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Let $n$ be a positive integer. Consider the sum $x_1 y_1+x_2 y_2+\cdots+x_n y_n$ for any $2 n$ numbers $a_i, b_i$ taking only the values 0 and 1 . Denote by $I(n)$ the number of $2 n$-tuples $\left(x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n\right)$ for which this sum is odd, and by $P(n)$ the number of those for which this sum is even. Prove that
$\frac{P(n)}{I(n)}=\frac{2^n+1}{2^n-1}$
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