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Translations of a cubic polynomial with integer roots

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

May 15, 2013
geometrygeometric transformationalgebrapolynomialalgebra proposed

Problem Statement

For a positive integer nn, a cubic polynomial p(x)p(x) is said to be nn-good if there exist nn distinct integers a1,a2,…,ana_1, a_2, \ldots, a_n such that all the roots of the polynomial p(x)+ai=0p(x) + a_i = 0 are integers for 1≤i≤n1 \le i \le n. Given a positive integer nn prove that there exists an nn-good cubic polynomial.
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