Problem 1. On the line $p$ are given points $M, A, B, C, D$ and $N$ in the given order, such that $\overline{A B}=\overline{B C}=\overline{C D}$. The distance between the midpoints of segments $A B$ and $C D$ is $32 \mathrm{~cm}$, and the distance between the midpoints of segments $M A$ and $D N$ is $57 \mathrm{~cm}$. How long is the segment $M N$?
Solution. From the fact that the distance between the midpoints of segments $A B$ and $C D$ is $32 \mathrm{~cm}$ and $\overline{A B}=\overline{B C}=\overline{C D}$, it follows that $2 \overline{B C}=32 \mathrm{~cm}$, which means the length of segment $\overline{A B}=16 \mathrm{~cm}$. Let $P$ be the midpoint of segment $M A$, and $Q$ be the midpoint of segment $D N$.

Since the distance between the midpoints of segments $M A$ and $D N$ is $57 \mathrm{~cm}$, it follows that the sum of the lengths of segments

$$
\overline{M P}+\overline{Q N}=57-3 \cdot 16=9 \mathrm{~cm}
$$

Thus, the length of segment $M N$ is

$$
\overline{M N}=\overline{M P}+\overline{P Q}+\overline{Q N}=57+9=66 \mathrm{~cm}
$$

$M \quad P \quad A \quad B \quad C \quad D \quad Q$