[Geometry]
Problem: A cylinder with radius $r$ and height $h$ has volume 1 and total surface area 12. Compute $\frac{1}{r}+\frac{1}{h}$.

Solution: Since $\pi r^{2} h=1$, we have $h=\frac{1}{\pi r^{2}}$ and $\pi r^{2}=\frac{1}{h}$. Consequently,

$$
2 \pi r h+2 \pi r^{2}=12 \Rightarrow(2 \pi r)\left(\frac{1}{\pi r^{2}}\right)+2\left(\frac{1}{h}\right)=12 \Rightarrow \frac{2}{r}+\frac{2}{h}=12 \Rightarrow \frac{1}{r}+\frac{1}{h}=\mathbf{6}
$$
The total surface area is $2 \pi r h+2 \pi r^{2}=12$ and the volume is $\pi r^{2} h=1$. Dividing, we obtain $\frac{12}{1}=\frac{2 \pi r h+2 \pi r^{2}}{\pi r^{2} h}=\frac{2}{r}+\frac{2}{h}$, thus $\frac{1}{r}+\frac{1}{h}=\frac{12}{2}=\mathbf{6}$.

Final answer: 6