a, b, c, d, e, f, g, h be distinct elements in the set -7,-5,-3,-2,2,4,6,13 . The minimum value of (a+b+c+d)^2+(e+f+g+h)^2 is

Step-by-step solution

aligned & Let a+b+c+d=x and e+f+g+h=y & x^2+y^2=(x+y)^2-2 x y & =(13+6+4+2-2-3-5-7)^2-2 x y & =64-2 x y aligned As we know AM GM aligned & x+y 2 x y & 8 2 x y & x y 16 aligned x^2+y^2 is minimum when x y is maximum. The maximum value of x y=16 . The minimum value of x^2+y^2=64-2 16=32 The minimum values of (a+b+c+d)^2+(e+f+g+h)=32