Problem

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Tags: AMC, USA(J)MO, USAMO, number theory



Determine whether or not there are any positive integral solutions of the simultaneous equations \begin{align*}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{align*}with distinct integers $x_1$, $x_2$, $\ldots$, $x_{1985}$.