Difference between revisions of "2002 AMC 12B Problems/Problem 14"
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| + | {{duplicate|[[2002 AMC 12B Problems|2002 AMC 12B #14]] and [[2002 AMC 10B Problems|2002 AMC 10B #18]]}} | ||
== Problem == | == Problem == | ||
Four distinct [[circle]]s are drawn in a [[plane]]. What is the maximum number of points where at least two of the circles intersect? | Four distinct [[circle]]s 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 == | == Solution == | ||
| − | For any given pair of circles, they can intersect at most <math>2</math> times. Since there are <math>{4\choose 2} = 6</math> pairs of circles, the maximum number of possible intersections is <math>6 \cdot 2 = 12</math>. We can construct such a situation as below, so the answer is <math>\mathrm{(D)}</math>. | + | For any given pair of circles, they can intersect at most <math>2</math> times. Since there are <math>{4\choose 2} = 6</math> pairs of circles, the maximum number of possible intersections is <math>6 \cdot 2 = 12</math>. We can construct such a situation as below, so the answer is <math>\boxed{\mathrm{(D)}\ 12}</math>. |
[[Image:2002_12B_AMC-14.png]] | [[Image:2002_12B_AMC-14.png]] | ||
== See also == | == See also == | ||
| + | {{AMC10 box|year=2002|ab=B|num-b=17|num-a=19}} | ||
{{AMC12 box|year=2002|ab=B|num-b=13|num-a=15}} | {{AMC12 box|year=2002|ab=B|num-b=13|num-a=15}} | ||
[[Category:Introductory Combinatorics Problems]] | [[Category:Introductory Combinatorics Problems]] | ||
Revision as of 15:15, 29 July 2011
- 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?
Solution
For any given pair of circles, they can intersect at most 
times. Since there are 
pairs of circles, the maximum number of possible intersections is 
. We can construct such a situation as below, so the answer is 
.
See also
| 2002 AMC 10B (Problems • Answer Key • Resources) | ||
| 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 (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 12 Problems and Solutions | |
