6. In a convex quadrilateral $A B C D$, angles $B$ and $D$ are equal, $C D=4 B C$, and the bisector of angle $A$ passes through the midpoint of side $C D$. What can the ratio $A D / A B$ be? (S. Berlov)
Answer: 2:3. Solution. Let $M$ be the midpoint of side $CD$. Consider point $K$ on ray $AB$, symmetric to point $D$ with respect to line $AM$. Since $\angle ABC = \angle ADM = \angle AKM$, then $BC \| KM$ and point $K$ lies on segment $AB$. Since $CM = DM = KM$, then $\angle DKC = 90^{\circ}$ and $KC \| AM$. Therefore, the sides of triangles $AKM$ and $KBC$ are respectively parallel, so they are similar with a coefficient $k = KM / BC = 2$, from which $AD = AK = 2KB$ and $AD: AB = 2: 3$.