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6

Part of 2016 Israel National Olympiad

Problems(1)

Israel 2016 Q6 - Ratio in a regular 12-gon

Source: Israel National Olympiad 2016 Q6

8/7/2019
Points A1,A2,A3,...,A12A_1,A_2,A_3,...,A_{12} are the vertices of a regular polygon in that order. The 12 diagonals A1A6,A2A7,A3A8,...,A11A4,A12A5A_1A_6,A_2A_7,A_3A_8,...,A_{11}A_4,A_{12}A_5 are marked, as in the picture below. Let XX be some point in the plane. From XX, we draw perpendicular lines to all 12 marked diagonals. Let B1,B2,B3,...,B12B_1,B_2,B_3,...,B_{12} be the feet of the perpendiculars, so that B1B_1 lies on A1A6A_1A_6, B2B_2 lies on A2A7A_2A_7 and so on.
Evaluate the ratio XA1+XA2+⋯+XA12B1B6+B2B7+⋯+B12B5\frac{XA_1+XA_2+\dots+XA_{12}}{B_1B_6+B_2B_7+\dots+B_{12}B_5}. https://i.imgur.com/DUuwFth.png
ratiogeometryregular polygon