In this paper, we present the results related to a problem posed by Andrzej Schinzel. Does the number $N_1(n)$ of integer solutions of the equation $$ \begin{align*}x_1+x_2+\cdots+x_n=x_1x_2\cdot\ldots\cdot x_n,\,\,x_1\ge x_2\ge\cdots\ge x_n\ge 1\end{align*} $$tend to infinity with n? Let a be a positive integer. We give a lower bound on the number of integer solutions, $N_a(n)$, to the equation $$ \begin{align*}x_1+x_2+\cdots+x_n=ax_1x_2\cdot\ldots\cdot x_n,\,\, x_1\ge x_2\ge\cdots\ge x_n\ge 1.\end{align*} $$We show that if $N_2(n)=1$, then the number $2n-3$ is prime. The average behavior of $N_2(n)$ is studied. We prove that the set $\{n:N_2(n)\le k,\,n\ge 2\}$ has zero natural density.

在本文中，我们提出了与 Andrzej Schinzel 提出的问题相关的结果。方程 $$ \begin{align*}x_1+x_2+\cdots+x_n=x_1x_2\cdot\ldots\cdot x_n,\,\,x_1\ge x_2\ge 的整数解的数量为$N_1(n)$吗\cdots\ge x_n\ge 1\end{align*} $$趋于无穷大，n？设a为正整数。我们为方程$$ \begin{align*}x_1+x_2+\cdots+x_n=ax_1x_2\cdot\ldots\cdot x_n,\,\给出整数解数量的下界$N_a(n) $ , x_1\ge x_2\ge\cdots\ge x_n\ge 1.\end{align*} $$我们证明，如果$N_2(n)=1$，则数字$2n-3$是素数。研究了$N_2(n)$的平均行为。我们证明集合$\{n:N_2(n)\le k,\,n\ge 2\}$ 的自然密度为零。