Difference between revisions of "2008 USAMO Problems/Problem 6"

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== Problem ==
 
== Problem ==
(''Sam Vandervelde'') At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form <math>2^k</math> for some positive integer <math>k</math>).
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(''[[Sam Vandervelde]]'') At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form <math>2^k</math> for some positive integer <math>k</math>).
  
 
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== Solution ==
 
== Solution ==
 
=== Solution 1 ===
 
=== Solution 1 ===

Revision as of 22:42, 2 May 2008

Problem

(Sam Vandervelde) At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $2^k$ for some positive integer $k$).

Solution

Solution 1

Solution 2

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

Resources

2008 USAMO (ProblemsResources)
Preceded by
Problem 5
Followed by
Last Problem
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All USAMO Problems and Solutions
  • <url>viewtopic.php?t=202908 Discussion on AoPS/MathLinks</url>
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