Question 47
Let $n \gt 1$ be an integer and $a_1, a_2, \dots, a_{n+1}$ be distinct positive integers. Does there exist a polynomial $p(x)$ with integer coefficients of degree at most $n$ that satisfies the following conditions?
* $\forall 1 \le i \lt j \le n+1 : \gcd(p(a_i), p(a_j)) \gt 1$
* $\forall 1 \le i \lt j \lt k \le n+1 : \gcd(p(a_i), p(a_j), p(a_k)) = 1$
* $\forall 1 \le i \lt j \le n+1 : \gcd(p(a_i), p(a_j)) \gt 1$
* $\forall 1 \le i \lt j \lt k \le n+1 : \gcd(p(a_i), p(a_j), p(a_k)) = 1$