The AM-GM inequality states that the arithmetic means of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list.
They are also only equal if every number in the list is the same.
In math symbols, AM-GM states that for any real numbers x1,x2,…,xn≥0,
nx1+x2+…+xn≥nx1x2…xn
where
nx1+x2+…+xn=nx1x2…xn
if and only if
x1=x2=…=xn.
Note: There are also other AM-GM inequalities, like the weighted variation and the Mean Inequality Chain, but I don't think they're worth covering at the high school level.
Here are some practice problems
Practice Problem
Find the maximum of 10−a−2a1 for all a>0.
Practice Problem
(1983 AIME Q9)
Find the minimum value of xsin(x)9x2sin2(x)+4 for 0<x<π.
Final Notes and Tips
Not super common, would keep in mind and memorize.