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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

But try the problem first…

Source

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.

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