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Maximizing red sum in even colouring of 2024-table

Source: Abelkonkurransen Finale 2024, Problem 3b

March 8, 2024
combinatoricscombinatorics proposed

Problem Statement

A 20242024-table is a table with two rows and 20242024 columns containg all the numbers 1,2,…,40481,2,\dots,4048. Such a table is evenly coloured if exactly half of the numbers in each row, and one number in each column, is coloured red. The red sum in an evenly coloured 20242024-table is the sum of all the red numbers in the table. Let NN be the largest number such that every 20242024-table has an even colouring with red sum ≥N\ge N. Determine NN, and find the number of 20242024-tables such that every even colouring of the table has red sum ≤N\le N.
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