Question 1
Let $x_1, x_2, \dots, x_{2024}$ be non-negative real numbers such that $x_1 \le x_2 \le \dots \le x_{2024}$, and $x_1^3 + x_2^3 + \dots + x_{2024}^3 = 2024$. Prove that
$$
\sum_{1 \le i \lt j \le 2024} (-1)^{i+j} x_i^2 x_j \ge -1012.
$$
$$
\sum_{1 \le i \lt j \le 2024} (-1)^{i+j} x_i^2 x_j \ge -1012.
$$