5. Given that $x_{1}, x_{2}, \cdots, x_{57}$ are all positive integers, and $x_{1}+x_{2}+\cdots+x_{57}=100$. Then the maximum value of $x_{1}^{2}+x_{2}^{2}+\cdots+x_{57}^{2}$ is $\qquad$
5. Solution For natural numbers $x_{i}, x_{j}$, if $x_{i}x_{i}^{2}+x_{j}^{2} .
\end{array}
$$

Thus, when the sum $x_{i}+x_{j}$ remains constant, the value is maximized by setting one of the numbers to 1 and the other to the sum minus 1, leading to
$$
\text { to } x_{1}=x_{2}=\cdots=x_{56}=1, x_{57}=44 \text {, when } x_{1}^{2}+x_{2}^{2}+\cdots
$$

is maximized, and the maximum value is
$$
\begin{array}{l}
x_{1}^{2}+x_{2}^{2}+\cdots+x_{50}^{2}+x_{57}^{2} \\
=56+44^{2}=1992 .
\end{array}
$$