Problem

Source: India IMOTC 2024 Day 1 Problem 3

Tags: number theory



Let $P(x) \in \mathbb{Q}[x]$ be a polynomial with rational coefficients and degree $d\ge 2$. Prove there is no infinite sequence $a_0, a_1, \ldots$ of rational numbers such that $P(a_i)=a_{i-1}+i$ for all $i\ge 1$. Proposed by Pranjal Srivastava and Rohan Goyal