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.
Also visit: Math Olympiad program of Cheenta
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.
Now, Means, is even.
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