Problem

Source: 2024 Turkey TST P6

Tags: inequalities



For a positive integer $n$ and real numbers $a_1, a_2, \dots ,a_n$ we'll define $b_1, b_2, \dots ,b_{n+1}$ such that $b_k=a_k+\max({a_{k+1},a_{k+2}})$ for all $1\leq k \leq n$ and $b_{n+1}=b_1$. (Also $a_{n+1}=a_1$ and $a_{n+2}=a_2$) Find the least possible value of $\lambda$ such that for all $n, a_1, \dots, a_n$ the inequality $$\lambda \Biggl[ \sum_{i=1}^n(a_i-a_{i+1})^{2024} \Biggr] \geq \sum_{i=1}^n(b_i-b_{i+1})^{2024}$$holds.