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 13:21, 4 September 2006
Problem
Let's say that an ordered triple of positive integers 
is 
-powerful if 
, 
, and 
is divisible by 
. For example, 
is 5-powerful.
- Determine all ordered triples (if any) which are
-powerful for all 
. - Determine all ordered triples (if any) which are 2004-powerful and 2005-powerful, but not 2007-powerful.
.
