Question

Lines $L_1,$ $L_2,$ $\ldots,$ $L_{100}$ are distinct. All lines $L_{4n},$ $n$ a positive integer, are parallel to each other. All lines $L_{4n-3},$ $n$ a positive integer, pass through a given point $A.$ The maximum number of points of intersection of pairs of lines from the complete set $\{L_1,L_2,\ldots,L_{100}\}$ is

$4350$

$4351$

$4900$

$4901$

$9851$

Answer: B

Concepts:counting intersections combinations double counting

Difficulty rating: 2320

Small Hint:

Start with $\binom{100}{2}$ intersections in general position

Big Hint:

Remove the pairs among the $25$ parallel lines and collapse the pairs among the $25$ concurrent lines to one point

Solution:

There are $25$ indices divisible by $4$ and $25$ congruent to $1\pmod4.$ Starting from $\binom{100}{2}=4950,$ the parallel group contributes no intersections, removing $\binom{25}{2}=300.$ The concurrent group’s $300$ pairs all give one point rather than $300$ distinct points, removing another $299.$ All remaining intersections can be chosen distinct, so the maximum is $4950-300-299=4351.$

Therefore, the correct answer is B.

1976 AMC 12 Problem 28

Answer: B

Difficulty rating: 2320

Small Hint:

Big Hint:

Solution:

Problem 28 in Other Years