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m two-sided cards C_i , with integers from 1-999 on each side

Source: Mediterranean Mathematical Olympiad 2022 P1 MMC

September 21, 2022
combinatorics

Problem Statement

Let S={1,...,999}S = \{1,..., 999\}. Determine the smallest integer mm. for which there exist mm two-sided cards C1C_1,..., CmC_m with the following properties: ∙\bullet Every card CiC_i has an integer from SS on one side and another integer from SS on the other side. ∙\bullet For all x,y∈Sx,y \in S with x≠yx\ne y, it is possible to select a card CiC_i that shows xx on one of its sides and another card CjC_j (with i≠ji \ne j) that shows yy on one of its sides.
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