Found problems: 1
2025 Bulgarian Spring Mathematical Competition, 12.3
Given integers \( m, n \geq 2 \), the points \( A_1, A_2, \dots, A_n \) are chosen independently and uniformly at random on a circle of circumference \( 1 \). That is, for each \( i = 1, \dots, n \), for any \( x \in (0,1) \), and for any arc \( \mathcal{C} \) of length \( x \) on the circle, we have $\mathbb{P}(A_i \in \mathcal{C}) = x$. What is the probability that there exists an arc of length \( \frac{1}{m} \) on the circle that contains all the points \( A_1, A_2, \dots, A_n \)?