2019 AMC 12A Problem 8

Below is the professionally curated solution for Problem 8 of the 2019 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2019 AMC 12A solutions, or check the answer key.

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Concepts:counting intersections casework

Difficulty rating: 1380

For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N?$

$14$

$16$

$18$

$19$

$21$

Solution:

With four lines, the number of intersection points ranges over the achievable configurations. All parallel gives $0;$ all concurrent gives $1.$

Working through the cases (parallel classes and points of concurrency), the achievable values are $0, 1, 3, 4, 5, 6;$ the value $2$ is impossible.

The sum is $0 + 1 + 3 + 4 + 5 + 6 = 19.$

Thus, the correct answer is D.

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