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set of 50 distinct positive integers

Source: Argentina 2001 OMA L3 p5

May 12, 2024
combinatoricsnumber theory

Problem Statement

All sets of 4949 distinct positive integers less than or equal to 100100 are considered. Leandro assigned each of these sets a positive integer less than or equal to 100100. Prove that there is a set LL of 5050 distinct positive integers less than or equal to 100100, such that for each number xx of LL the number that Leandro assigned to the set of 4949 numbers L−{x}L-\{ x\} is different from xx.
Clarification: L−{x}L-\{x\} denotes the set that results from removing the number xx from LL.
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