INMO 2025 Problem 4

Let $n\ge 3$ be a positive integer. Find the largest real number $t_n$ as a function of $n$ such that the inequality $$\max\left(|a_1+a_2|, |a_2+a_3|, \dots ,|a_{n-1}+a_{n}| , |a_n+a_1|\right) \ge t_n \cdot \max(|a_1|,|a_2|, \dots ,|a_n|)$$ holds for all real numbers $a_1, a_2, \dots , a_n$ .

Proposed by Rohan Goyal and Rijul Saini