BMO 2023 | Problem 2 - 10 May 2023 - Balkan Mathematical Olympiad

Problem 2.
In triangle $ABC$ , the incircle touches sides $BC,CA,AB$ at $D,E,F$ respectively. Assume there exists a point $X$ on the line $EF$ such that $\angle{XBC} = \angle{XCB} = 45^{\circ}.$ Let $M$ be the midpoint of the arc $BC$ on the circumcircle of $ABC$ not containing $A$ . Prove that the line $MD$ passes through $E$ or $F$ .

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