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Difference between revisions of "2003 AMC 12A Problems/Problem 9"

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== Problem ==
 
== Problem ==
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A set <math>S</math> of points in the <math>xy</math>-plane is symmetric about the orgin, both coordinate axes, and the line <math>y=x</math>. If <math>(2,3)</math> is in <math>S</math>, what is the smallest number of points in <math>S</math>?
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A set <math>S</math> of points in the <math>xy</math>-plane is symmetric about the origin, both coordinate axes, and the line <math>y=x</math>. If <math>(2,3)</math> is in <math>S</math>, what is the smallest number of points in <math>S</math>?
  
 
<math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 16 </math>
 
<math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 16 </math>
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== Solution ==
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If <math>(2,3)</math> is in <math>S</math>, then <math>(3,2)</math> is also, and quickly we see that every point of the form <math>(\pm 2, \pm 3)</math> or <math>(\pm 3, \pm 2)</math> must be in <math>S</math>. Now note that these <math>8</math> points satisfy all of the symmetry conditions. Thus the answer is <math>\boxed{\mathrm{(D)}\ 8}</math>.
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== See Also ==
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{{AMC12 box|year=2003|ab=A|num-b=8|num-a=10}}
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[[Category:Introductory Algebra Problems]]
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{{MAA Notice}}

Latest revision as of 19:35, 30 December 2014

Problem

A set $S$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y=x$. If $(2,3)$ is in $S$, what is the smallest number of points in $S$?

$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 16$

Solution

If $(2,3)$ is in $S$, then $(3,2)$ is also, and quickly we see that every point of the form $(\pm 2, \pm 3)$ or $(\pm 3, \pm 2)$ must be in $S$. Now note that these $8$ points satisfy all of the symmetry conditions. Thus the answer is $\boxed{\mathrm{(D)}\ 8}$.

See Also

2003 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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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