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geometry problem

Source: Netherlands TST for IMO 2017 day 3 problem 1

February 1, 2018
geometry

Problem Statement

A circle ω\omega with diameter AKAK is given. The point MM lies in the interior of the circle, but not on AKAK. The line AMAM intersects ω\omega in AA and QQ. The tangent to ω\omega at QQ intersects the line through MM perpendicular to AKAK, at PP. The point LL lies on ω\omega, and is such that PLPL is tangent to ω\omega and L≠QL\neq Q. Show that K,LK, L, and MM are collinear.
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