Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 1996 based on Greatest Positive Integer.
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.
Inequality
Greatest integer
Integers
Answer: is 340.
AIME I, 1996, Question 2
Elementary Number Theory by Sierpinsky
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.
