1987 AMC 12 Problem 10

Attempt Problem 10 of the 1987 AMC 12 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 1987 AMC 12 solutions, or check the answer key.

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

10.

How many ordered triples

(a,b,c)

of nonzero real numbers have the property that each number is the product of the other two?

1

2

3

4

5

Translate the condition into

a=bc,\ b=ca,\ c=ab

Multiply the equations and use the nonzero condition to determine the possible magnitudes

Multiplying

a=bc,\ b=ca,\ c=ab

gives

abc=(abc)^2.

Since none of the variables is zero,

abc=1.

Also

abc=a^2=b^2=c^2,

so each variable is

1

-1.

Product

1

allows either no negative signs or exactly two, giving

1+3=4

ordered triples.

Thus the correct answer is D.

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