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subtraction of subsets

Source: Swiss 2004

October 11, 2006
inductioncombinatorics unsolvedcombinatorics

Problem Statement

Let A1,...,AnA_{1}, ..., A_{n} be different subsets of an nn-element set XX. Show that there exists x∈Xx\in X such that the sets A1−{x},A2−{x},...,An−{x}A_{1}-\{x\}, A_{2}-\{x\}, ..., A_{n}-\{x\} are all different.
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