7th grade third round Bulgaria 2007

[MM]

In my opinion that's a very good geometrical porblem.
Let M an N be the midpoints of the cathetuses AC and BC in right-angled triangle ABC. A line through M parallel to AB intersects the angle bisector of [tex]\angle BAC[/tex] in point P and a line through N parallel to AB intersects the angle bisector of [tex]\angle ABC[/tex] in point Q. If the angle bisectors of [tex]\angle BAC[/tex] and [tex]\angle ABC[/tex] intesects each other in point O prove that [tex]CO>PQ>\frac{CO}{2}[/tex].