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4 numbers with sum more than n/2

Source: PAMO 2023 P4

May 22, 2023
combinatorics

Problem Statement

Manzi has nn stamps and an album with 1010 pages. He distributes the nn stamps in the album such that each page has a distinct number of stamps. He finds that, no matter how he does this, there is always a set of 44 pages such that the total number of stamps in these 44 pages is at least n2\frac{n}{2}. Determine the maximum possible value of nn.
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