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Difference between revisions of "2008 AMC 12B Problems/Problem 19"

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We need only concern ourselves with the imaginary portions of <math>f(1)</math> and <math>f(i)</math> (both of which must be 0). These are:
 
We need only concern ourselves with the imaginary portions of <math>f(1)</math> and <math>f(i)</math> (both of which must be 0). These are:
  
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<math>1) f(1) = i+i\textrm{Im}(\alpha)+i\textrm{Im}(\gamma)</math>
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Im (f(1)) = i+i\textrm{Im}(\alpha)+i\textrm{Im}(\gamma)<math>
  
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<math>2) f(i) = -i+i\textrm{Re}(\alpha)+i\textrm{Im}(\gamma)</math>
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Im (f(i)) = -i+i\textrm{Re}(\alpha)+i\textrm{Im}(\gamma)</math>
  
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Since <math>\textrm{Im}(\gamma)</math> 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 <math>\alpha</math> must be <math>-1</math>, and equation 2 tells us that the real part of <math>\alpha</math> must be <math>i/i = 1</math>. Therefore, <math>\alpha = 1-i</math>. There are no restrictions on <math>\textrm{Re}(\gamma)</math>, so to minimize <math>\gamma</math>'s absolute value, we let <math>\textrm{Re}(\gamma) = 0</math>.
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Let <math>p=\textrm{Im}(\gamma)</math> and <math>q=\textrm{Re}{\gamma},</math> then we know <math>\textrm{Im}(\alpha)=-p-1</math> and <math>\textrm{Re}(\alpha)=1-p.</math> Therefore <cmath>|\alpha|+|\gamma|=\sqrt{(1-p)^2+(-1-p)^2}+\sqrt{q^2+p^}=\sqrt{2p^2+2}+\sqrt{p^2+q^2},</cmath> which reaches its minimum <math>\sqrt 2</math> when <math>p=q=0.</math> So the answer is <math>\boxed B.</math>
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<math>| \alpha | + |\gamma | = |1-i| + |0| = \sqrt{2} \Rightarrow \boxed{B}</math>.
 
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2008|ab=B|num-b=18|num-a=20}}
 
{{AMC12 box|year=2008|ab=B|num-b=18|num-a=20}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 12:40, 9 September 2016

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:

Im (f(1)) = i+i\textrm{Im}(\alpha)+i\textrm{Im}(\gamma)$Im (f(i)) = -i+i\textrm{Re}(\alpha)+i\textrm{Im}(\gamma)$

Let $p=\textrm{Im}(\gamma)$ and $q=\textrm{Re}{\gamma},$ then we know $\textrm{Im}(\alpha)=-p-1$ and $\textrm{Re}(\alpha)=1-p.$ Therefore 

\[|\alpha|+|\gamma|=\sqrt{(1-p)^2+(-1-p)^2}+\sqrt{q^2+p^}=\sqrt{2p^2+2}+\sqrt{p^2+q^2},\] (Error compiling LaTeX. Unknown error_msg)

which reaches its minimum $\sqrt 2$ when $p=q=0.$ So the answer is $\boxed B.$

See Also

2008 AMC 12B (Problems • Answer Key • Resources)
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

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