1997 OIM Problems/Problem 4

Problem

Let $n$ be a positive integer. Let us consider the sum $x_1y_1 + x_2y_2 + \cdots + x_ny_n$ , where the values that the variables $x_1, x_2, \cdots , x_n, y_1, y_2, \cdots , y_n$ can take are only 0 and 1. Let $O(n)$ be the number of $2n$ -tuples $(x_1, x_2, \cdots , x_n, y_1, y_2, \cdots , y_n)$ for which the value of the sum is an odd number and let $E(n)$ be the number of $2n$ -coordinates $(x_1, x_2, \cdots , x_n, y_1, y_2, \cdots , y_n)$ for which the sum is an even value. Prove that