2008 AMC 12B Problem 19

Question

A function

f

is defined by

f(z) = (4 + i)z^2 + \alpha z + \gamma

for all complex numbers

z,

where

\alpha

and

\gamma

are complex numbers and

i^2 = -1.

Suppose that

f(1)

and

f(i)

are both real. What is the smallest possible value of

|\alpha| + |\gamma|?

1

\sqrt{2}

2

2\sqrt{2}

4

Write

\alpha = a + bi

and

\gamma = c + di;

requiring

f(1)

and

f(i)

to be real gives two linear conditions

They force

a = 1 - d

and

b = -1 - d,

so

|\alpha| + |\gamma|

= \sqrt{2 + 2d^2} + \sqrt{c^2 + d^2};

minimize over

c, d

Let

\alpha = a + bi

and

\gamma = c + di.

Then

f(1) = (4 + a + c)

+ (1 + b + d)i

and

f(i) = (-4 - b + c)

+ (-1 + a + d)i.

Both being real forces

1 + b + d = 0

and

-1 + a + d = 0,

i.e.

a = 1 - d

and

b = -1 - d.

Hence

\begin{aligned} &|\alpha| + |\gamma| \\ &= \sqrt{(1 - d)^2 + (1 + d)^2} \\ &\quad {}+ \sqrt{c^2 + d^2} \\ &= \sqrt{2 + 2d^2} \\ &\quad {}+ \sqrt{c^2 + d^2}, \end{aligned}

which is smallest when

c = d = 0,

giving

\sqrt{2}.

Thus, the correct answer is B.

Problem 19 in other years