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Problem 2. Let $ABC$ be a triangle and let points $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Let points $K$ and $L$ be chosen strictly inside triangles $BMC$ and $BNC$, respectively, such that $K$ lies strictly inside triangle $ABL$ and $L$ lies strictly inside triangle $AKC$. Suppose that $$\angle KBA = \angle ACL, \quad \angle LBK = \angle LNC, \quad \text{and} \quad \angle LCK = \angle BMK.$$ Let $O$ be the circumcentre of triangle $AKL$. Prove that $OM = ON$.
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