Statement: For positive reals $a_1$, ..., $a_n$: AM = $\frac{a_1+...+a_n}{n}$ >= GM = ($a_1$...$a_n$)^$\frac{1}{n}$ >= HM = $\frac{n}{1/a_1+...+1/a_n}$

Equality holds iff $a_1$ = $a_2$ = ... = $a_n$.

Application 1: Minimize sum given product If abc = k (constant), min(a+b+c) = 3k^$\frac{1}{3}$, at a=b=c=k^$\frac{1}{3}$.

Application 2: Maximize product given sum If a+b+c = s (constant), max(abc) = $\frac{s}{3}$^3, at a=b=c=s/3.

Application 3: Min of x + k/x For x > 0: x + k/x >= 2*sqrt(k). Min at x = sqrt(k).

Application 4: Weighted AM-GM To optimize $xy^2$ given x+2y=c: split as x+y+y and apply AM-GM to three terms.

Common mistakes:

1. Applying to non-positive quantities (AM-GM requires positive values)
2. Forgetting to check equality condition
3. Not matching the split with the target expression
4. Using AM-GM when the constraint is not sum-type