Problem 3. (15 points) In parallelogram $A B C D$, side $A D$ is divided into equal parts by points $A_{1}, A_{2}, \ldots, A_{2020}$. Point $E_{1}$ is the intersection point of lines $B A_{1}$ and $A C$. Determine what fraction of diagonal $A C$ segment $A E_{1}$ constitutes.
Solution.

![](https://cdn.mathpix.com/cropped/2024_05_06_6d5fca905c76f828a3d8g-31.jpg?height=348&width=617&top_left_y=400&top_left_x=731)

Fig. 1

The points $A_{1}, A_{2}, \ldots, A_{2020}$ divide the side $A D$ into 2021 equal parts (Fig. 1). Connect points $B$ and $A_{1}$, and then draw lines parallel to $B A_{1}$ through points $A_{1}, A_{2}, \ldots, A_{2020}, D$, which also divide the side $B C$ into 2021 equal parts. These lines cut off equal segments $C E_{2021}, \ldots, E_{1} E_{2}, \ldots, A E_{1}$ on the line $A C$. From the equality of triangles $\triangle A E_{1} A_{1}$ and $\Delta C E_{2021} C_{1}$ by side and the adjacent angles, it follows that $C E_{2021}=A E_{1}$. Therefore, the constructed parallel lines divide $A C$ into 2022 equal parts. Hence, $A E_{1}=\frac{1}{2022} A C$.

$$
\text { Answer. } \frac{1}{2022} \text {. }
$$