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# Trigonometry and greatest integer | AIME I, 1997 | Question 11

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1997 based on Trigonometry and greatest integer.

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1997 based on Trigonometry and greatest integer.

## Trigonometry and greatest integer – AIME I, 1997

Let x=$\frac{\displaystyle\sum_{n=1}^{44}cos n}{\displaystyle\sum_{n=1}^{44}sin n}$, find greatest integer that does not exceed 100x.

• is 107
• is 241
• is 840
• cannot be determined from the given information

### Key Concepts

Trigonometry

Greatest Integer

Algebra

AIME I, 1997, Question 11

Plane Trigonometry by Loney

## Try with Hints

here $\displaystyle\sum_{n=1}^{44}cosn+\displaystyle\sum_{n=1}^{44}sin n$

=$\displaystyle\sum_{n=1}^{44}sinn+\displaystyle\sum_{n=1}^{44}sin(90-n)$

=$2^\frac{1}{2}\displaystyle\sum_{n=1}^{44}cos(45-n)$

=$2^\frac{1}{2}\displaystyle\sum_{n=1}^{44}cosn$

$\displaystyle\sum_{n=1}^{44}sin n=(2^\frac{1}{2}-1)\displaystyle\sum_{n=1}^{44}cosn$

$\Rightarrow x=\frac{1}{2^\frac{1}{2}-1}$

$\Rightarrow x= 2^\frac{1}{2}+1$

$\Rightarrow 100x=(100)(2^\frac{1}{2}+1)$=241.