XLVIII OM - I - Problem 7

Calculate the upper limit of the volume of tetrahedra contained within a sphere of a given radius $ R $, one of whose edges is the diameter of the sphere.
Let $ABCD$ be one of the considered tetrahedra. Assume that the edge $AB$ is the diameter of the given sphere. Denote the distance from point $C$ to the line $AB$ by $w$, and the distance from point $D$ to the plane $ABC$ by $h$. The volume of the tetrahedron $ABCD$ is given by

Of course, $w \leq R$, $h \leq R$, and therefore $V \leq \frac{1}{3} R^3$.
The tetrahedron $ABCD$ was chosen arbitrarily; hence, the upper bound of the volumes of the considered tetrahedra does not exceed the number $\frac{1}{3} R^3$.
When $ABC$ is an isosceles right triangle with hypotenuse $AB$, then $w = R$. When, on the other hand, $ABD$ is an isosceles right triangle with hypotenuse $AB$, lying in a plane perpendicular to the plane $ABC$, then also $h = R$. When both these conditions are met, the equality $V = \frac{1}{3} R^3$ holds.
Conclusion: the upper bound of the considered volumes is equal to $\frac{1}{3} R^3$.