Question 9
We call a point sequence $(A_0, A_1, \dots, A_n)$ interesting, if the abscissa and ordinate are equal for each $A_i$, and the slopes of segment $OA_0, OA_1, \dots, OA_n$ strictly increase ($O$ is the origin), and the area of each $\triangle OA_iA_{i+1}$ ($0 \le i \le n-1$) is $\frac{1}{2}$.
For a point sequence $(A_0, A_1, \dots, A_n)$, insert a point $A$ adjacent to two points $A_i, A_{i+1}$ satisfying $\overrightarrow{OA} = \overrightarrow{OA_i} + \overrightarrow{OA_{i+1}}$, then we call thus obtained new point sequence $(A_0, \dots, A_i, A, A_{i+1}, \dots, A_n)$ an expansion of $(A_0, A_1, \dots, A_n)$.
Let $(A_0, A_1, \dots, A_n)$ and $(B_0, B_1, \dots, B_m)$ be any two interesting point sequences. Prove that if $A_0 = B_0$ and $A_n = B_m$, then we can expand both point sequences to some same point sequence $(C_0, C_1, \dots, C_k)$.
(posed by Qu Zhenhua)
For a point sequence $(A_0, A_1, \dots, A_n)$, insert a point $A$ adjacent to two points $A_i, A_{i+1}$ satisfying $\overrightarrow{OA} = \overrightarrow{OA_i} + \overrightarrow{OA_{i+1}}$, then we call thus obtained new point sequence $(A_0, \dots, A_i, A, A_{i+1}, \dots, A_n)$ an expansion of $(A_0, A_1, \dots, A_n)$.
Let $(A_0, A_1, \dots, A_n)$ and $(B_0, B_1, \dots, B_m)$ be any two interesting point sequences. Prove that if $A_0 = B_0$ and $A_n = B_m$, then we can expand both point sequences to some same point sequence $(C_0, C_1, \dots, C_k)$.
(posed by Qu Zhenhua)