jmc

Question

Katie has a list of real numbers such that the sum of the numbers on her list is equal to the sum of the squares of the numbers on her list. Compute the largest possible value of the arithmetic mean of her numbers.

Accepted Answer

Let the numbers in the list be x_1, x_2, , x_n. Then by the trivial inequality, (x_1 - 1)^2 + (x_2 - 1)^2 + + (x_n - 1)^2 0.Expanding, we get (x_1^2 + x_2^2 + + x_n^2) - 2(x_1 + x_2 + + x_n) + n 0.Since x_1^2 + x_2^2 + + x_n^2 = x_1 + x_2 + + x_n, x_1 + x_2 + + x_n n,so x_1 + x_2 + + x_nn 1. Equality occurs when all the x_i are equal to 1, so the largest possible arithmetic mean is 1.