Difference between revisions of "2008 AMC 12B Problems/Problem 19"
(New page: ==Problem 19== 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>\gamm...) |
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==Solution== | ==Solution== | ||
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+\alpha</math> | <math>1) f(1) = i+\alpha</math> | ||
Revision as of 22:19, 1 March 2008
Problem 19
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
We need only concern ourselves with the imaginary portions of and (both of which must be 0). These are:
Equation 1 tells us that the imaginary part of must be 
, and equation two tells us that the real part of must be 
. Therefore, 
. There are no restrictions on gamma, so to minimize it's absolute value, we let 
.
, answer choice B.
