I need help proving the following:
if a,b,c,... are n real positive number such that a+b+c+... = 1, then
a^2 + b^2 + c^2 + ... >=1/n
denote all these numbers , , ... , with the property :
Now note that when , we have , and it follows that :
This is not a proof in itself : it is there to help you grasp what this problem means. All you have to prove is that there is no solution for where .
Here is an idea : say . Therefore, the following holds : , and so on following the same logic with , , .... Take the squares of these inequalities, and see where you get You will very likely stumble upon a difficulty, that you will probably be able to express in mathematical terms and that will lead to the solution to your problem
EDIT : forget this stuff, I got it wrong. Hang on, I'm thinking of something ...
Yeah, I think I got it. Say you got your numbers, all equal to each other (so they all equal ). If you add some value to one number, you need to report the opposite of this value onto the other numbers to balance the equality. Now which values of don't allow any balancing of the other numbers that satisfies the sum of the squares being less than ? Try to express this in mathematical terms. You should find that has no solution ... your problem follows. I might post a proof, I'm actually as stumped as you on this one
There is one small thing I didn't get in your proof:
by saying , you're assuming that: and and . But if A is the average of a,b and c, then A may be smaller or larger than a, b or c. Would you please explain how did you get the inequality?