Let $A_{1}, ..., A_{n}$ be different subsets of an $n$-element set $X$. Show that there exists $x\in X$ such that the sets $A_{1}-\{x\}, A_{2}-\{x\}, ..., A_{n}-\{x\}$ are all different.
Source: Swiss 2004
Tags: induction, combinatorics unsolved, combinatorics
Let $A_{1}, ..., A_{n}$ be different subsets of an $n$-element set $X$. Show that there exists $x\in X$ such that the sets $A_{1}-\{x\}, A_{2}-\{x\}, ..., A_{n}-\{x\}$ are all different.