Difference between revisions of "2017 OIM Problems/Problem 6"
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== Problem == | == Problem == | ||
Let <math>n > 2</math> be an even positive integer and <math>a_1 < a_2 < \cdots < a_n</math> real numbers such that | Let <math>n > 2</math> be an even positive integer and <math>a_1 < a_2 < \cdots < a_n</math> real numbers such that | ||
| − | <math>a_{k+1} -a_k \le 1</math> for all <math>k</math> with <math>1 \le k le n-1</math>. Let <math>A</math> be the set of pairs <math>(i, j)</math> with <math>1 \le i < j \le n</math> and <math>j - i</math> even, and let <math>B</math> be the set of pairs <math>(i, j)</math> with <math>1 \le i < j le n</math> and <math>j - i</math> odd. Show that | + | <math>a_{k+1} -a_k \le 1</math> for all <math>k</math> with <math>1 \le k le n-1</math>. Let <math>A</math> be the set of pairs <math>(i, j)</math> with <math>1 \le i < j \le n</math> and <math>j - i</math> even, and let <math>B</math> be the set of pairs <math>(i, j)</math> with <math>1 \le i < j \le n</math> and <math>j - i</math> odd. Show that |
<cmath>\prod_{(i,j)\in A}^{}(a_j-a_i)>\prod_{(i,j)\in B}^{}(a_j-a_i)</cmath> | <cmath>\prod_{(i,j)\in A}^{}(a_j-a_i)>\prod_{(i,j)\in B}^{}(a_j-a_i)</cmath> | ||
Latest revision as of 14:50, 14 December 2023
Problem
Let 
be an even positive integer and 
real numbers such that
for all 
with 
. Let 
be the set of pairs 
with 
and 
even, and let 
be the set of pairs with and odd. Show that
![\[\prod_{(i,j)\in A}^{}(a_j-a_i)>\prod_{(i,j)\in B}^{}(a_j-a_i)\]](http://latex.artofproblemsolving.com/5/1/8/518ee51c6441e7bc32a51edbbb852deb017b4ad1.png)
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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