Question

The lines $L_1, L_2, \ldots, L_{20}$ are distinct. For $n = 1, 2, 3, \ldots, 10$, all the lines $L_{2n-1}$ are parallel to each other, and all the lines $L_{2n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set $\\{L_1, L_2, \ldots, L_{20}\\}$ is equal to:

Answer 1

Given:
- Lines $L_{2n-1}$ ($n = 1, 2, \dots, 10$) are parallel to each other.
- Lines $L_{2n}$ ($n = 1, 2, \dots, 10$) pass through a common point $P$.

Step 1: Points of Intersection between $L_{2n-1}$ and $L_{2m}$
Since all $L_{2n-1}$ lines are parallel, they do not intersect among themselves. Similarly, all $L_{2n}$ lines pass through the point $P$, so they intersect at $P$ and do not form additional intersection points among themselves.

However, each line $L_{2n-1}$ intersects each line $L_{2m}$ exactly once (since they are not parallel), leading to:
$10 \times 10 = 100 \text{ intersection points}$

Step 2: Points of Intersection among $L_{2n}$ Lines
All $L_{2n}$ lines pass through the common point $P$. Therefore, there is exactly one intersection point among these lines at $P$.

Step 3: Total Number of Points of Intersection
The total number of points of intersection is given by:
$100 + 1 = 101$

Conclusion: The maximum number of points of intersection of pairs of lines from the set $\\{L_1, L_2, \dots, L_{20}\\}$ is 101.