2003 AMC 12A Problems/Problem 9

Problem

A set $S$ of points in the $xy$ -plane is symmetric about the orgin, 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 $(3,2)$ is also, and quickly we see that every point of the form $(\pm 2, \pm 3)$ or $(\pm 3, \pm 2)$ must be in $S$ . Now note that these $8$ points satisfy all of the symmetry conditions. Thus the answer is $D$ .

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2003 AMC 12A Problems/Problem 9

Problem

Solution