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n students each having r positive integers, nr integers all different, classes

Source: KJMO 2011 p8

May 4, 2019
number theorycombinatoricscombinationSubsets

Problem Statement

There are nn students each having rr positive integers. Their nrnr positive integers are all different. Prove that we can divide the students into kk classes satisfying the following conditions: (a) k≤4rk \le 4r (b) If a student AA has the number mm, then the student BB in the same class can't have a number ℓ\ell such that (m−1)!<ℓ<(m+1)!+1(m - 1)! < \ell < (m + 1)! + 1
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