2003 AMC 12A Problem 9

Below is the professionally curated solution for Problem 9 of the 2003 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 12A solutions, or check the answer key.

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Concepts:transformation symmetry

Difficulty rating: 1350

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 $S?$

$1$

$2$

$4$

$8$

$16$

Solution:

Reflecting across $y=x$ gives $(3,2),$ and reflecting across the axes gives all points $(\pm2,\pm3)$ and $(\pm3,\pm2).$

There are $8$ such points, and this set is already symmetric about the origin, both axes, and $y=x.$

Thus, the correct answer is D.

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