2024 IMO Shortlist C2

Let $n$ be a positive integer. The integers $1,2,3,\ldots,n^2$ are to be written in the cells of an $n\times n$ board, each integer in exactly one cell and each cell containing exactly one integer. For every integer $d$ with $d\mid n$, the $d$-division of the board is the division of the board into $(n/d)^2$ nonoverlapping sub-boards, each of size $d\times d$, such that each cell is contained in exactly one $d\times d$ sub-board.

We say that $n$ is a cool number if the integers can be written on the $n\times n$ board such that, for each integer $d$ with $d\mid n$ and $1<d<n$, in the $d$-division of the board, the sum of the integers written in each $d\times d$ sub-board is not a multiple of $d$.

Determine all even cool numbers.

设 $n$ 为正整数。将整数 $1,2,3,\ldots,n^2$ 写入一个 $n\times n$ 棋盘的格子中，每个整数恰好写入一个格子，每个格子也恰好含有一个整数。对每个满足 $d\mid n$ 的整数 $d$，棋盘的 $d$-划分指把棋盘分成 $(n/d)^2$ 个互不重叠的 $d\times d$ 子棋盘，且每个格子恰好属于一个这样的子棋盘。

若可以这样填写 $n\times n$ 棋盘，使得对每个满足 $d\mid n$ 且 $1<d<n$ 的整数 $d$，在棋盘的 $d$-划分中，每个 $d\times d$ 子棋盘内所写整数之和都不是 $d$ 的倍数，则称 $n$ 是 cool number。

确定所有偶数 cool number。