Question 107
Let $n \ge 3$ be an integer and $a_1, a_2, \dots, a_n$ be real numbers. For each $1 \le k \le n$ the real numbers $b_1, b_2, \dots, b_{n+1}$ are defined by
$$
b_k = \frac{a_k + \max\{a_{k+1}, a_{k+2}\}}{2}
$$
and $b_{n+1} = b_1$ ($a_{n+1} = a_1$ and $a_{n+2} = a_2$). Find the smallest $\lambda$ such that the inequality
$$
\lambda \left[ \sum_{i=1}^{n} (a_i - a_{i+1})^{2024} \right] \geq \sum_{i=1}^{n} (b_i - b_{i+1})^{2024}
$$
is held for each $n \ge 3$ and all real numbers $a_1, a_2, \dots, a_n$.
$$
b_k = \frac{a_k + \max\{a_{k+1}, a_{k+2}\}}{2}
$$
and $b_{n+1} = b_1$ ($a_{n+1} = a_1$ and $a_{n+2} = a_2$). Find the smallest $\lambda$ such that the inequality
$$
\lambda \left[ \sum_{i=1}^{n} (a_i - a_{i+1})^{2024} \right] \geq \sum_{i=1}^{n} (b_i - b_{i+1})^{2024}
$$
is held for each $n \ge 3$ and all real numbers $a_1, a_2, \dots, a_n$.