  In this note we revisit the classical geometric-arithmetic mean inequality
and find a formula for the difference of the arithmetic and the geometric means
of given $n\in\mathbb N$ nonnegative numbers $x_1,x_2,\dots,x_n$. The formula
yields new stronger versions of the geometric-arithmetic mean inequality. We
also find a second version of a strong geometric-arithmetic mean inequality and
show that all inequalities are optimal in some sense. Anther striking novelty
is, that the equality in all new inequalities holds not only in the case when
all $n$ numbers are equal, but also in other cases.
