the equation (x-n)^m+(x-n^2)^m+(x-n^3)^m+...............+(x-n^m)^m (m is odd integer) has 1)all real roots 2)1 real root and (n-1) imaginary roots 3)no real root 4)one real root and (m-1) imaginary roots

let f(x) = 

f(x) is an equation of degree m and has ATMOST m roots.

differentiate the given equation it will become : 

since m is odd m-1 is even and the obtained equation f1(x)  is an even degree equation such that f1(x) > 0

this implies f(x) is always increasing . so it will cut x- axis only once. hence f(x) will have only 1 real solution.

Answer is (4).