Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

m (→‎See also: box)
(2 intermediate revisions by the same user not shown)
Line 9: Line 9:
  
 
{{solution}}
 
{{solution}}
 +
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==

Revision as of 09:13, 16 January 2020

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

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 $S_k= a^k+b^k+c^k$. Since a ,b ,c are roots of P(x), P(x)=0 is the characteristic equation of $s_k$. So : $s_{k+3}-(a+b+c)s_{k+2}+(ab+bc+ca)s_{k+1}-(abc)s_k=0$. So clearly if $a+b+c \vert s_k, s_{k+1}, a+b+c \vert s_{k+3}$. This proves (b).

See also

2005 Canadian MO (Problems)
Preceded by
Problem 4 		1 • 2 • 3 • 4 • 5 Followed by
Last Question
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2005_Canadian_MO_Problems/Problem_5&oldid=114870"