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International Zhautykov Olympiad 2011 - Problem 5

Source:

January 17, 2011
number theorygreatest common divisorEulerfunctionmodular arithmeticnumber theory unsolved

Problem Statement

Let nn be integer, n>1.n>1. An element of the set M={1,2,3,…,n2−1}M=\{ 1,2,3,\ldots,n^2-1\} is called good if there exists some element bb of MM such that ab−bab-b is divisible by n2.n^2. Furthermore, an element aa is called very good if a2−aa^2-a is divisible by n2.n^2. Let gg denote the number of good elements in MM and vv denote the number of very good elements in M.M. Prove that v2+v≤g≤n2−n.v^2+v \leq g \leq n^2-n.
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