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

Correct Answer: 101

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.