2003 AMC 12A Problems/Problem 9

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 $\boxed{\mathrm{(D)}\ 8}$ .

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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2003 AMC 12A Problems/Problem 9

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