Problem

Source: 2007 All-Ukrainian Correspondence MO of magazine ''In the World of Mathematics'', grades 5-11 p11

Tags: ratio, geometry, circumcircle, Ukraine Correspondence



Denote by $B_1$ and $C_1$, the midpoints of the sides $AB$ and $AC$ of the triangle $ABC$. Let the circles circumscribed around the triangles $ABC_1$ and $AB_1C$ intersect at points $A$ and $P$, and let the line $AP$ intersect the circle circumscribed around the triangle $ABC$ at points $A$ and $Q$. Find the ratio $\frac{AQ}{AP}$.