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2024 Romanian Master of Mathematics, 6
Tags: algebra , polynomial , number theory , analytic number theory , complex roots
A polynomial $P$ with integer coefficients is [i]square-free[/i] if it is not expressible in the form $P = Q^2R$, where $Q$ and $R$ are polynomials with integer coefficients and $Q$ is not constant. For a positive integer $n$, let $P_n$ be the set of polynomials of the form $$1 + a_1x + a_2x^2 + \cdots + a_nx^n$$ with $a_1,a_2,\ldots, a_n \in \{0,1\}$. Prove that there exists an integer $N$ such that for all integers $n \geq N$, more than $99\%$ of the polynomials in $P_n$ are square-free. [i]Navid Safaei, Iran[/i]