IMO Shortlist 2015 problem G3

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Let ABC be a triangle with \angle{C} = 90^{\circ}, and let H be the foot of the altitude from C. A point D is chosen inside the triangle CBH so that CH bisects AD. Let P be the intersection point of the lines BD and CH. Let \omega be the semicircle with diameter BD that meets the segment CB at an interior point. A line through P is tangent to \omega at Q. Prove that the lines CQ and AD meet on \omega.