12th European Girls' Mathematical Olympiad (EGMO 2023)

Problem 1.
There are $n \ge 3$ positive real numbers $a_1, a_2, \dots, a_n$ . For each $1 \le i \le n$ we let $b_i = \frac{a_{i-1} + a_{i+1}}{a_i}$ (here we define $a_0$ to be $a_n$ and $a_{n+1}$ to be $a_1$ ). Assume that for all $i$ and $j$ in the range $1$ to $n$ , we have $a_i \le a_j$ if and only if $b_i \le b_j$ .
Prove that $a_1 = a_2 = \dots = a_n$ .