Question 13
A7 (RUS). For a polynomial $P$ of degree $2000$ with distinct real coefficients let $M(P)$ be the set of all polynomials that can be produced from $P$ by permutation of its coefficients. A polynomial $P$ will be called $n$-independent if $P(n)=0$ and we can get from any $Q$ in $M(P)$ a polynomial $Q_1$ such that $Q_1(n)=0$ by interchanging at most one pair of coefficients of $Q$. Find all integers $n$ for which $n$-independent polynomials exist.