Fomin S.V.

Two identical correctly working 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_39480ed8b0d60d88e837g-18.jpg?height=280&width=518&top_left_y=1466&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

$1 / 2(M+m)$.