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Math Prize 2022 Problem 18

Source:

October 12, 2022

Problem Statement

Let AA be the locus of points (α,β,γ)(\alpha, \beta, \gamma) in the αβγ\alpha\beta\gamma-coordinate space that satisfy the following properties:
(I) We have α\alpha, β\beta, γ>0\gamma > 0. (II) We have α+β+γ=π\alpha + \beta + \gamma = \pi. (III) The intersection of the three cylinders in the xyzxyz-coordinate space given by the equations \begin{eqnarray*} y^2 + z^2 & = & \sin^2 \alpha \\ z^2 + x^2 & = & \sin^2 \beta \\ x^2 + y^2 & = & \sin^2 \gamma \end{eqnarray*} is nonempty.
Determine the area of AA.
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