3.3. [6] Misha stands at the center of a meadow in the shape of a circle with a radius of 100 m. Every minute, he takes a step of 1 m. Before each step, Misha informs Eli of the direction he has chosen. Eli can force him to take a step in the opposite direction of the one he chose. Can Misha act in such a way that at some point he will leave the meadow, or can Eli always prevent him?

Answer. Yes.
Solution. From the second step onwards, Misha can each time choose a direction perpendicular to the line segment connecting the center $O$ of the meadow and the point where he is currently located. Let Misha have taken a step from point $A$, where $O A=\sqrt{a}$, to point $B$. By the Pythagorean theorem, $O B=\sqrt{a+1}$. In this way, after the $n$-th step, Misha will be at a distance of $\sqrt{n}$ m from the center. After 10001 steps, Misha will leave the meadow.