2002 AMC 12B Problem 14

Attempt Problem 14 of the 2002 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 12B solutions, or check the answer key.

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14.

Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?

8

9

10

12

16

Two distinct circles meet in at most

2

points

There are

\binom{4}{2}

pairs of circles; multiply by

2

Each pair of circles meets in at most

2

points, and there are

\binom{4}{2}=6

pairs, giving at most

6\cdot2=12

intersection points.

The bound is attainable by taking four circles in general position so that every pair crosses twice and no three pass through the same point.

Thus, the correct answer is D.

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