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April 28, 2020

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.

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


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.

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