National problem · Geometry

Let \(ABC\) be a triangle with \(\angle CAB = 60^\circ\). Denote by \(O\) and \(I\) the centres of the circle circumscribed about it and of the circle inscribed in it, respectively, and let \(A'\) be the point symmetric to \(A\) with respect to the line \(BC\). Prove that the line \(OI\) bisects the angle \(AOA'\).

The angle at \(A\) is \(60^\circ\); the point \(A'\) is the mirror image of \(A\) in the line \(BC\).

Serbian National Competition (Drzavno takmicenje) 2024, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source

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