Problem

Source: Uzbekistan National Olympiad, 2014, P5

Tags: geometry, 3D geometry, pyramid, circumcircle, geometry proposed



Let $PA_1A_2...A_{12} $ be the regular pyramid, $ A_1A_2...A_{12} $ is regular polygon, $S$ is area of the triangle $PA_1A_5$ and angle between of the planes $A_1A_2...A_{12} $ and $ PA_1A_5 $ is equal to $ \alpha $. Find the volume of the pyramid.