Difference between revisions of "SFMT"
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== Example Problems == | == Example Problems == | ||
=== Problem 1 === | === Problem 1 === | ||
− | Given nonnegative | + | Given nonnegative integers <math>x, y, z</math> such that <math>x+y+z=16,</math> find the maximum value <math>x^3y^2z^3.</math> |
+ | |||
=== Solution 1 === | === Solution 1 === | ||
− | By the new theorem, we know that <math>\dfrac{x}{3}=\dfrac{y}{2}=\dfrac{z}{3},</math> so x=6, y=4, z=6. Plugging it in, our answer is <math>746496.</math> | + | By the new theorem, we know that <math>\dfrac{x}{3}=\dfrac{y}{2}=\dfrac{z}{3},</math> so <math>x=6, y=4, z=6.</math> Plugging it in, our answer is <math>746496.</math> |
+ | |||
+ | == 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. |
Latest revision as of 01:10, 27 September 2021
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.