Question 15
There are 2020 points $\{A_i = (x_i, y_i) : i = 1, \dots, 2020\}$ on the plane, satisfying
$$
\begin{aligned}
0 &= x_1 \lt x_2 \lt \dots \lt x_{2020}, \\
0 &= y_{2020} \lt y_{2019} \lt \dots \lt y_1.
\end{aligned}
$$
Denote $O = (0,0)$, the origin. Sequentially construct edges $OA_1, A_1A_2, \dots, A_{2019}A_{2020}, A_{2020}O$ to form a 2021-gon $C$. Tao wants to paint the entire region $C$ in black. On each turn, he chooses a point $(x, y)$, pays $xy$ dollars, and paints the entire region $\{(x', y') : 0 \le x' \le x, 0 \le y' \le y\}$ in black. Prove that Tao can always paint the entire region $C$ in black while spending at most $4|C|$ dollars, where $|C|$ is the area of $C$.
$$
\begin{aligned}
0 &= x_1 \lt x_2 \lt \dots \lt x_{2020}, \\
0 &= y_{2020} \lt y_{2019} \lt \dots \lt y_1.
\end{aligned}
$$
Denote $O = (0,0)$, the origin. Sequentially construct edges $OA_1, A_1A_2, \dots, A_{2019}A_{2020}, A_{2020}O$ to form a 2021-gon $C$. Tao wants to paint the entire region $C$ in black. On each turn, he chooses a point $(x, y)$, pays $xy$ dollars, and paints the entire region $\{(x', y') : 0 \le x' \le x, 0 \le y' \le y\}$ in black. Prove that Tao can always paint the entire region $C$ in black while spending at most $4|C|$ dollars, where $|C|$ is the area of $C$.