Problem 10. The sum of the surface areas of the polyhedra into which a parallelepiped is divided by sections is equal to the sum of the surface area of the parallelepiped and the areas of the internal surfaces. The sum of the areas of the internal surfaces is equal to twice the sum of the areas of the sections.

Let's find the maximum possible area of a section passing through the diagonal $XY$ of an arbitrary parallelepiped with edges $a \leq b \leq c$. The section is a parallelogram $ZXTY$, whose vertices lie on opposite edges of the parallelepiped. The area of the parallelogram is the product of the length of the diagonal $XY$ and the distance from point $Z$ to $XY$.

Consider the projection of the parallelepiped onto a plane perpendicular to the diagonal $XY$. From the diagram, it is clear that the distance from point $Z$ to the broken line $ABC$, and thus to the diagonal $XY$, is greatest if

$Z$ coincides with one of the vertices $A, B$, or $C$.

This means that the section passes through one of the edges of the parallelepiped. Therefore, the section with the largest area is one of the diagonal sections. All these sections are rectangles. Let's find the largest of their areas:

$$
S_{1}=a \sqrt{b^{2}+c^{2}}, \quad S_{2}=b \sqrt{a^{2}+c^{2}}, \quad \text{and} \quad S_{3}=c \sqrt{b^{2}+a^{2}}
$$

From the condition $a \leq b \leq c$, it follows that

$a^{2} b^{2}+a^{2} c^{2} \leq c^{2} b^{2}+a^{2} c^{2}$, and $a^{2} b^{2}+c^{2} b^{2} \leq c^{2} b^{2}+a^{2} c^{2}$. Therefore, $S_{1} \leq S_{3}$ and $S_{2} \leq S_{3}$. This means that the section with the largest area passes through the largest edge. According to the condition, the largest length is the edge $AA_{1}$, so the sections $AA_{1}C_{1}C$ and $BB_{1}D_{1}D$ have the largest area:

$5 \sqrt{4^{2}+3^{2}}=25$. The sum of the surface areas of the polyhedra into which the parallelepiped is divided by these sections (see the diagram) is

$$
2\left(A A_{1} \cdot A B + A A_{1} \cdot A D + A B \cdot A D\right) + 4 \cdot 25 = 194
$$
Answer: 194.


[^0]:    ${ }^{1}$ This statement is a particular case of Cauchy's inequality.