SFMT
Theorem
Given nonnegative real numbers and that is fixed and all the terms inside the sum are nonnegative, the maximum value of is when
Proof
The weighted AM-GM Inequality states that if are nonnegative real numbers, and are nonnegative real numbers (the "weights") which sum to 1, then We let and in this inequality. We get that Dividing both sides and then taking to the nth power, we get Then we can multiply both sides to get The equality case for the weighted AM-GM inequality is when all the terms such that is not are equal, or in this case
Example Problems
Problem 1
Given nonnegative integers such that find the maximum value
Solution 1
By the new theorem, we know that so Plugging it in, our answer is
Name
This theorem has been widely known for a while now. The AoPS user solasky rediscovered it and put a name to it. You can call it whatever you want.