Cube inscribed in a Sphere

[msmith12]

Can anyone help me with this problem?

Suppose a sphere is colored in two colors: 10% of the surface is white, and the remaining part is black. Prove that there is a cube inscribed in the sphere such that all vertices are black

thanks
~matt

[quark]

This may have many solutions, provided the white paint is continuous, but I came up with this rather crude method. What you have to prove is that the diameter of the circle formed by the projection of white painted segment is lesser than the side of the cube.

If the surface area painted in white is 10% then length of the arc with the same radius that of the circle is also 10% of the circle circumference. Now you can get the angle intended by the arc at center. Now calculate the diameter of the projected circle.