AM-GM

In Human Language

Namely,

Any set of nonnegative real numbers comes with an inequality
Average Mean > Geometric Mean.
Equality Condition: Average Mean = Geometric Mean only if all numbers in the set are equal

AM-GM has a weighted form, given by using weighted average.
For example, the weighted arithmetic mean of x:y = 3:1 is

and its weighted geometric mean is


The weighted AM-GM inequality states that IF
1. The numbers, are nonnegeative and real
2. The weights, are nonnegative and real, and add up to 1
Then  
BOOM!

Cauchy's Induction Proof
This proof uses induction to prove the unweighted theorem holds for all n=2^k and n-1
Plus I translated it, so definitely a friendly proof :)

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Complete Proofs by Convexity
2 slightly monstrous Proofs by Convexity are here on the official AOPS page
They work for the Weighted Form too :)

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Proof by Rearrangement
This is a ghastly condensed proof on the AOPS official page
Here you will also find the original Cauchy's proof by induction :)

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Here's a Bunch of half-baked ideas at the Stack Exchange
Really creative :)

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This leads to a webpage showing I know how to change background colors for webpages:)