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?

2003 AMC 12A Problem 9

Attempt Problem 9 of the 2003 AMC 12A below, then check your answer against the professionally curated solution 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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

A set

S

of points in the

xy

-plane is symmetric about the origin, both coordinate axes, and the line

y = x.

(2, 3)

is in

S,

what is the smallest number of points in

S?

1

2

4

8

16

Reflecting

(2,3)

over

y=x

gives

(3,2)

The symmetries produce every sign choice and coordinate swap

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.