jmc

Question

Let x_1, x_2, , x_n be nonnegative real numbers such that x_1 + x_2 + + x_n = 1 and x_1^2 + x_2^2 + + x_n^2 1100.Find the smallest possible value of n.

Accepted Answer

By QM-AM, x_1^2 + x_2^2 + + x_n^2n x_1 + x_2 + + x_nn.Then 1n x_1^2 + x_2^2 + + x_n^2n 1100n.Hence, 1n^2 1100n,and n 100. For n = 100, we can take x_i = 1100 for all i, so the smallest such n is 100.