jmc

Question

Let x, y, z, v, w be positive real numbers such that x^2 + y^2 + z^2 + v^2 + w^2 = 2016. Let M be the maximum value of xz + 2yz + 3zv + 7zw,and let x_M, y_M, z_M, v_M, w_M be the values of x, y, z, v, w, respectively, that produce the maximum value of M. Find M + x_M + y_M + z_M + v_M + w_M.

Accepted Answer

Note that xz + 2yz + 3zv + 7zw = z(x + 2y + 3v + 7w). By Cauchy-Schwarz, align* x + 2y + 3v + 7w & (1 + 4 + 9 + 49)(x^2 + y^2 + v^2 + w^2) &= 63 (x^2 + y^2 + v^2 + w^2) &= 3 7(2016 - z^2), align*so z(x + 2y + 3v + 7w) 3z 7(2016 - z^2) = 3 7z^2 (2016 - z^2). By AM-GM, z^2 (2016 - z^2) ( z^2 + (2016 - z^2)2 )^2 = 1008^2,so 3 7z^2 (2016 - z^2) 3 7 1008^2 = 3024 7.Equality occurs when x:y:v:w = 1:2:3:7, z^2 = 1008, and x^2 + y^2 + z^2 + v^2 + w^2 = 2016, which leads to x = 4, y = 8, z = 12 7, v = 12, and w = 28. Thus, M + x_M + y_M + z_M + v_M + w_M = 3024 7 + 4 + 8 + 12 7 + 12 + 28 = 52 + 3036 7.