2018 IMO Shortlist N2

Let $n>1$ be a positive integer. Each cell of an $n \times n$ table contains an integer. Suppose that the following conditions are satisfied: (i) Each number in the table is congruent to 1 modulo $n$; (ii) The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^{2}$. Let $R_{i}$ be the product of the numbers in the $i^{\text {th }}$ row, and $C_{j}$ be the product of the numbers in the $j^{\text {th }}$ column. Prove that the sums $R_{1}+\cdots+R_{n}$ and $C_{1}+\cdots+C_{n}$ are congruent modulo $n^{4}$. (Indonesia)

令 $n>1$ 为正整数。 $n \times n$ 表的每个单元格都包含一个整数。假设满足以下条件： (i) 表中的每个数字都与 1 模 $n$ 全等； (ii) 任何行中的数字之和以及任何列中的数字之和都等于 $n$ 模 $n^{2}$。令 $R_{i}$ 为 $i^{\text {th }}$ 行中数字的乘积，$C_{j}$ 为 $j^{\text {th }}$ 列中数字的乘积。证明 $R_{1}+\cdots+R_{n}$ 和 $C_{1}+\cdots+C_{n}$ 模 $n^{4}$ 全等。 (印度尼西亚)