IMO 1967 LL MON32

Origin: MON

Problem

Determine the volume of the body obtained by cutting the ball of radius R by the trihedron with vertex in the center of that ball if its dihedral angles are \alpha, \beta, \gamma.

Solution

Let us denote by V the volume of the given body, and by Va, Vb, Vc the volumes of the parts of the given ball that lie inside the dihe- dra of the given trihedron. It holds that Va = 2R3\alpha/3, Vb = 2R3\beta/3, Vc = 2R3\gamma/3. It is easy to see that 2(Va+Vb+Vc) = 4V +4\piR3/3, from which it follows that O A A′ C C′ B B′ V = 1 3R3(\alpha + \beta + \gamma −\pi).