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Sum of some numbers of a set is equal to the sum of some of the rests

Source: BMO 2023 Problem 4

May 10, 2023

BMO 2023combinatorics

Problem Statement

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.
Romania

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