Eötvös 1935/1 solution

Problem 1

x₁, x₂, ... , x_n is any sequence of positive reals and y_i is any permutation of x_i. Show that ∑ x_i/y_i ≥ n.

Solution

By AM/GM (∑ x_i/y_i)/n ≥ geometric mean of x_i/y_i which is 1 since ∏ x_i = ∏ y_i.