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This essential trains for: Math Kangaroo 9-10, AMC-10, AMC-12, AIME.

For any non-negative real numbers x, y, it is true that:

equation

Using square roots instead of x, y :

equation

equation

equation

equation

where we notice that the left hand side is equal to the arithmetic mean of a, b and the right hand side is equal to the geometric mean of a, b .

Observation: note that, since this inequality has been derived by assuming that a perfect square is always positive, it is valid only for real numbers. Complex numbers with a non-zero imaginary part are not guaranteed to fulfill this inequality.

This inequality is sometimes useful in proving the existence of solutions of Diophantine equations.

AM-GM inequality with more than two variables:

In general, for a set of non-negative real variables:

equation

the following inequality is true:

equation