2019 AMC 10A Problems/Problem 14
- The following problem is from both the 2019 AMC 10A #14 and 2019 AMC 12A #8, so both problems redirect to this page.
Contents
Problem
For a set of four distinct lines in a plane, there are exactly distinct points that lie on two or more of the lines. What is the sum of all possible values of
?
Solution
It is possible to obtain ,
,
,
,
, and
points of intersection, as demonstrated in the following figures:
It is clear that the maximum number of possible intersections is , since each pair of lines can intersect at most once. We now prove that it is impossible to obtain two intersections.
We proceed by contradiction. Assume a configuration of four lines exists such that there exist only two intersection points. Let these intersection points be and
. Consider two cases:
Case 1: No line passes through both and
Then, since an intersection is obtained by an intersection between at least two lines, two lines pass through each of and
. Then, since there can be no additional intersections, no line that passes through
can intersect a line that passes through
, and so each line that passes through
must be parallel to every line that passes through
. Then the two lines passing through
are parallel to each other by transitivity of parallelism, so they coincide, contradiction.
Case 2: There is a line passing through and
Then there must be a line passing through
, and a line
passing through
. These lines must be parallel. The fourth line
must pass through either
or
. Without loss of generality, suppose
passes through
. Then since
and
cannot coincide, they cannot be parallel. Then
and
cannot be parallel either, so they intersect, contradiction.
All possibilities have been exhausted, and thus we can conclude that two intersections is impossible. Our answer is given by the sum .
Solution 2(cheap)
We know that the minimum is 0 but we do not know the maximum, so we check the answer choices for something 0+1+2+...+k. Only 19 satisfies this. ~mathboy282
See Also
2019 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2019 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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