IMO 2024 Shortlist C2
Let n be a positive integer. The integers 1, 2, 3, ..., n2 are to be written in the cells of an n ˆ n board such that ea...
Category: Combinatorics
Problem
Let n be a positive integer. The integers 1, 2, 3, ..., n2 are to be written in the cells of an n ˆ n board such that each integer is written in exactly one cell and each cell contains exactly one integer. For every integer d with d | n, the d-division of the board is the division of the board into pn{dq2 nonoverlapping sub-boards, each of size d ˆ d, such that each cell is contained in exactly one d ˆ d sub-board. We say that n is a cool number if the integers can be written on the nˆn board such that, for each integer d with d | n and 1 ă d ă n, in the d-division of the board, the sum of the integers written in each d ˆ d sub-board is not a multiple of d. Determine all even cool numbers. (Türkiye)