2012/005) If ab + bc + cd +da = 16, what will at least a^2 + b^2 + c^2 + d^2

we have (a-b)^2 = a^2 + b^2 - 2ab
(b-c)^2 = b^2 + c^2 - 2bc
(c-d)^2 = c^2 + d^2 - 2cd
(d-a) ^2 = d^2+a^2 - 2ad

adding (a-b)^2 + (b-c)^2 + (c-d)^2 - (d-a)^2 = 2(a^2 +b^2 +c^2 + d^2) - 2(ab+bc+cd+da)

or 2(a^2 +b^2 +c^2 + d^2) = (a-b)^2 + (b-c)^2 + (c-d)^2 - (d-a)^2 + 2(ab+bc+cd+da)

or (a^2 +b^2 +c^2 + d^2) = ((a-b)^2 + (b-c)^2 + (c-d)^2 - (d-a)^2) /2+ (ab+bc+cd+da)
= ((a-b)^2 + (b-c)^2 + (c-d)^2 - (d-a)^2) /2+ 16
clearly is lowest when a= b=c=d and value= 16