Problem

Source: Indian IMOTC 2013, Team Selection Test 1, Problem 3

Tags: geometry, geometric transformation, algebra, polynomial, algebra proposed



For a positive integer $n$, a cubic polynomial $p(x)$ is said to be $n$-good if there exist $n$ distinct integers $a_1, a_2, \ldots, a_n$ such that all the roots of the polynomial $p(x) + a_i = 0$ are integers for $1 \le i \le n$. Given a positive integer $n$ prove that there exists an $n$-good cubic polynomial.