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a really nice polynomial problem

Source: Iranian TST 2018, third exam day 1, problem 3

April 18, 2018
polynomialInteger PolynomialIranian TSTnumber theoryIran

Problem Statement

n>1n>1 and distinct positive integers a1,a2,…,an+1a_1,a_2,\ldots,a_{n+1} are  given. Does there exist a polynomial p(x)∈Z[x]p(x)\in\Bbb{Z}[x] of degree  ≤n\le n that satisfies the following conditions? a. ∀1≤i<j≤n+1:gcd⁡(p(ai),p(aj))>1\forall_{1\le i < j\le n+1}: \gcd(p(a_i),p(a_j))>1 b. ∀1≤i<j<k≤n+1:gcd⁡(p(ai),p(aj),p(ak))=1\forall_{1\le i < j < k\le n+1}: \gcd(p(a_i),p(a_j),p(a_k))=1
Proposed by Mojtaba Zare
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