Difference between revisions of "2003 AMC 12A Problems/Problem 9"
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== Solution == | == Solution == | ||
| − | If <math>(2,3)</math> is in <math>S</math>, then <math>(3,2)</math> is also, | + | If <math>(2,3)</math> is in <math>S</math>, then its reflection in the line <math>y = x</math>, i.e. <math>(3,2)</math>, is also in <math>S</math>. Now by reflecting these points in both coordinate axes, we quickly deduce 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>. Moreover, by drawing out this set of <math>8</math> points, we observe that it satisfies all of the required symmetry conditions, so no more points need to be added to <math>S</math>. Accordingly, the smallest possible number of points in <math>S</math> is precisely <math>\boxed{\mathrm{(D)}\ 8}</math>. |
== See Also == | == See Also == | ||
Latest revision as of 15:17, 20 June 2025
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 its reflection in the line 
, i.e. 
, is also in . Now by reflecting these points in both coordinate axes, we quickly deduce that every point of the form 
or 
must be in . Moreover, by drawing out this set of 
points, we observe that it satisfies all of the required symmetry conditions, so no more points need to be added to . Accordingly, the smallest possible number of points in is precisely 
.
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 | |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
