2016 AMC 12A Problem 24

Question

There is a smallest positive real number

a

such that there exists a positive real number

b

such that all the roots of the polynomial

x^3-ax^2+bx-a

are real. In fact, for this value of

a

the value of

b

is unique. What is this value of

b?

8

9

10

11

12

If the roots are

r,s,t,

then

r+s+t=a

and

rst=a

Apply AM-GM to

r,s,t;

the minimum

a

forces

r=s=t

By Vieta’s formulas, the real roots

r,s,t

satisfy

r+s+t=a,

rs+st+tr=b,

and

rst=a,

so

r+s+t=rst.

The positive product means either all three roots are positive or two are negative. In the latter case write the roots as

-u,-v,w

with

u,v,w\gt0.

The positive sum forces

w\gt u+v,

and then

b=uv-w(u+v)\lt0,

a contradiction. Hence all three roots are positive.

By the AM-GM inequality,

27rst\le(r+s+t)^3=(rst)^3,

so

a=rst\ge 3\sqrt3,

with equality if and only if

r=s=t=\sqrt3.

At this smallest

a,

\begin{gathered} b=rs+st+tr\\ =3r^2=3\cdot 3=9. \end{gathered}

Thus, the correct answer is B.

Problem 24 in other years