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

## Check the Answer

But try the problem first…

Answer: is 241.

AIME I, 1997, Question 11

Plane Trigonometry by Loney

## Try with Hints

First hint

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

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

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

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

Second Hint

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

\(\Rightarrow x=\frac{1}{2^\frac{1}{2}-1}\)

\(\Rightarrow x= 2^\frac{1}{2}+1\)

Final Step

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

## Other useful links

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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