[Geometry]
Problem: A bubble in the shape of a hemisphere of radius 1 is on a tabletop. Inside the bubble are five congruent spherical marbles, four of which are sitting on the table and one which rests atop the others. All marbles are tangent to the bubble, and their centers can be connected to form a pyramid with volume $V$ and with a square base. Compute $V$.

Solution: The first step is to compute the radius $r$ of one of the marbles. The diagram below shows a cross-section through the centers of two diagonally opposite marbles.

<img_3908>

Triangle $B Q R$ appears to be equilateral, and in fact, it is. Reflect the diagram in the tabletop $\overline{A C}$ to obtain six mutually tangent congruent circles inside a larger circle:

<img_3444>

Because the circles are congruent, their centers are equidistant from $B$, and the distances between adjacent centers are equal. So $Q$ can be obtained as the image of $R$ under a rotation of $360^{\circ} / 6=60^{\circ}$ counterclockwise around $B$. Then $P Q=r \Rightarrow B Q=B R=2 r \Rightarrow B D=$ $3 r$, hence $r=1 / 3$. Notice too that the height of the pyramid is simply the radius $r$ and the diagonal of the square base is twice the altitude of the equilateral triangle $B Q R$, that is, $2 \cdot \frac{r \sqrt{3}}{2}=r \sqrt{3}$. So the area of the base is $3 r^{2} / 2$. Thus the volume of the pyramid is $(1 / 3)\left(3 r^{2} / 2\right)(r)=r^{3} / 2$. Because $r=1 / 3$, the volume is $\mathbf{1} / \mathbf{5 4}$.

Final answer: $\frac{1}{54}$