RMM 2024 Problem 6

A polynomial $P$ with integer coefficients is square-free if it is not expressible in the form $P=Q^{2} R$, where $Q$ and $R$ are polynomials with integer coefficients and $Q$ is not constant. For a positive integer $n$, let $\mathcal{P}_{n}$ be the set of polynomials of the form

$$1+a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}$$

with $a_{1}, a_{2}, \ldots, a_{n} \in\{0,1\}$. Prove that there exists an integer $N$ so that, for all integers $n \geq N$, more than $99 \%$ of the polynomials in $\mathcal{P}_{n}$ are square-free.

Each problem is worth 7 marks. Time allowed: $4 \frac{1}{2}$ hours.