Problem 825

10.494.

Since the number of ways to write 58 as the sum of 40 positive integers is finite, the minimum and maximum values of $x_{1}^{2}+x_{2}^{2}+\cdots+x_{40}^{2}$ exist.
Assume without loss of generality that $x_{1} \leqslant x_{2} \leqslant \cdots \leqslant x_{40}$.
If $x_{1}>1$, then $x_{1}+x_{2}=\left(x_{1}-1\right)+\left(x_{2}+1\right)$, and
$$\begin{array}{l} \left(x_{1}-1\right)^{2}+\left(x_{2}+1\right)^{2} \\ =x_{1}^{2}+x_{2}^{2}+2\left(x_{2}-x_{1}\right)+2>x_{1}^{2}+x_{2}^{2} . \end{array}$$

Therefore, when $x_{1}>1$, $x_{1}$ can be gradually adjusted to 1, at which point $x_{1}^{2}+x_{2}^{2}+\cdots+x_{40}^{2}$ will increase.

Similarly, $x_{2}, x_{3}, \cdots, x_{39}$ can be gradually adjusted to 1, at which point $x_{1}^{2}+x_{2}^{2}+\cdots+x_{40}^{2}$ will increase.

Thus, when $x_{1}, x_{2}, \cdots, x_{39}$ are all 1 and $x_{40}=19$, $x_{1}^{2}+$ $x_{2}^{2}+\cdots+x_{40}^{2}$ reaches its maximum value, i.e.,
$$A=\underbrace{1^{2}+1^{2}+\cdots+1^{2}}_{39 \text { terms }}+19^{2}=400 \text {. }$$

If there exist two numbers $x_{i}$ and $x_{j}$ such that
$$\begin{array}{l} x_{j}-x_{i} \geqslant 2 \quad(1 \leqslant i<j \leqslant 40), \text { then } \\ \left(x_{i}+1\right)^{2}+\left(x_{j}-1\right)^{2} \\ =x_{i}^{2}+x_{j}^{2}-2\left(x_{j}-x_{i}-1\right)<x_{i}^{2}+x_{j}^{2} . \end{array}$$

This indicates that in $x_{1}, x_{2}, \cdots, x_{39}, x_{40}$, if the difference between any two numbers is greater than 1, then by increasing the smaller number by 1 and decreasing the larger number by 1, $x_{1}^{2}+x_{2}^{2}+\cdots+x_{40}^{2}$ will decrease.

Therefore, when $x_{1}^{2}+x_{2}^{2}+\cdots+x_{40}^{2}$ reaches its minimum value, the difference between any two numbers in $x_{1}, x_{2}, \cdots, x_{40}$ does not exceed 1.

Thus, when $x_{1}=x_{2}=\cdots=x_{22}=1$ and $x_{23}=x_{24}=\cdots=x_{40}=2$, $x_{1}^{2}+x_{2}^{2}+\cdots+x_{40}^{2}$ reaches its minimum value, i.e.,
$$B=\underbrace{1^{2}+1^{2}+\cdots+1^{2}}_{22 \text { terms }}+\underbrace{2^{2}+2^{2}+\cdots+2^{2}}_{18 \text { terms }}=94 \text {. }$$

Hence, $A+B=494$.