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2003 IMO Problems/Problem 1

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$S$ is the set $\set{1, 2, 3, . . . , 1000000}$ (Error compiling LaTeX. ! Undefined control sequence.). Show that for any subset $A$ of $S$ with $101$ elements we can find $100$ distinct elements $x_i$ of $S$, such that the sets $\set{a + x_i \mid a \in A}$ (Error compiling LaTeX. ! Undefined control sequence.) are all pairwise disjoint.

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