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. The book, Challenge and Thrill of Pre-College Mathematics is very useful for preparation of RMO. West Bengal RMO 2015 Problem 3 Solution has been written for RMO preparation series.
Show that there are infinitely many triples of positive integers, such that .
Suppose we have found one such triplet (x, y, z). Then . Multiply to both sides where a is an arbitrary integer.
Clearly we have
Hence if (x, y, z) is a triple then is another such triple. Since a can be any arbitrary integer, hence we have found infinitely many such triplets provided we have found at least one
To find one such triple, we use the following intuition: set x, y, z as some powers of 2 such that . Then r must be of the form 12k. Finally, their sum must be . This r+1 must be divisible by 31.
Let and we get . Since 12 and 31 are coprime there is integer solution to this linear diophantine equation (by Bezoat's theorem). We can solve this linear diophantine equation by euclidean algorithm.
Hence we use this to form an equation:
Hence we have found one such triple : (from which we have shown earlier that infinitely more can be generated)