Problem

Source: China North MO 2019, Problem 3

Tags: algebra



$n(n\geq2)$ is a given intenger, and $a_1,a_2,...,a_n$ are real numbers. For any $i=1,2,\cdots ,n$, $$a_i\neq -1,a_{i+2}=\frac{a_i^2+a_i}{a_{i+1}+1}.$$Prove: $a_1=a_2=\cdots=a_n$. (Note: $a_{n+1}=a_1,a_{n+2}=a_2$.)