# The quick brown fox and the lazy duck

Calculus Level 2

A fox moving at a constant speed of $s$$\text{ m/s}$ wants to eat a duck that is swimming at $1 \text{ m/s}$ in a pond. The duck can only fly when he’s on land, and the fox can’t swim.

What is the maximal value of $s$ such that wherever the animals start, the duck can reach land without being caught by the fox?

Assumptions:

• The pond is a perfectly round circle.
• The fox can start anywhere except inside the pond, but he can never enter the pond.
• The duck must start somewhere on the pond.
• If the duck reaches the edge of the pond at the exact same time as the fox, the fox catches the duck.
• If the duck reaches land and the fox is not there, he instantly flies away and the fox loses his meal.
• The duck and the fox see each other and take decisions in real time.
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