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G(n)/U(n) = (2^n + 1)(2^n - 1) for even, odd sum of products of 0s and 1s

Source: 10th QEDMO p3 Juniors (8-10. 12. 2011) https://artofproblemsolving.com/community/c1512515_qedmo_200507

May 16, 2021
combinatoricsoddEvenSum

Problem Statement

Let nn be a positive integer. Let G(n)G (n) be the number of x1,...,xn,y1,...,yn∈{0,1}x_1,..., x_n, y_1,...,y_n \in \{0,1\}, for which the number x1y1+x2y2+...+xnynx_1y_1 + x_2y_2 +...+ x_ny_n is even, and similarly let U(n)U (n) be the number for which this sum is odd. Prove that G(n)U(n)=2n+12n−1.\frac{G(n)}{U(n)}= \frac{2^n + 1}{2^n - 1}.
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