AM-GM Inequality

In the case of two variables, the Arithmetic Mean – Geometric Mean (AM-GM) inequality – √ab ≤ (a+b)/2 – is a consequence of a specific case of the Isoperimetric theorem:

Among all rectangles of a given area the square has the least perimeter.

Or, equivalently,

Among all rectangles of a given perimeter the square has the largest area.

This duality of the formulation carries over to the AM-AG inequality.

For positive a, b that satisfy a + b = 2, ab ≤ 1.

Equivalently,

For positive a, b that satisfy ab = 1, a + b ≥ 2.

While trivial, it is often useful while solving problems to keep this interpretation in mind. Here’s one example from the 1935 Moscow Mathematical Olympiad:

Find all real solutions of the following system:

x + y = 2
xy – z² = 1.

Solution

Since x + y = 2, xy ≤ 1 so that

1 = xy – z² ≤ 1 – z² < 1,

unless z = 0. To avoid a contradiction (1 < 1), we have to accept z = 0 as the only possibility. But then xy = 1 and x + y = 2, implying x = y = 1.