IR-TST 2012 Problem 14

Do there exist $2000$ real numbers (not necessarily distinct) such that all of them are not zero and if we put any group containing $1000$ of them as the roots of a monic polynomial of degree $1000$ , the coefficients of the resulting polynomial (except the coefficient of $x^{1000}$ ) be a permutation of the $1000$ remaining numbers?

Proposed by Morteza Saghafian