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Sums of maximums of remainders

Source: AIME II 2013, Problem 14

April 4, 2013
inductionmodular arithmeticAMCnumber theoryAIMEpattern finding

Problem Statement

For positive integers nn and kk, let f(n,k)f(n,k) be the remainder when nn is divided by kk, and for n>1n>1 let F(n)=max⁡1≤k≤n2f(n,k)F(n) = \displaystyle\max_{1 \le k \le \frac{n}{2}} f(n,k). Find the remainder when ∑n=20100F(n)\displaystyle\sum_{n=20}^{100} F(n) is divided by 10001000.
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