Difference between revisions of "2005 Canadian MO Problems/Problem 5"

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==Problem==
 
==Problem==
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Let's say that an ordered triple of positive integers <math>(a,b,c)</math> is <math>n</math>-''powerful'' if <math>a \le b \le c</math>, <math>\gcd(a,b,c) = 1</math>, and  <math>a^n + b^n + c^n</math> is divisible by <math>a+b+c</math>. For example, <math>(1,2,2)</math> is 5-powerful.
 
Let's say that an ordered triple of positive integers <math>(a,b,c)</math> is <math>n</math>-''powerful'' if <math>a \le b \le c</math>, <math>\gcd(a,b,c) = 1</math>, and  <math>a^n + b^n + c^n</math> is divisible by <math>a+b+c</math>. For example, <math>(1,2,2)</math> is 5-powerful.
  
 
* Determine all ordered triples (if any) which are <math>n</math>-powerful for all <math>n \ge 1</math>.
 
* Determine all ordered triples (if any) which are <math>n</math>-powerful for all <math>n \ge 1</math>.
 
* Determine all ordered triples (if any) which are 2004-powerful and 2005-powerful, but not 2007-powerful.
 
* Determine all ordered triples (if any) which are 2004-powerful and 2005-powerful, but not 2007-powerful.
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==Solution==
 
==Solution==
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==See also==
 
==See also==
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*[[2005 Canadian MO]]
 
*[[2005 Canadian MO]]
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[[Category:Olympiad Number Theory Problems]]

Revision as of 14:21, 4 September 2006

Problem

Let's say that an ordered triple of positive integers $(a,b,c)$ is $n$-powerful if $a \le b \le c$, $\gcd(a,b,c) = 1$, and $a^n + b^n + c^n$ is divisible by $a+b+c$. For example, $(1,2,2)$ is 5-powerful.

  • Determine all ordered triples (if any) which are $n$-powerful for all $n \ge 1$.
  • Determine all ordered triples (if any) which are 2004-powerful and 2005-powerful, but not 2007-powerful.

Solution

See also