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2020 PUMaC Combinatorics A7

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January 1, 2022
combinatorics

Problem Statement

Let ff be defined as below for integers n≥0n \ge 0 and a0,a1,...a_0, a_1, ... such that ∑i≥0ai\sum_{i\ge 0}a_i is finite: f(n;a0,a1,...)={a2020, n=0∑i≥0aif(n−1;a0,...,ai−1,ai−1,ai+1+1,ai+2,...)/∑i≥0ain>0f(n; a_0, a_1, ...) = \begin{cases} a_{2020}, & \text{ $n = 0$} \\ \sum_{i\ge 0} a_i f(n-1;a_0,...,a_{i-1},a_i-1,a_{i+1}+1,a_{i+2},...)/ \sum_{i\ge 0}a_i & \text{$n > 0$} \end{cases}. Find the nearest integer to f(20202;2020,0,0,...)f(2020^2; 2020, 0, 0, ...).
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