Question 41
Find the smallest positive real number $p \ (\le 1)$ such that the inequality
$$
\sum_{i=1}^{2024} x_i (y_{2025-i} - y_{2024-i}) \ge 1 - p
$$
holds for all real numbers $0 \le x_1 \le x_2 \le \dots \le x_{2024} \le 1$ and $0 = y_0 \le y_1 \le y_2 \le \dots \le y_{2024} \le 1$ satisfying
$$
\sum_{i=1}^{2024} x_i = \sum_{i=1}^{2024} y_i = 2024p.
$$
$$
\sum_{i=1}^{2024} x_i (y_{2025-i} - y_{2024-i}) \ge 1 - p
$$
holds for all real numbers $0 \le x_1 \le x_2 \le \dots \le x_{2024} \le 1$ and $0 = y_0 \le y_1 \le y_2 \le \dots \le y_{2024} \le 1$ satisfying
$$
\sum_{i=1}^{2024} x_i = \sum_{i=1}^{2024} y_i = 2024p.
$$