KöMaL 2009 A472

Call a finite sequence ( $p_1$($x$),...,$p_k$($x$) ) of polynomials with integer coefficients Euclidean if there exist polynomials $q_1(x),\ldots,q_k(x)$ 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 $r_1(x),\ldots,r_k(x)$ with integer coefficients for which $p_i$($x$)=$r_i$($x$)$d$($x$) for every 1$\le i \le k$. Prove that whenever $p_1(x),\ldots,p_n(x)$ are polynomials with integer coefficients and every two of them form a Euclidean pair, then the sequence $\big(p_1(x),\ldots,p_n(x)\big)$ is Euclidean as well.