Try this beautiful problem from the Pre-RMO, 2019 based on Greatest Integer.
Find the greatest integer not exceeding the sum \(\sum_{n=1}^{1599}\frac{1}{\sqrt{n}}\)
Largest integer
Divisibility
Integer
But try the problem first...
Answer: is 78.
PRMO, 2019, Question 22
Elementary Number Theory by David Burton
First hint
\(\int\limits_1^{1600}\frac{1}{x}{d}x < \sum_{x=1}^{1599}\frac{1}{\sqrt{n}}\)
\(< 1+\sum_{n=1}^{1599}\frac{1}{\sqrt{x}}{d}x\)
Second Hint
or, \([2\sqrt{x}]_{1}^{1600}< \sum_{n=1}^{1599}\frac{1}{\sqrt{n}}\)
\(< 1+|2{\sqrt{x}}|_1^{1599}\)
Final Step
or, 78<\(\sum_{n=1}^{1599}\frac{1}{\sqrt{n}} <2\sqrt{1599}-1\)
or, 78 < \(\sum_{n=1}^{1599}\frac{1}{\sqrt{n}}\)<78.97
or,\([\sum_{n=1}^{1599}\frac{1}{\sqrt{n}}]\)=78.
Math Olympiad Program... taught by Olympians and researchers
From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:
