Source

I got this problem from Rustan Leino, who got it from Koen Claessen.

I solved it and wrote up my solution.

Problem

A duck is in circular pond. The duck wants to swim ashore, because it wants to fly off and this particular duck is unable to start flying from the water. There's also a fox, on the shore. The fox wants to eat the duck, but this particular fox can't swim, so it can only hope to catch the duck when the duck reaches the shore. The fox can run four times faster than the duck can swim. Is there always a way for the duck to escape?

Solution
Reveal

Yes, there is.

Let $r$ be the pond's radius, let $p$ be the pond's center, and let $C$ be the circle centered at $p$ with radius $0.23r.$

The duck should swim to a point on $C,$ then swim along $C$ in a direction away from the fox. Note that when traveling in a circle, one's speed in radians per unit time is one's regular speed divided by the radius of the circle one is traveling along. So, since the circle the duck travels along has a radius $1/0.23 > 4$ times smaller than than the circle the fox travels along, the duck's radial speed is greater than the fox's. Thus, the duck can eventually reach a point radially opposite the fox.

Once the duck is radially opposite the fox, it should swim directly toward the closest point along the shore. The duck has to travel $0.77r$ to reach the shore, while the fox has to travel $\pi r$ to intercept the duck at that point. Fortunately for the duck, $0.77 < \pi/4,$ so even though the fox goes four times as fast as the duck, the duck reaches the shore first and can safely escape.