Exercise 2. A prisoner is at the center of a circle with a radius of 10 meters. Each minute, he announces a direction he wants to take, and it is up to the guard to choose between this direction and the opposite direction. The prisoner then moves one meter in the direction chosen by the guard. Does the prisoner have a strategy that guarantees he will eventually leave the circle, even if the guard tries to prevent it?
## Solution

We will show that the prisoner has a winning strategy that allows him to exit the circle. Let $O$ be the center of the circle and $P_{n}$ the prisoner's position after the $n^{\text{th}}$ minute. All distances will be expressed in meters.

The strategy the prisoner should adopt is as follows. In the first minute, he indicates any direction, which will bring him to a point $P_{1}$ such that $O P_{1}=1$. Then, at the $(n+1)^{\text{th}}$ minute, the prisoner, now at $P_{n}$, will indicate to the guard a direction perpendicular to the line $O P_{n}$; regardless of the guard's choice, the prisoner will end up at a point $P_{n+1}$ such that $P_{n} P_{n+1}=1$ and such that the triangle $O P_{n} P_{n+1}$ is a right triangle at $P_{n}$; in particular, the distance $O P_{n+1}$ will not depend on the guard's choice and will be strictly greater than $O P_{n}$.

Thus, we have found a strategy that allows the prisoner to strictly increase his distance from the center of the circle every minute. This is not enough to conclude that he will be able to exit the circle; indeed, just because the distance $O P_{n}$ is strictly increasing does not mean it will necessarily exceed 10. (Similarly, the sequence $u_{n}=2-\frac{1}{n}$ is strictly increasing, but none of its terms are greater than 10). To conclude, we need to explicitly calculate the distance $O P_{n}$; since $O P_{1}^{2}=1$ and by the Pythagorean theorem, $O P_{n+1}^{2}=O P_{n}^{2}+1$, we deduce that for all $n, O P_{n}^{2}=n$ and thus $O P_{n}=\sqrt{n}$. In particular, $O P_{100}=10$; the prisoner therefore exits the circle at the $100^{\text{th}}$ minute.

A few remarks after correction. At each minute, the prisoner must announce a direction to the guard. By direction, we mean any ray originating from the current position $P_{n}$ of the prisoner. In particular, there are not just four, nor even eight possible directions (up, down, left, right, and the four directions inclined at $45^{\circ}$ relative to the previous ones), but an infinity. Moreover, it can be proven that no strategy involving only movements in the eight directions previously mentioned allows the prisoner to exit the circle.

On the other hand, if you claim that a strategy allows the prisoner to exit the circle regardless of the guard's choices, you must rigorously prove it, and it is not enough for it to seem to work. Many of you have thus proposed strategies and shown that the first steps of these strategies allow the prisoner to move away from the center of the circle; then they said (without proving it) that by repeating the same movements, the prisoner will continue to gradually move away from the center until he exits the circle. In fact, in almost all cases, these strategies did not work. Therefore, do not forget to rigorously prove everything you assert. And if you are not yourself convinced that what you say is true, you will not convince the grader.