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Inequality with n towns and m two way airlines

Source: Serbia NMO 2010 problem 1

March 11, 2011
inequalitiesfloor functioncombinatorics

Problem Statement

Some of nn towns are connected by two-way airlines. There are mm airlines in total. For i=1,2,⋯ ,ni = 1, 2, \cdots, n, let did_i be the number of airlines going from town ii. If 1≤di≤20101\le d_i \le 2010 for each i=1,2,⋯ ,2010i = 1, 2,\cdots, 2010, prove that \displaystyle\sum_{i=1}^n d_i^2\le 4022m- 2010n Find all nn for which equality can be attained.
Proposed by Aleksandar Ilic
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