2008 AMC 12B Problems/Problem 19
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:
\begin{align*} \textrm{Im}(f(1)) & = i+i\textrm{Im}(\alpha)+i\textrm{Im}(\gamma) \\ \textrm{Im}(f(i)) & = -i+i\textrm{Re}(\alpha)+i\textrm{Im}(\gamma) \end{align*}
Let and
then we know
and
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 when
So the answer is
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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