Difference between revisions of "2008 AMC 12B Problems/Problem 19"
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− | ==Problem | + | ==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> | ||
− | ==Solution== | + | ==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: | ||
− | < | + | <cmath>\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*}</cmath> | ||
− | <math> | + | 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> |
− | + | ==Solution 2:== | |
− | <math> | + | <math>f(1)=4+i+\alpha+\gamma</math> |
+ | |||
+ | <math>f(i)=-4-i+\alpha \cdot i +\gamma</math> | ||
+ | |||
+ | Since <math>f(1)</math> and <math>f(i)</math> are both real we get, | ||
+ | <cmath>\alpha+\gamma=-i</cmath> | ||
+ | <cmath>\alpha \cdot i+\gamma=i</cmath> | ||
+ | |||
+ | 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> | ||
+ | |||
+ | 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 19:28, 26 September 2023
Contents
Problem
A function is defined by
for all complex numbers
, where
and
are complex numbers and
. Suppose that
and
are both real. What is the smallest possible value of
?
Solution 1:
We need only concern ourselves with the imaginary portions of and
(both of which must be 0). These are:
Let and
then we know
and
Therefore
which reaches its minimum
when
by the Trivial Inequality. Thus, the answer is
Solution 2:
Since and
are both real we get,
Solving, we get ,
can be anything, to minimize the value we set
, so then the answer is
. Thus, the answer is
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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