[Geometry]
Problem: Let $T=125$. Cube $\mathcal{C}_{1}$ has volume $T$ and sphere $\mathcal{S}_{1}$ is circumscribed about $\mathcal{C}_{1}$. For $n \geq 1$, the sphere $\mathcal{S}_{n}$ is circumscribed about the cube $\mathcal{C}_{n}$ and is inscribed in the cube $\mathcal{C}_{n+1}$. Let $k$ be the least integer such that the volume of $\mathcal{C}_{k}$ is at least 2019. Compute the edge length of $\mathcal{C}_{k}$.

Solution: In general, let cube $\mathcal{C}_{n}$ have edge length $x$. Then the diameter of sphere $\mathcal{S}_{n}$ is the space diagonal of $\mathcal{C}_{n}$, which has length $x \sqrt{3}$. This in turn is the edge length of cube $\mathcal{C}_{n+1}$. Hence the edge lengths of $\mathcal{C}_{1}, \mathcal{C}_{2}, \ldots$ form an increasing geometric sequence with common ratio $\sqrt{3}$ and volumes of $\mathcal{C}_{1}, \mathcal{C}_{2}, \ldots$ form an increasing geometric sequence with common ratio $3 \sqrt{3}$. With $T=125$, the edge length of $\mathcal{C}_{1}$ is 5 , so the sequence of edge lengths of the cubes is $5,5 \sqrt{3}, 15, \ldots$, and the respective sequence of the volumes of the cubes is $125,375 \sqrt{3}, 3375, \ldots$. Hence $k=3$, and the edge length of $\mathcal{C}_{3}$ is $\mathbf{1 5}$.

Final answer: 15