2008 AMC 12B Problem 19

Below is the professionally curated solution for Problem 19 of the 2008 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 12B solutions, or check the answer key.

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Concepts:complex number optimization

Difficulty rating: 1990

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 $|\alpha| + |\gamma|?$

$1$

$\sqrt{2}$

$2$

$2\sqrt{2}$

$4$

Solution:

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 $|\alpha| + |\gamma| = \sqrt{(1 - d)^2 + (1 + d)^2} + \sqrt{c^2 + d^2} = \sqrt{2 + 2d^2} + \sqrt{c^2 + d^2},$ which is smallest when $c = d = 0,$ giving $\sqrt{2}.$

Thus, the correct answer is B.

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