Difference between revisions of "2005 Canadian MO Problems/Problem 5"
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+ | Partial Solution: | ||
+ | Consider P(x)=(x-a)(x-b)(x-c). | ||
+ | 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>. | ||
+ | So : | ||
+ | <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 \vert s_k, s_{k+1}, a+b+c \vert s_{k+3}</math>. | ||
+ | This proves (b). | ||
==See also== | ==See also== |
Latest revision as of 08:13, 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 |