Problem A. 472. (January 2009)
A. 472. Call a finite sequence (p_{1}(x),...,p_{k}(x)) of polynomials with integer coefficients Euclidean if there exist polynomials with integer coefficients such that d(x)=q_{1}(x)p_{1}(x)+...+q_{k}(x)p_{k}(x) is a common divisor of p_{1}(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 16 February 2009.
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