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problem involving tangencyand midpoints

Source: Netherlands TST for IMO 2017 day 1 problem 4

February 1, 2018
geometry

Problem Statement

Let ABCABC be a triangle, let MM be the midpoint of ABAB, and let NN be the midpoint of CMCM. Let XX be a point satisfying both ∠XMC=∠MBC\angle XMC = \angle MBC and ∠XCM=∠MCB\angle XCM = \angle MCB such that XX and BB lie on opposite sides of CMCM. Let ω\omega be the circumcircle of triangle AMXAMX. (a)(a) Show that CMCM is tangent to ω\omega. (b)(b) Show that the lines NXNX and ACAC intersect on ω\omega
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