1.2. Inside a rectangular parallelepiped $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, there are two spheres such that the first one touches all three faces containing $A$, and the second one touches all three faces containing $C^{\prime}$. Additionally, the spheres touch each other. The radii of the spheres are 10 and 11, respectively. The lengths of the edges $A B$ and $A D$ are 25 and 26. Find the length of the edge $A A^{\prime}$.
### 1.2. Answer: 41.

Let's introduce coordinates with the center at $A$; the axes will be directed along $A B, A D$, and $A A^{\prime}$. Denote the centers of the spheres as $O_{1}$ and $O_{2}$, and let $A A^{\prime}=x$. Then $O_{1}=(10,10,10)$, $O_{2}=(14,15, x-11)$, and $O_{1} O_{2}=21$. We obtain $4^{2}+5^{2}+(x-21)^{2}=21^{2}$, which simplifies to $x^{2}-42 x+41=0$. The solution $x=1$ is not valid because the spheres are inside the parallelepiped.