MathDB

5

Part of 2005 Canada National Olympiad

Problems(1)

powerful numbers

Source: Canada 2005

6/26/2009
Let's say that an ordered triple of positive integers (a,b,c)(a,b,c) is nn-powerful if a≤b≤c,gcd⁡(a,b,c)=1a\le b\le c,\gcd (a,b,c)=1 and an+bn+cna^n+b^n+c^n is divisible by a+b+ca+b+c. For example, (1,2,2)(1,2,2) is 55-powerful. a)a) Determine all ordered triples (if any) which are nn-powerful for all n≥1n\ge 1. b)b) Determine all ordered triples (if any) which are 20042004-powerful and 20052005-powerful, but not 20072007-powerful.
number theory unsolvednumber theory