Baltic Way 1991 · Problem 18

Solution:

No, it is not. Any tetrahedron that does not contain the centre of the sphere as an internal point has a height drawn to one of its faces less than or equal to the radius of the sphere. As each of the faces of the tetrahedron is contained in a circle with radius not greater than $1$, its area cannot exceed $\frac{3 \sqrt{3}}{4}$. Thus, the volume of such a tetrahedron must be less or equal than $\frac{1}{3} \cdot 1 \cdot \frac{3 \sqrt{3}}{4} = \frac{\sqrt{3}}{4} < \frac{1}{2}$.