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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

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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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

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.