2005 AMC 12B Problem 12

Attempt Problem 12 of the 2005 AMC 12B 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 2005 AMC 12B solutions, or check the answer key.

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12.

The quadratic equation

x^2 + mx + n = 0

has roots that are twice those of

x^2 + px + m = 0,

and none of

m,

n,

and

p

is zero. What is the value of

\dfrac{n}{p}?

1

2

4

8

16

Let

r_1, r_2

be the roots of

x^2 + px + m = 0

; the other equation has roots

2r_1, 2r_2

Write

m, n, p

in terms of

r_1 r_2

and

r_1 + r_2

using Vieta’s formulas

Let

r_1

and

r_2

be the roots of

x^2 + px + m = 0,

m = r_1 r_2

and

p = -(r_1 + r_2).

The roots of

x^2 + mx + n = 0

are

2r_1

and

2r_2,

n = 4r_1 r_2

and

m = -2(r_1 + r_2).

Then

n = 4m

and

m = 2p,

which gives

p = \dfrac{m}{2},

\dfrac{n}{p} = \dfrac{4m}{\tfrac{m}{2}} = 8.

Thus, the correct answer is D.

Problem 12 in other years