2005 Canadian MO Problems/Problem 5
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.
This problem needs a solution. If you have a solution for it, please help us out by. Partial Solution: Consider P(x)=(x-a)(x-b)(x-c). Let . Since a ,b ,c are roots of P(x), P(x)=0 is the characteristic equation of . So : . So clearly if . This proves (b).
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