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Putnam 2020 B4

Source: 81st William Lowell Putnam Competition

February 22, 2021
PutnamPutnam 2020

Problem Statement

Let nn be a positive integer, and let VnV_n be the set of integer (2n+1)(2n+1)-tuples v=(s0,s1,⋯ ,s2n−1,s2n)\mathbf{v}=(s_0,s_1,\cdots,s_{2n-1},s_{2n}) for which s0=s2n=0s_0=s_{2n}=0 and ∣sj−sj−1∣=1|s_j-s_{j-1}|=1 for j=1,2,⋯ ,2nj=1,2,\cdots,2n. Define q(v)=1+∑j=12n−13sj, q(\mathbf{v})=1+\sum_{j=1}^{2n-1}3^{s_j}, and let M(n)M(n) be the average of 1q(v)\frac{1}{q(\mathbf{v})} over all v∈Vn\mathbf{v}\in V_n. Evaluate M(2020)M(2020).
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