Difference between revisions of "1997 OIM Problems/Problem 4"

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

Latest revision as of 18:29, 23 December 2023

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

\[\frac{E(n)}{O(n)} =\frac{2^n+1}{2^n-1}\]

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

https://www.oma.org.ar/enunciados/ibe12.htm