AMC12 2019 A

AMC12 2019 A · Q8

AMC12 2019 A · Q8. It mainly tests Casework, Geometry misc.

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 $N$?

对于平面内四条不同的直线集合，有恰好 $N$ 个不同的点位于两条或更多条直线上。所有可能的 $N$ 值之和是多少？

(A) 14 14

(B) 16 16

(D) 19 19

(E) 21 21

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): There are several cases to consider. If all four lines are concurrent, then there is 1 intersection point. If three of the lines are concurrent and the fourth line is parallel to one of those three, then there are 3 intersection points. If three of the lines are concurrent and the fourth line is parallel to none of those three, then there are 4 intersection points. In the remaining cases no three lines are concurrent. If they are all parallel, then there are 0 intersection points. If only three of them are parallel, then there are again 3 intersection points. If two of them are parallel but no three are mutually parallel, then there are either again 4 intersection points, if the other two lines are parallel to each other; or 5 intersection points, if the other two lines intersect. In the final case, every line intersects every other line, giving 6 points of intersection. These are all the cases, so the requested sum is $1+3+4+0+5+6=19$.

答案（D）：有几种情况需要考虑。如果四条直线都共点，则有 1 个交点。如果其中三条直线共点，且第四条直线与这三条中的某一条平行，则有 3 个交点。如果其中三条直线共点，而第四条直线与这三条都不平行，则有 4 个交点。在其余情况下，不存在三条直线共点。若四条直线都互相平行，则有 0 个交点。如果只有三条直线互相平行，则同样有 3 个交点。如果有两条直线平行，但不存在三条直线两两平行，则：若另外两条直线也互相平行，则仍有 4 个交点；若另外两条直线相交，则有 5 个交点。最后一种情况是，每条直线都与其他每条直线相交，从而得到 6 个交点。这些就是全部情况，因此所求的和为 $1+3+4+0+5+6=19$。

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