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ak mod 2012 > bk mod 2012

Source: 2012 USAJMO Day 2 #5

April 25, 2012
2012 USAJMO

Problem Statement

For distinct positive integers a,b<2012a, b<2012, define f(a,b)f(a, b) to be the number of integers kk with 1≤k<20121\le k<2012 such that the remainder when akak divided by 20122012 is greater than that of bkbk divided by 20122012. Let SS be the minimum value of f(a,b)f(a, b), where aa and bb range over all pairs of distinct positive integers less than 20122012. Determine SS.
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