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>). | + | (''[[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 
for some positive integer 
).
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 (Problems • Resources) | ||
| Preceded by Problem 5 |
Followed by Last Problem | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAMO Problems and Solutions | ||
- <url>viewtopic.php?t=202908 Discussion on AoPS/MathLinks</url>
