15.1. Let $A^{\prime}$ be the image of point $A$ under a parallel translation by vector $\overrightarrow{M N}$. Then $A^{\prime} N = A M$, so the length of the path $A M N B$ is $A^{\prime} N + N B + M N$. Since the length of segment $M N$ is constant, we need to find the point $N$ for which the sum $A^{\prime} N + N B$ is minimal. It is clear that this sum is minimal when point $N$ lies on the segment $A^{\prime} B$, i.e., point $N$ is the intersection of the shore closest to point $B$ and the segment $A^{\prime} B$.
15.1. Where should the bridge $M N$ be built across the river separating villages $A$ and $B$, so that the path $A M N B$ from $A$ to $B$ is the shortest? (The riverbanks are considered parallel lines, and the bridge is perpendicular to the banks.)