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modulo 2553 distance d(m, n) <= 36 , |S| >= 70

Source: 2010 Thailand Mathematical Olympiad day 1 p8

March 12, 2021
number theorydistance

Problem Statement

Define the modulo 25532553 distance d(x,y)d(x, y) between two integers x,yx, y to be the smallest nonnegative integer dd equivalent to either x−yx - y or y−xy - x modulo 25532553. Show that, given a set S of integers such that ∣S∣≥70|S| \ge 70, there must be m,n∈Sm, n \in S with d(m,n)≤36d(m, n) \le 36.
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