Difference between revisions of "2013 AMC 12A Problems/Problem 25"
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+ | == Problem== | ||
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+ | Let <math>f : \mathbb{C} \to \mathbb{C} </math> be defined by <math> f(z) = z^2 + iz + 1 </math>. How many complex numbers <math>z </math> are there such that <math> \text{Im}(z) > 0 </math> and both the real and the imaginary parts of <math>f(z)</math> are integers with absolute value at most <math> 10 </math>? | ||
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+ | <math> \textbf{(A)} \ 399 \qquad \textbf{(B)} \ 401 \qquad \textbf{(C)} \ 413 \qquad \textbf{(D}} \ 431 \qquad \textbf{(E)} \ 441 </math> | ||
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+ | ==Solution== | ||
+ | |||
Suppose <math>f(z)=z^2+iz+1=c=a+bi</math>. We look for <math>z</math> with <math>\text{Im}(z)>0</math> such that <math>a,b</math> are integers where <math>|a|, |b|\leq 10</math>. | Suppose <math>f(z)=z^2+iz+1=c=a+bi</math>. We look for <math>z</math> with <math>\text{Im}(z)>0</math> such that <math>a,b</math> are integers where <math>|a|, |b|\leq 10</math>. | ||
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So there are <math>231+168=399</math> in total. | So there are <math>231+168=399</math> in total. | ||
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+ | == See also == | ||
+ | {{AMC12 box|year=2013|ab=A|num-b=24|after=Last Question}} |
Revision as of 18:56, 22 February 2013
Problem
Let be defined by . How many complex numbers are there such that and both the real and the imaginary parts of are integers with absolute value at most ?
$\textbf{(A)} \ 399 \qquad \textbf{(B)} \ 401 \qquad \textbf{(C)} \ 413 \qquad \textbf{(D}} \ 431 \qquad \textbf{(E)} \ 441$ (Error compiling LaTeX. Unknown error_msg)
Solution
Suppose . We look for with such that are integers where .
First, use the quadratic formula:
Generally, consider the imaginary part of a radical of a complex number: , where .
.
Now let , then , , .
Note that if and only if . The latter is true only when we take the positive sign, and that ,
or , , or .
In other words, for all , satisfies , and there is one and only one that makes it true. Therefore we are just going to count the number of ordered pairs such that , are integers of magnitude no greater than , and that .
When , there is no restriction on so there are pairs;
when , there are pairs.
So there are in total.
See also
2013 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last Question |
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 |