Problem A. 472. (January 2009)

A. 472. Call a finite sequence (p₁(x),...,p_k(x)) of polynomials with integer coefficients Euclidean if there exist polynomials with integer coefficients such that d(x)=q₁(x)p₁(x)+...+q_k(x)p_k(x) is a common divisor of p₁(x),...,p_k(x), i.e. there are polynomials with integer coefficients for which p_i(x)=r_i(x)d(x) for every 1ik. Prove that whenever are polynomials with integer coefficients and every two of them form a Euclidean pair, then the sequence is Euclidean as well.

(5 pont)

Deadline expired on February 16, 2009.

Statistics:

3 students sent a solution.
5 points:	Backhausz Tibor, Nagy 235 János, Tomon István.

Problems in Mathematics of KöMaL, January 2009