Difference between revisions of "2003 AMC 12A Problems/Problem 9"
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== Problem == | == Problem == | ||
| − | A set <math>S</math> of points in the <math>xy</math>-plane is symmetric about the | + | 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> | ||
== Solution == | == Solution == | ||
| − | 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>D</math>. | + | 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>. |
| + | |||
| + | == See Also == | ||
| + | {{AMC12 box|year=2003|ab=A|num-b=8|num-a=10}} | ||
| + | |||
| + | [[Category:Introductory Algebra Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 19:35, 30 December 2014
Problem
A set 
of points in the 
-plane is symmetric about the origin, both coordinate axes, and the line 
. If 
is in , what is the smallest number of points in ?
Solution
If is in , then 
is also, and quickly we see that every point of the form 
or 
must be in . Now note that these 
points satisfy all of the symmetry conditions. Thus the answer is 
.
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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
