IMO 2018 Shortlist N2
Let n ą 1 be a positive integer. Ea h ell of an n ˆ n table ontains an integer. Suppose that the following onditions are...
Category: Number Theory
Problem
Let n ą 1 be a positive integer. Ea h ell of an n ˆ n table ontains an integer.
Suppose that the following onditions are satis ed:
piq Ea h number in the table is ongruent to 1 modulo n;
piiq The sum of numbers in any row, as well as the sum of numbers in any olumn, is ongruent
to n modulo n2
.
Let Ri be the produ t of the numbers in the ith row, and Cj be the produ t of the numbers in
the jth olumn. Prove that the sums R1 ¨¨¨Rn and C1 ¨¨¨Cn are ongruent modulo n4
.
(Indonesia)