IMO Shortlist 2018 problem N2

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Dodao/la: arhiva
3. listopada 2019.

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 number in the j^{\text{th}} column. Prove that the sums R_1+\hdots R_n and C_1+\hdots C_n are congruent modulo n^4.