Problem

Source: 2023 Taiwan TST Round 1 Independent Study 2-A

Tags: algebra, polynomial, Taiwan



Given some monic polynomials $P_1, \ldots, P_n$ with real coefficients, for any real number $y$, let $S_y$ be the set of real number $x$ such that $y = P_i(x)$ for some $i = 1, 2, ..., n$. If the sets $S_{y_1}, S_{y_2}$ have the same size for any two real numbers $y_1, y_2$, show that $P_1, \ldots, P_n$ have the same degree. Proposed by usjl