1st Swedish Mathematical Society Problems 1961

1.  Let S be the system of equations (1) y(x4 - y2 + x2) = x, (2) x(x4 - y2 + x2) = 1. Take S' to be the system of equations (1) and x·(1) - y·(2) (or y = x2). Show that S and S' do not have the same set of solutions and explain why. 2.  Show that x1/xn + x2/xn-1 + x3/xn-2 + ... + xn/x1 ≥ n for any positive reals x1, x2, ... , xn.