Problem A. 429. (May 2007)
A. 429. Find all pairs f(x), g(x) of polynomials with integer coefficients satisfying
(Proposed by Katalin Gyarmati)
Deadline expired on 15 June 2007.
Solution. Deriving the equation,
By Eisenstein's criterion, the polynomial 2007x206+2 is irreducible. Hence, one of the two factors is constant. Since gcd(2007,2)=1, this constant must be 1 or -1.
If g'(x)=1 then g(x)=x+c with some integer c and f(x)=(x-c)2007+2(x-c)+1.
If g'(x)=-1 then g(x)=-x+c and f(x)=(-x+c)2007+2(-x+c)+1.
If f'(g(x))=1 then g'(x)=2007x2006+2. The degree of polynomial g(x) is 2007, and it attains infinitely many distinct values. So f'(g(x))=1 is possible only if f'(x)=1. Then f(x)=x+c and g(x)=x2007+2x+1-c.
Finally, if f'(g(x))=-1 then, similarly to the previous case, f'(x)=-1, f(x)=-x+c and g(x)=-x2007-2x-1+c.