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

See Also

2003 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
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All AMC 12 Problems and Solutions

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