2007 AIME I Problems/Problem 3: Difference between revisions
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== Solution == | == Solution == | ||
Squaring, we find that <math> | Squaring, we find that <math>(9 + bi)^2 = 81 + 18bi - b^2</math>. Cubing and ignoring the real parts of the result, we find that <math>(81 + 18bi - b^2)(9 + bi) = \ldots + (9\cdot 18 + 81)bi - b^3i</math>. | ||
Setting these two equal, we get that <math> | Setting these two equal, we get that <math>18bi = 243bi - b^3i</math>, so <math>b(b^2 - 225) = 0</math> and <math>b = -15, 0, 15</math>. Since <math>b > 0</math>, the solution is <math>015</math>. | ||
== See also == | == See also == | ||
{{AIME box|year=2007|n=I|num-b=2|num-a=4}} | {{AIME box|year=2007|n=I|num-b=2|num-a=4}} | ||
[[Category:Intermediate | [[Category:Intermediate Algebra Problems]] | ||
Revision as of 21:21, 30 November 2007
Problem
The complex number 
is equal to 
, where 
is a positive real number and 
. Given that the imaginary parts of 
and 
are the same, what is equal to?
Solution
Squaring, we find that 
. Cubing and ignoring the real parts of the result, we find that 
.
Setting these two equal, we get that 
, so 
and 
. Since 
, the solution is 
.
See also
| 2007 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
