Problem

Source: 2009 China Western Mathematics Olympiad

Tags: algebra, polynomial, geometry, geometric transformation, absolute value, algebra proposed



Let $M$ be the set of the real numbers except for finitely many elements. Prove that for every positive integer $n$ there exists a polynomial $f(x)$ with $\deg f = n$, such that all the coefficients and the $n$ real roots of $f$ are all in $M$.