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AIME I Algebra Arithmetic Math Olympiad USA Math Olympiad

Greatest Positive Integer | AIME I, 1996 | Question 2

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 1996 based on Greatest Positive Integer.

Positive Integer – AIME I, 1996


For each real number x, Let [x] denote the greatest integer that does not exceed x,find number of positive integers n is it true that \(n \lt 1000\) and that \([log_{2}n]\) is a positive even integer.

  • is 107
  • is 340
  • is 840
  • cannot be determined from the given information

Key Concepts


Inequality

Greatest integer

Integers

Check the Answer


Answer: is 340.

AIME I, 1996, Question 2

Elementary Number Theory by Sierpinsky

Try with Hints


here Let \([log_{2}n]\)=2k for k is an integer

\(\Rightarrow 2k \leq log_{2}n \lt 2k+1\)

\(\Rightarrow 2^{2k} \leq n \lt 2^{2k+1}\) and \(n \lt 1000\)

\(\Rightarrow 4 \leq n \lt 8\)

\(16 \leq n \lt 32\)

\(64 \leq n \lt 128\)

\(256 \leq n \lt 512\)

\(\Rightarrow 4+16+64+256\)=340.

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Categories
AIME I Algebra Arithmetic Math Olympiad USA Math Olympiad

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


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.

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