Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

m
(Improved explanation)
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
−
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>?
+
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>\boxed{\mathrm{(D)}\ 8}</math>.
+
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 ==
Line 11: Line 11:
  
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]
 +
{{MAA Notice}}

Latest revision as of 15:17, 20 June 2025

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 its reflection in the line $y = x$, i.e. $(3,2)$, is also in $S$. Now by reflecting these points in both coordinate axes, we quickly deduce that every point of the form $(\pm 2, \pm 3)$ or $(\pm 3, \pm 2)$ must be in $S$. Moreover, by drawing out this set of $8$ points, we observe that it satisfies all of the required symmetry conditions, so no more points need to be added to $S$. Accordingly, the smallest possible number of points in $S$ is precisely $\boxed{\mathrm{(D)}\ 8}$.

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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_12A_Problems/Problem_9&oldid=251694"