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Pairing cards such that numbers differ by 1

Source: Tuymaada 2014, Day 2, Problem 1, Senior League

July 12, 2014
combinatorics unsolvedcombinatoricsTuymaada

Problem Statement

There is an even number of cards on a table; a positive integer is written on each card. Let aka_k be the number of cards having kk written on them. It is known that an−an−1+an−2−⋯≥0a_n-a_{n-1}+a_{n-2}- \cdots \ge 0 for each positive integer nn. Prove that the cards can be partitioned into pairs so that the numbers in each pair differ by 11.
(A. Golovanov)
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