1959 AMC 12 Problem 44

Attempt Problem 44 of the 1959 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 1959 AMC 12 solutions, or check the answer key.

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44.

The roots of

x^2+bx+c=0

are both real and greater than

1.

Let

s=b+c+1.

Then

s:

may be less than zero

may be equal to zero

must be greater than zero

must be less than zero

must be between

-1

and

1

Answer: C

Concepts:Vieta’s Formulas inequality quadratic

Difficulty rating: 1360

Small hint:

Call the two roots

u

and

v

and express

b,c

using Vieta’s formulas

Big hint:

Factor

1-(u+v)+uv

Solution:

If the roots are

u,v>1,

then

b=-(u+v)

and

c=uv.

Therefore

\begin{aligned} s&=1-u-v+uv\\ &=(u-1)(v-1)>0. \end{aligned}

Thus

s

must be greater than zero, and the correct answer is C.

Problem 44 in other years