$\underline{\text { Fomin S.V. }}$

Two correctly functioning identical clocks are hanging on the wall. One shows Moscow time, the other shows local time. The minimum distance between the ends of their hour hands is $m$, and the maximum distance is $M$. Find the distance between the centers of these clocks.
In the notation of the figure, $\overrightarrow{A B}=\overrightarrow{C D}+(\overrightarrow{A C}+\overrightarrow{D B})$.

![](https://cdn.mathpix.com/cropped/2024_05_06_b2c6d4775835e38abfe3g-27.jpg?height=271&width=518&top_left_y=1345&top_left_x=770)

The vector $\boldsymbol{c}=\overrightarrow{C D}$ is constant, and the vector $\boldsymbol{a}=\overrightarrow{A C}+\overrightarrow{D B}$ has a constant length and rotates uniformly around a circle. The length of the vector $\boldsymbol{c}+\boldsymbol{a}$ reaches its maximum and minimum when the vectors $\boldsymbol{c}$ and $\boldsymbol{a}$ are collinear. Therefore, the length of the vector $\boldsymbol{c}$ is equal to half the sum of the minimum and maximum values of the length of the vector $\boldsymbol{c}+\boldsymbol{a}$.

## Answer

$1 / 2(M+m)$.