IR-TST 2018 Problem 15

$n>1$ and distinct positive integers $a_1,a_2,\ldots,a_{n+1}$ are given. Does there exist a polynomial $p(x)\in\Bbb{Z}[x]$ of degree $\le n$ that satisfies the following conditions? a. $\forall_{1\le i < j\le n+1}: \gcd(p(a_i),p(a_j))>1$ b. $\forall_{1\le i < j < k\le n+1}: \gcd(p(a_i),p(a_j),p(a_k))=1$ Proposed by Mojtaba Zare