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On a sphere of radius 11, points $A, A 1, B, B 1, C$ and $C 1$ are located. Lines $A A 1, B B 1$ and $C C 1$ are pairwise perpendicular and intersect at point $M$, which is at a distance of $\sqrt{59}$ from the center of the sphere. Find $A A 1$, given that $B B 1=18$, and point $M$ divides the segment $C C 1$ in the ratio $(8+\sqrt{2}):(8-\sqrt{2})$.

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Let $C M=(8+\sqrt{2}) x, C 1 M=(8-\sqrt{2}) x$. Consider the section of the sphere by a plane passing through its center $O$ and the line $C C 1$. We obtain a circle of radius 11 with center $O$ and chord $C C 1$. Extend the segment $O M$ to intersect the circle at points $D$ and $E$. By the intersecting chords theorem,

$$
C M \cdot C 1 M=D M \cdot E M, (8+\sqrt{2})(8-\sqrt{2}) \times 2=(11+\sqrt{59})(11-\sqrt{59}), 62 \times 2=62
$$

from which $x=1$. Therefore,

$$
C M=8+\sqrt{2}, C 1 M=8-\sqrt{2}, C C 1=16
$$

Consider the section of the sphere by a plane passing through the lines $C C 1$ and $B B 1$. Let $B M=y$. Then

$$
B 1 M=B B 1-B M=18-y, B M \cdot B 1 M=C M \cdot C 1 M
$$

If $B M<B 1 M$, then from the equation

$$
y(18-y)=(8+\sqrt{2})(8-\sqrt{2})=62
$$

we find that $B M=y=9-\sqrt{19}$. Then $B 1 M=9+\sqrt{19}$. Let $K$ and $N$ be the midpoints of the chords $C C 1$ and $B B 1$ respectively. Then

$$
M K=C 1 K-C 1 M=8-(8-\sqrt{2})=\sqrt{2}
$$

$$
M N=B 1 M-B 1 N=9+\sqrt{19}-9=\sqrt{19}.
$$

Let $P$ be the orthogonal projection of the center $O$ onto the plane of the lines $C C 1$ and $B B 1$. Then $M K P N$ is a rectangle. Therefore,

$$
P M=\sqrt{M K^{2}+M N^{2}}=\sqrt{2+19}=\sqrt{21}
$$

From the right triangle $O P M$ we find that

$$
O P=\sqrt{O M^{2}-P^{2}}=\sqrt{59-21}=\sqrt{38}
$$

Consider the section of the sphere by a plane passing through the lines $A A 1$ and $B B 1$. Let $Q$ be the orthogonal projection of the center $O$ onto this plane, and $F$ be the midpoint of $A A 1$. Denote $A F=z$. Then

$$
\begin{gathered}
Q F=M N=\sqrt{19}, N Q=O P=\sqrt{38}, Q A=Q B=\sqrt{N Q^{2}+N B^{2}}=\sqrt{38+81}=\sqrt{119} \\
\frac{1}{2} A A 1=A F=\sqrt{Q A^{2}-Q F^{2}}=\sqrt{119-19}=10
\end{gathered}
$$

Therefore, $A A 1=20$.

## Answer

20.00