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>- | + | 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 19:29, 23 December 2023
Problem
Let 
be a positive integer. Let us consider the sum 
, where the values that the variables 
can take are only 0 and 1. Let 
be the number of 
-tuples 
for which the value of the sum is an odd number and let 
be the number of -coordinates for which the sum is an even value. Prove that
![\[\frac{E(n)}{O(n)} =\frac{2^n+1}{2^n-1}\]](http://latex.artofproblemsolving.com/8/4/0/8403b9434d1ef72113fa323651b612f0f6c20017.png)
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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