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Found problems: 1

2021 Brazil National Olympiad, 6
Tags: bicentric polygon , convex polygon , inscribed circle , circumscribed , tangent circle , geometry
Let \(n \geq 5\) be integer. The convex polygon \(P = A_{1} A_{2} \ldots A_{n}\) is bicentric, that is, it has an inscribed and circumscribed circle. Set \(A_{i+n}=A_{i}\) to every integer \(i\) (that is, all indices are taken modulo \(n\)). Suppose that for all \(i, 1 \leq i \leq n\), the rays \(A_{i-1} A_{i}\) and \(A_{i+2} A_{i+1}\) meet at the point \(B_{i}\). Let \(\omega_{i}\) be the circumcircle of \(B_{i} A_{i} A_{i+1}\). Prove that there is a circle tangent to all \(n\) circles \(\omega_{i}\), \(1 \leq i \leq n\).