Problem

Source: ELMO Shortlist 2023 A3

Tags: Elmo, algebra



Does there exist an infinite sequence of integers \(a_0\), \(a_1\), \(a_2\), \(\ldots\) such that \(a_0\ne0\) and, for any integer \(n\ge0\), the polynomial \[P_n(x)=\sum_{k=0}^na_kx^k\]has \(n\) distinct real roots? Proposed by Amol Rama and Espen Slettnes