Difference between revisions of "2005 Canadian MO Problems/Problem 5"
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Partial Solution: | Partial Solution: | ||
| − | Consider P(x)=(x-a)(x-b)(x-c). | + | Consider P(x)=(x-a)(x-b)(x-c). |
| − | Let <math>S_k= a^k+b^k+c^k</math>. | + | Let <math>S_k= a^k+b^k+c^k</math>. |
Since a ,b ,c are roots of P(x), P(x)=0 is the characteristic equation of <math>s_k</math>. \\ | Since a ,b ,c are roots of P(x), P(x)=0 is the characteristic equation of <math>s_k</math>. \\ | ||
| − | So : | + | So : |
<math>s_{k+3}-(a+b+c)s_{k+2}+(ab+bc+ca)s_{k+1}-(abc)s_k=0</math>. \\ | <math>s_{k+3}-(a+b+c)s_{k+2}+(ab+bc+ca)s_{k+1}-(abc)s_k=0</math>. \\ | ||
| − | So clearly if <math>a+b+c \ | + | So clearly if <math>a+b+c \vert s_k, s_{k+1}, a+b+c \vert s_{k+3}</math>. \\ |
This proves (b). | This proves (b). | ||
Revision as of 08:11, 16 January 2020
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.
.Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
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).
See also
| 2005 Canadian MO (Problems) | ||
| Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 | Followed by Last Question |
