2008 AMC 12B Problems/Problem 19

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.

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2008 AMC 12B Problems/Problem 19

Problem 19

Solution