As a writer; especially one that focuses on solving problems, I thought it would be a good idea to start a series where I teach different concepts and ways of thinking to approach problem solving. The beauty of competitions and Olympiads is that even when you know the techniques for solving the problem, the problem can still prove to be extra-ordinarily difficult. However, knowing what to typically expect is a great start, though it should be kept in mind that there will almost always be multiple ways to solve a problem. It is sometimes difficult to judge which method is better, for different methods may provide different insights about the problem. Without further ado, we begin this short article on probably one of the most famous inequality — the Arithmetic Mean - Geometric mean inequality.
First, we start with the statement.
In other words, the arithmetic mean is greater or equal to the geometric mean for non-negative numbers. There are many proofs for this inequality; though we will not prove it ourselves. The curious reader may see a variety of proofs here. Or if brave, can prove it themselves.
