Question 127
An acute angle with vertex $A$ and size $\alpha$ is given on a plane. Points $B_0$ and $B_1$ are chosen on different sides of the angle in such a way that $\angle AB_0B_1 = \beta$. Whenever points $B_0, B_1, \dots, B_{n-1}$ are defined, the next point $B_n$ on side $AB_{n-2}$ is allowed to be defined in such a way that $B_n \neq B_{n-2}$ and $B_{n-1}B_n = B_{n-2}B_{n-1}$. Prove that this process cannot last infinitely and determine the largest index $n$ (depending on $\alpha$ and $\beta$) for which $B_n$ can be defined.