1985 USAMO 第 1 题

Determine whether or not there are any positive integral solutions of the simultaneous equations

$$$$
\begin{aligned} x_1^2 +x_2^2 +\cdots +x_{1985}^2 & = y^3,\\ x_1^3 +x_2^3 +\cdots +x_{1985}^3 & = z^2 \end{aligned}
$$$$

with distinct integers $x_1,x_2,\cdots,x_{1985}$.

确定联立方程组是否存在正积分解

$$

$$

\begin{对齐} x_1^2 +x_2^2 +\cdots +x_{1985}^2 & = y^3,\\ x_1^3 +x_2^3 +\cdots +x_{1985}^3 & = z^2 \end{对齐}

$$

$$

具有不同的整数 $x_1,x_2,\cdots,x_{1985}$。