The second stage examination of INMO, the Regional Mathematical Olympiad (RMO) is a three hour examination with six problems. The problems under each topic involve high level of difficulty and sophistication. West Bengal RMO 2015 Problem 6 Solution has been written for RMO preparation series. The book, Challenge and Thrill of Pre-College Mathematics is very useful for preparation of RMO.

## Solution

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We can write where denotes the integral part of

Now, we can say that as

Let where is odd. Then

All such integers, must satisfy the property where is a non-negative integer.

Then

Now, Means, is even.

So

Or, is an integer.

Hence, is an integer for all positive reals satisfying

for some non-negative integer

As takes infinitely many values, number of such positive real numbers is also infinite.

This completes the proof.

**Key Idea: ** Greatest Integer Function

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