5. A race on an unpredictable distance is held as follows. On a circular running track 1 kilometer long, two points $A$ and $B$ are chosen randomly (using a spinning arrow), after which the athletes run from $A$ to $B$ along the shorter arc. Find the median value of the length of this arc, that is, such an $m$ that the length of the arc will exceed $m$ with a probability of exactly $50 \%$.

(A. A. Tesler)
Solution. Let's choose the origin and the positive direction on the path, then the pairs $(A, B)$ can be identified with pairs of numbers from $[0,1)$ (when measured in kilometers). The probability that $(A, B)$ belongs to some subset of $[0,1) \times[0,1)$ is equal to the area of this subset. The length of the shorter arc is $|A-B|$ when $|A-B| \leqslant \frac{1}{2}$ and $(1-|A-B|)$ otherwise. The region where this length does not exceed a number $0 \leqslant m \leqslant \frac{1}{2}$ is shown in the figure, its area is $2 m$. For $m=0.25$

![](https://cdn.mathpix.com/cropped/2024_05_06_e219a7e95d389188eca9g-2.jpg?height=322&width=362&top_left_y=136&top_left_x=1555)
we get a probability of $50 \%$.

Another solution. Suppose point $A$ is chosen arbitrarily. Note that with a probability of $50 \%$, point $B$ lies on the semicircle with midpoint $A$ - which is equivalent to the arc length being no more than $250 \mathrm{m}$.

Answer: $m=250 \mathrm{m}$.