Τρίτη, 17 Οκτωβρίου 2017

2004 IMO Shortlist G5

Let $A_1A_2 … A_n$ be a regular $n$-gon. The points $B_1,..., B_{n-1}$ are defined as follows:
If $i =1$ or $ i = n – 1$, then $B_i $ is the midpoint of the side $A_iA_{i+1}$,
If $i \ne 1, i \ne n – 1$ and $S$ is the intersection point of $A_iA_{i+1}$ and $A_nA_i$, then $B_i$  is the intersection point of the bisector of the angle $A_iSA_{i+1}$ with $A_iA_{i+1}$.

Prove the equality $ \angle  A_1B_1A_n + \angle  A_1B_2A_n + … +\angle  A_1B_{n-1}A_n = 180^\circ$.


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