Several real numbers (not necessarily are different) are chosen. The sum of these numbers is 10. Is it possible that the sum of the squares of these numbers is less than one-millionth? Justify your answer.
Any hints or help would be greatly appreciated. Thanks!
Square each number, each of them is , so its square is .
Then add up all of these square times. See what you get, and check if it is possible that the sum you just computed is less than one-millionth.
And the point of looking at all numbers equal is that you can show that the more they differ, the larger the sum of squares will be. The case with all numbers equal is the smallest sum of squares. If that sum can't be "less than one-millionth" then the other sums certainly can't be.
To everyone that's posted, I will give thanks when I figure this out but I don't see the picture everyone is trying to paint here.