jmc

Question

For a certain positive integer n, there exist real numbers x_1, x_2, , x_n such that align* x_1 + x_2 + x_3 + + x_n &= 1000, x_1^4 + x_2^4 + x_3^4 + + x_n^4 &= 512000. align*Find the smallest positive integer n for which this is possible.

Accepted Answer

By Cauchy-Schwarz, (1^2 + 1^2 + + 1^2)(x_1^2 + x_2^2 + + + x_n^2) (x_1 + x_2 + + x_n)^2 = 1000^2,so x_1^2 + x_2^2 + + x_n^2 1000^2n. Again by Cauchy-Schwarz, (1^2 + 1^2 + + 1^2)(x_1^4 + x_2^4 + + + x_n^4) (x_1^2 + x_2^2 + + x_n^2)^2,so n 512000 1000^4n^2.Then n^3 1000^4512000 = 1000^3512 = 5^9,so n 125. For n = 125, we can take x_1 = x_2 = = x_125 = 8, so the smallest such n is 125.