Difference between revisions of "2014 AIME I Problems/Problem 7"
(→Solution) |
(→Solution) |
||
| Line 7: | Line 7: | ||
Multiplying both the numerator and denominator of this fraction by <math>\mathrm{cis}{(-\beta)}</math> gives us: | Multiplying both the numerator and denominator of this fraction by <math>\mathrm{cis}{(-\beta)}</math> gives us: | ||
| − | <math>\dfrac{w - z}{z} = \dfrac{1}{10}\mathrm{cis}{\alpha - \beta} - 1</math>. | + | <math>\dfrac{w - z}{z} = \dfrac{1}{10}\mathrm{cis}{(\alpha - \beta)} - 1</math>. |
== See also == | == See also == | ||
{{AIME box|year=2014|n=I|num-b=6|num-a=8}} | {{AIME box|year=2014|n=I|num-b=6|num-a=8}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 11:08, 15 March 2014
Problem 7
Let 
and 
be complex numbers such that 
and 
. Let 
. The maximum possible value of 
can be written as 
, where 
and 
are relatively prime positive integers. Find 
. (Note that 
, for 
, denotes the measure of the angle that the ray from 
to makes with the positive real axis in the complex plane.
Solution
Let 
and 
. Then, 
.
Multiplying both the numerator and denominator of this fraction by 
gives us:
.
See also
| 2014 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
