Difference between revisions of "2002 AMC 12B Problems/Problem 14"

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==Solution 2==
 
==Solution 2==
Because a pair or circles can intersect at most <math>2</math> times, the first circle can intersect the second at <math>2</math> points, the third can intersect the first two at <math>4</math> points, and the fourth can intersect the first three at <math>6</math> points. This means that our answer is <math>2+4+6=\boxed{\mathrm{(D)}\ 12}</math>
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Because a pair or circles can intersect at most <math>2</math> times, the first circle can intersect the second at <math>2</math> points, the third can intersect the first two at <math>4</math> points, and the fourth can intersect the first three at <math>6</math> points. This means that our answer is <math>2+4+6=\boxed{\mathrm{(D)}\ 12}.</math>
  
 
== See also ==
 
== See also ==

Revision as of 23:26, 1 January 2016

The following problem is from both the 2002 AMC 12B #14 and 2002 AMC 10B #18, so both problems redirect to this page.

Problem

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

$\mathrm{(A)}\ 8 \qquad\mathrm{(B)}\ 9 \qquad\mathrm{(C)}\ 10 \qquad\mathrm{(D)}\ 12 \qquad\mathrm{(E)}\ 16$

Solution 1

For any given pair of circles, they can intersect at most $2$ times. Since there are ${4\choose 2} = 6$ pairs of circles, the maximum number of possible intersections is $6 \cdot 2 = 12$. We can construct such a situation as below, so the answer is $\boxed{\mathrm{(D)}\ 12}$.

2002 12B AMC-14.png

Solution 2

Because a pair or circles can intersect at most $2$ times, the first circle can intersect the second at $2$ points, the third can intersect the first two at $4$ points, and the fourth can intersect the first three at $6$ points. This means that our answer is $2+4+6=\boxed{\mathrm{(D)}\ 12}.$

See also

2002 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
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
2002 AMC 12B (ProblemsAnswer KeyResources)
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 12 Problems and Solutions

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