Problem 3. (15 points) In parallelogram $A B C D$, side $A D$ is divided into equal parts by points $A_{1}, A_{2}, \ldots, A_{2022}$. 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-35.jpg?height=348&width=628&top_left_y=454&top_left_x=725)

Fig. 1

Points $A_{1}, A_{2}, \ldots, A_{2022}$ divide side $A D$ into 2023 equal parts (Fig. 1). Connect points $B$ and $A_{1}$, and then draw lines parallel to $B A_{1}$ through points $A_{1}, A_{2}, \ldots, A_{2022}, D$, which also divide side $B C$ into 2023 equal parts. These lines cut off equal segments $C E_{2023}, \ldots, E_{1} E_{2}, \ldots, A E_{1}$ on line $A C$. From the congruence of triangles $\triangle A E_{1} A_{1}$ and $\triangle C E_{2023} C_{1}$ by side and the angles adjacent to it, it follows that $C E_{2023}=A E_{1}$. Therefore, the constructed parallel lines divide $A C$ into 2024 equal parts. Hence, $A E_{1}=\frac{1}{2024} A C$.

Answer. $\frac{1}{2024}$.