Question 40
Let $v \gt 1$, a positive integer. Each cell of a $v \times v$ table contains one integer. Suppose that the following conditions are satisfied:
a. Each number in the table is congruent to $1$ modulo $v$.
b. The sum of numbers in any row, as well the sum of numbers in any column is congruent to $v$ modulo $v^2$.
Let $\Gamma_i$ be the product of numbers in the $i^{\text{th}}$ row, and $\Sigma_j$ be the product of numbers in the $j^{\text{th}}$ column. Prove that the sums $\Gamma_1 + \Gamma_2 + \dots + \Gamma_v$ and $\Sigma_1 + \Sigma_2 + \dots + \Sigma_v$ are congruent modulo $v^4$. (IMO 2018 shortlist)
a. Each number in the table is congruent to $1$ modulo $v$.
b. The sum of numbers in any row, as well the sum of numbers in any column is congruent to $v$ modulo $v^2$.
Let $\Gamma_i$ be the product of numbers in the $i^{\text{th}}$ row, and $\Sigma_j$ be the product of numbers in the $j^{\text{th}}$ column. Prove that the sums $\Gamma_1 + \Gamma_2 + \dots + \Gamma_v$ and $\Sigma_1 + \Sigma_2 + \dots + \Sigma_v$ are congruent modulo $v^4$. (IMO 2018 shortlist)