Difference between revisions of "1997 AIME Problems/Problem 11"
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== Problem == | == Problem == | ||
− | Let <math>x=\frac{\sum_{n=1}^{44} \cos n^\circ}{\sum_{n=1}^{44} \sin n^\circ}</math>. What is the greatest integer that does not exceed <math>100x</math>? | + | Let <math>x=\frac{\sum_{n=1}^{44} \cos n^\circ}{\sum_{n=1}^{44} \sin n^\circ}</math>. What is the [[greatest integer function|greatest integer]] that does not exceed <math>100x</math>? |
== Solution == | == Solution == | ||
− | {{ | + | Manipulating the [[numerator]], |
+ | |||
+ | <cmath>\begin{eqnarray*} | ||
+ | \sum_{n=1}^{44} \cos n &=& \sum_{n=1}^{44} \cos n + \sum_{n=1}^{44} \sin n - \sum_{n=1}^{44} \sin n\\ | ||
+ | &=& \sum_{n=1}^{44} \sin n + \sin(90-n) - \sum_{n=1}^{44} \sin n\\ | ||
+ | \end{eqnarray*}</cmath> | ||
+ | |||
+ | Using the identity <math>\sin a + \sin b = 2\sin \frac{a+b}2 \cos \frac{a-b}{2}</math> <math>\Longrightarrow \sin x + \sin (90-x) = 2 \sin 45 \cos (45-x) = \sqrt{2} \cos (45-x)</math>, that first [[summation]] reduces to | ||
+ | |||
+ | <cmath>\begin{eqnarray*} | ||
+ | \sum_{n=1}^{44} \cos n &=& \sqrt{2}\sum_{n=1}^{44} \cos(45-n) - \sum_{n=1}^{44} \sin n\\ | ||
+ | &=& \sqrt{2}\sum_{n=1}^{44} \cos n - \sum_{n=1}^{44} \sin n\\ | ||
+ | (\sqrt{2} - 1)\sum_{n=1}^{44} \cos n &=& \sum_{n=1}^{44} \sin n\\ | ||
+ | \end{eqnarray*}</cmath> | ||
+ | |||
+ | This is the [[ratio]] we are looking for! This reduces is <math>\frac{1}{\sqrt{2} - 1} = \sqrt{2} + 1</math>, and <math>\lfloor 100(\sqrt{2} + 1)\rfloor = \boxed{241}</math>. | ||
== See also == | == See also == | ||
{{AIME box|year=1997|num-b=10|num-a=12}} | {{AIME box|year=1997|num-b=10|num-a=12}} | ||
+ | |||
+ | [[Category:Intermediate Trigonometry Problems]] |
Revision as of 19:16, 23 November 2007
Problem
Let . What is the greatest integer that does not exceed
?
Solution
Manipulating the numerator,
Using the identity
, that first summation reduces to
This is the ratio we are looking for! This reduces is , and
.
See also
1997 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |