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Difference between revisions of "2013 IMO Problems/Problem 6"

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==Problem==
 
Let <math>n \ge 3</math> be an integer, and consider a circle with <math>n + 1</math> equally spaced points marked on it. Consider all labellings of these points with the numbers <math>0, 1, ... , n</math> such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called beautiful if, for any four labels <math>a < b < c < d</math> with <math>a + d = b + c</math>, the chord joining the points labelled <math>a</math> and <math>d</math> does not intersect the chord joining the points labelled <math>b</math> and <math>c</math>.
 
Let <math>n \ge 3</math> be an integer, and consider a circle with <math>n + 1</math> equally spaced points marked on it. Consider all labellings of these points with the numbers <math>0, 1, ... , n</math> such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called beautiful if, for any four labels <math>a < b < c < d</math> with <math>a + d = b + c</math>, the chord joining the points labelled <math>a</math> and <math>d</math> does not intersect the chord joining the points labelled <math>b</math> and <math>c</math>.
  
 
Let <math>M</math> be the number of beautiful labelings, and let N be the number of ordered pairs <math>(x, y)</math> of positive integers such that <math>x + y \le n</math> and <math>\gcd(x, y) = 1</math>. Prove that <cmath>M = N + 1.</cmath>
 
Let <math>M</math> be the number of beautiful labelings, and let N be the number of ordered pairs <math>(x, y)</math> of positive integers such that <math>x + y \le n</math> and <math>\gcd(x, y) = 1</math>. Prove that <cmath>M = N + 1.</cmath>
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==Solution==
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{{solution}}
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==See Also==
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*[[2013 IMO]]
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{{IMO box|year=2013|num-b=5|after=Last Problem}}

Latest revision as of 00:32, 19 November 2023

Problem

Let $n \ge 3$ be an integer, and consider a circle with $n + 1$ equally spaced points marked on it. Consider all labellings of these points with the numbers $0, 1, ... , n$ such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called beautiful if, for any four labels $a < b < c < d$ with $a + d = b + c$, the chord joining the points labelled $a$ and $d$ does not intersect the chord joining the points labelled $b$ and $c$.

Let $M$ be the number of beautiful labelings, and let N be the number of ordered pairs $(x, y)$ of positive integers such that $x + y \le n$ and $\gcd(x, y) = 1$. Prove that \[M = N + 1.\]

Solution

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See Also

2013 IMO (Problems) • Resources
Preceded by
Problem 5 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Last Problem
All IMO Problems and Solutions
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