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2002 AMC 10B Problem 18

Problem 18 of 25IntermediateCombinatorics

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

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Solution

Any two distinct circles intersect in at most 22 points. There are (42)=6\binom{4}{2} = 6 pairs of circles, giving at most 6⋅2=126\cdot 2 = 12 intersection points. This maximum is achievable by a configuration where every pair of circles crosses twice, so the answer is 12.12. Thus, the correct answer is D.
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Tagged: counting intersections · counting pairs

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