Strange problem

[ins-]

Let PA and PB are the tangents to the circle k. The line through P intersects k at the points C and D (C is between P and D). A line through C, parallel to PA intersects k at the point E. BE intersects PA at the point F and Q is the intersection point of k and DF. If M and N are (respectively) intersection points of BC and QE with the line PA - prove that AM=AN.

[nnosipov]

Проверил непосредственным вычислением. Всё сошлось. В процессе вычислений бросилось в глаза следующее: точка $Q$ не зависит от положения точки $C$, при этом $\angle QOP$ равен утроенному $\angle AOP$ ($O$ --- центр данной окружности). Скорее всего, это можно обнаружить и геометрически, но это уже для любителей геометрии.

[ins-]

10x for the time spent. I saw some solutions - it is an eight grade problem but sometimes people are able to see how to solve it sometimes - not. It depends on the person who solve he problem and his/her taste and previous experience. Hope you like the problem. If you are interested I can post some links where the problem is solved.