1969 AMC 12 Problem 5

Attempt Problem 5 of the 1969 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 1969 AMC 12 solutions, or check the answer key.

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

If a number

N,

N\ne0,

diminished by four times its reciprocal, equals a given real constant

R,

then, for this given

R,

the sum of all such possible values of

N

is:

\dfrac1R

R

4

\dfrac14

-R

Rewrite

N-\frac{4}{N}=R

as a quadratic in

N

Use the sum of the roots of that quadratic

Multiplying

N-\frac{4}{N}=R

N

gives

N^2-RN-4=0.

The sum of its two roots is

R

by Vieta’s formulas. Both roots are nonzero because their product is

-4.

Therefore, the correct answer is B.

Problem 5 in other years