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k boxes with the same number of balls - ILL 1990 CZS3

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September 18, 2010
combinatoricscountingpigeonhole principleIMO ShortlistIMO Longlist

Problem Statement

Let p,kp, k and xx be positive integers such that p≥kp \geq k and x<[p(p−k+1)2(k−1)]x < \left[ \frac{p(p-k+1)}{2(k-1)} \right], where [q][q] is the largest integer no larger than qq. Prove that when xx balls are put into pp boxes arbitrarily, there exist kk boxes with the same number of balls.
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