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Greatest Integer | PRMO 2019 | Question 22

Try this beautiful problem from the Pre-RMO, 2019 based on Greatest Integer. You may use sequential hints to solve the problem.

Try this beautiful problem from the Pre-RMO, 2019 based on Greatest Integer.

Greatest integer – PRMO 2019

Find the greatest integer not exceeding the sum $\sum_{n=1}^{1599}\frac{1}{\sqrt{n}}$

• is 107
• is 78
• is 840
• cannot be determined from the given information

Key Concepts

Largest integer

Divisibility

Integer

PRMO, 2019, Question 22

Elementary Number Theory by David Burton

Try with Hints

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.