,0)(0, 0) in the x−yx-y plane. At any given time, Alireza can move from the point (x,y)(x, y) to the point (x+1,y)(x + 1, y) or the point (x,y+1)(x, y + 1). However, he cannot move to any point of the form (x,y)(x, y) where y≡2x  (mod    5)y \equiv 2x\,\, (\mod \,\,5). Let pkp_k be the number of paths Alireza can take starting from the point (0,0)(0, 0) to the point (k+1,2k+1)(k + 1, 2k + 1). Evaluate the sum ∑k=1∞pk5k.\sum^{\infty}_{k=1} \frac{p_k}{5^k}..
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