3. In parallelogram $A B C D$ with side $A B=1$, point $M$ is the midpoint of side $B C$, and angle $A M D$ is 90 degrees. Find the side $B C$.
Answer: 2. First solution. Draw the median $M N$ in triangle $A M D$. $A B M N$ is a parallelogram since $B M$ and $A N$ are equal and parallel, thus $M N = B C = 1$. But the median $M N$ is half the hypotenuse $A D$ in the right triangle $A M D$, from which it follows that $B C = A D = 2$. Second solution. Extend the segment $A M$ and side $C D$ until they intersect at point $K$. Consider triangle $A D K$. Segment $M C$ is parallel to the base $A D$ and equal to half of it. Therefore, $M C$ is the midline, $A M = M K$, and $K D = 2 C D = 2$. But in the considered triangle, the median $D M$ coincides with the altitude, so it is isosceles and $A D = 2 A B = 2$.