2008 AMC 12B Problems/Problem 19

Revision as of 22:24, 1 March 2008 by Castle (talk | contribs)

Problem 19

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 $| \alpha | + |\gamma |$

$\textbf{(A)} \; 1 \qquad \textbf{(B)} \; \sqrt {2} \qquad \textbf{(C)} \; 2 \qquad \textbf{(D)} \; 2 \sqrt {2} \qquad \textbf{(E)} \; 4 \qquad$

Solution

We need only concern ourselves with the imaginary portions of $f(1)$ and $f(i)$ (both of which must be 0). These are:

$1) f(1) = i+\alpha_{imaginary}+\gamma_{imaginary}$

$2) f(i) = -i+i\alpha_{real}+\gamma_{imaginary}$

Since $\gamma_{imaginary}$ appears in both equations, we let it equal 0 to simplify the equations. This yields two single-variable equations. Equation 1 tells us that the imaginary part of $\alpha$ must be $-i$, and equation two tells us that the real part of $\alpha$ must be $i/i = 1$. Therefore, $\alpha = 1-i$. There are no restrictions on $\gamma_{real}$, so to minimize $\gamma$'s absolute value, we let $\gamma_{real} = 0$.


$| \alpha | + |\gamma | = |1-i| + |0| = \sqrt{2}$, answer choice B.

See Also

2008 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 18
Followed by
Problem 20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions