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38th Austrian Mathematical Competition 2007

Source: 2nd round

February 7, 2009
geometry unsolvedgeometry

Problem Statement

Let n>4 n > 4 be a non-negative integer. Given is the in a circle inscribed convex n n-gon A_0A_1A_2\dots A_{n \minus{} 1}A_n (A_n \equal{} A_0) where the side A_{i \minus{} 1}A_i \equal{} i (for 1≤i≤n 1 \le i \le n). Moreover, let ϕi \phi_i be the angle between the line A_iA_{i \plus{} 1} and the tangent to the circle in the point Ai A_i (where the angle ϕi \phi_i is less than or equal 90o 90^o, i.e. ϕi \phi_i is always the smaller angle of the two angles between the two lines). Determine the sum \Phi \equal{} \sum_{i \equal{} 0}^{n \minus{} 1} \phi_i of these n n angles.
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