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

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==Problem 19==
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==Problem==
 
A function <math>f</math> is defined by <math>f(z) = (4 + i) z^2 + \alpha z + \gamma</math> for all complex numbers <math>z</math>, where <math>\alpha</math> and <math>\gamma</math> are complex numbers and <math>i^2 = - 1</math>. Suppose that <math>f(1)</math> and <math>f(i)</math> are both real. What is the smallest possible value of <math>| \alpha | + |\gamma |</math> ?
 
A function <math>f</math> is defined by <math>f(z) = (4 + i) z^2 + \alpha z + \gamma</math> for all complex numbers <math>z</math>, where <math>\alpha</math> and <math>\gamma</math> are complex numbers and <math>i^2 = - 1</math>. Suppose that <math>f(1)</math> and <math>f(i)</math> are both real. What is the smallest possible value of <math>| \alpha | + |\gamma |</math> ?
  
 
<math>\textbf{(A)} \; 1 \qquad \textbf{(B)} \; \sqrt {2} \qquad \textbf{(C)} \; 2 \qquad \textbf{(D)} \; 2 \sqrt {2} \qquad \textbf{(E)} \; 4 \qquad</math>
 
<math>\textbf{(A)} \; 1 \qquad \textbf{(B)} \; \sqrt {2} \qquad \textbf{(C)} \; 2 \qquad \textbf{(D)} \; 2 \sqrt {2} \qquad \textbf{(E)} \; 4 \qquad</math>
  
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==Solution==
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==Solution 1:==
 
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+\textrm{Im}(\alpha)+\textrm{Im}(\gamma)</math>
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<cmath>\begin{align*}
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\text{Im}(f(1)) & = i+i\text{Im}(\alpha)+i\text{Im}(\gamma) \\
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\text{Im}(f(i)) & = -i+i\text{Re}(\alpha)+i\text{Im}(\gamma)
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\end{align*}</cmath>
  
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<math>2) f(i) = -i+i\textrm{Re}(\alpha)+i\textrm{Im}(\gamma)</math>
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Let <math>p=\text{Im}(\gamma)</math> and <math>q=\text{Re}{(\gamma)},</math> then we know <math>\text{Im}(\alpha)=-p-1</math> and <math>\text{Re}(\alpha)=1-p.</math> Therefore <cmath>|\alpha|+|\gamma|=\sqrt{(1-p)^2+(-1-p)^2}+\sqrt{q^2+p^2}=\sqrt{2p^2+2}+\sqrt{p^2+q^2},</cmath> which reaches its minimum <math>\sqrt 2</math> when <math>p=q=0</math> by the Trivial Inequality. Thus, the answer is <math>\boxed B.</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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==Solution 2:==
  
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<math>| \alpha | + |\gamma | = |1-i| + |0| = \sqrt{2} \Rightarrow \boxed{B}</math>.
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<math>f(1)=4+i+\alpha+\gamma</math>
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<math>f(i)=-4-i+\alpha \cdot i +\gamma</math>
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Since <math>f(1)</math> and <math>f(i)</math> are both real we get,
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<cmath>\alpha+\gamma=-i</cmath>
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<cmath>\alpha \cdot i+\gamma=i</cmath>
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Solving, we get <math>\alpha=1-i</math>, <math>\gamma</math> can be anything, to minimize the value we set <math>\gamma=0</math>, so then the answer is <math>\sqrt{1^2+1^2}=\sqrt{2}</math>. Thus, the answer is <math>\boxed{B}</math>
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By: Quaratinium
  
 
==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}}

Latest revision as of 18:28, 26 September 2023

Problem

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 1:

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

\begin{align*} \text{Im}(f(1)) & = i+i\text{Im}(\alpha)+i\text{Im}(\gamma) \\ \text{Im}(f(i)) & = -i+i\text{Re}(\alpha)+i\text{Im}(\gamma) \end{align*}

Let $p=\text{Im}(\gamma)$ and $q=\text{Re}{(\gamma)},$ then we know $\text{Im}(\alpha)=-p-1$ and $\text{Re}(\alpha)=1-p.$ Therefore \[|\alpha|+|\gamma|=\sqrt{(1-p)^2+(-1-p)^2}+\sqrt{q^2+p^2}=\sqrt{2p^2+2}+\sqrt{p^2+q^2},\] which reaches its minimum $\sqrt 2$ when $p=q=0$ by the Trivial Inequality. Thus, the answer is $\boxed B.$

Solution 2:

$f(1)=4+i+\alpha+\gamma$

$f(i)=-4-i+\alpha \cdot i +\gamma$

Since $f(1)$ and $f(i)$ are both real we get, \[\alpha+\gamma=-i\] \[\alpha \cdot i+\gamma=i\]

Solving, we get $\alpha=1-i$, $\gamma$ can be anything, to minimize the value we set $\gamma=0$, so then the answer is $\sqrt{1^2+1^2}=\sqrt{2}$. Thus, the answer is $\boxed{B}$

By: Quaratinium

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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