16. On a cylindrical screw that is $3 \mathrm{~cm}$ long, the radius of the base measures $\sqrt{30} / \pi$ millimeters. A tiny insect walks on the surface of the screw, moving along the thread. Knowing that the thread makes exactly 30 turns around the screw, how many centimeters will the insect travel to move from one end of the screw to the other?
(A) 32
(B) 30
(C) 33
(D) 36
(E) 31
Problem 16. The answer is (A).

![](https://cdn.mathpix.com/cropped/2024_04_17_1bf40f01494a5cda69b2g-4.jpg?height=705&width=301&top_left_y=65&top_left_x=1430)

b

When the cyclist and the runner meet for the first time, they have collectively covered the entire distance between $A$ and $B$. We also know that the runner has covered $12 \mathrm{~km}$, and that the distance covered by the cyclist, who moves three times faster, is triple; we can conclude that $A$ and $B$ are $48 \mathrm{~km}$ apart, and that the cyclist has covered $36 \mathrm{~km}$ up to that point.

If the second meeting occurs after the runner has covered an additional $x \mathrm{~km}$, the cyclist will have covered an additional $2 \cdot 12 + x$. The speed is always three times greater, so $24 + x = 3x$. Solving this, we get $x = 12$. The second meeting therefore occurs $24 \mathrm{~km}$ from $B$, and thus also $24 \mathrm{~km}$ from $A$.

(Problem proposed by A. Dal Zotto - S. Pelizzola - R. Zanotto)