Let $n,k$ be positive integers such that $n^k>(k+1)!$ and consider the set \[ M=\{(x_1,x_2,\ldots,x_n)\dvd x_i\in\{1,2,\ldots,n\},\ i=\overline{1,k}\}. \] Prove that if $A\subset M$ has $(k+1)!+1$ elements, then there are two elements $\{\alpha,\beta\}\subset A$, $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_n)$, $\beta=(\beta_1,\beta_2,\ldots,\beta_n)$ such that \[ (k+1)! \left| (\beta_1-\alpha_1)(\beta_2-\alpha_2)\cdots (\beta_k-\alpha_k) \right. .\]