BMO 2023 | Problem 4 - 10 May 2023 - Balkan Mathematical Olympiad

Problem 4.
Find the greatest integer $k\leq 2023$ for which the following holds: whenever Alice colours exactly $k$ numbers of the set $\{1,2,\dots, 2023\}$ in red, Bob can colour some of the remaining uncoloured numbers in blue, such that the sum of the red numbers is the same as the sum of the blue numbers.

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