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2013 Japan Mathematical Olympiad Finals Problem 1

Source:

February 13, 2013
combinatorics proposedcombinatorics

Problem Statement

Let n, kn,\ k be positive integers with n≥kn\geq k. There are nn persons, each person belongs to exactly one of group 11, group 2, ⋯2,\ \cdots, group kk and more than or equal to one person belong to any groups. Show that n2n^2 sweets can be delivered to nn persons in such way that all of the following condition are satisfied.
∙\bullet At least one sweet are delivered to each person.
∙\bullet aia_i sweet are delivered to each person belonging to group i (1≤i≤k).i\ (1\leq i\leq k).
∙\bullet If 1≤i<j≤k1\leq i<j\leq k, then ai>aj.a_i>a_j.
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