1980 AMC 12 Problem 8

Attempt Problem 8 of the 1980 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 1980 AMC 12 solutions, or check the answer key.

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

How many pairs

(a,b)

of nonzero real numbers satisfy the equation

\frac1a+\frac1b=\frac1{a+b}?

none

1

2

one pair for each

b\ne0

two pairs for each

b\ne0

Clear the denominators, noting that

a,

b,

and

a+b

must be nonzero

Treat the resulting homogeneous quadratic as an equation in the ratio

\frac{a}{b}

Clearing denominators gives

(a+b)^2=ab,

a^2+ab+b^2=0.

Since

b\ne0,

let

t=\frac{a}{b}.

Then

t^2+t+1=0,

whose discriminant is

1-4=-3.

Thus there are no real pairs.

Therefore, the correct answer is A.

Problem 8 in other years