Statement: For positive reals a_1, ..., a_n: AM = (a_1+...+a_n)/n >= GM = (a_1*...*a_n)^(1/n) >= HM = 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^(1/3), at a=b=c=k^(1/3).

Application 2: Maximize product given sum If a+b+c = s (constant), max(abc) = (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