Question 125
Let $P_1(x), \dots, P_n(x)$ be monic polynomials with real coefficients. For every real number $y$, define the set
$$
S_y = \{z \in \mathbb{R} \mid \text{there exists some } i \in \{1, \dots, n\} \text{ such that } y = P_i(z)\}.
$$
Prove that if for any two distinct real numbers $y_1, y_2$ the sets $S_{y_1}, S_{y_2}$ have the same number of elements, then $P_1, \dots, P_n$ all have the same degree.
$$
S_y = \{z \in \mathbb{R} \mid \text{there exists some } i \in \{1, \dots, n\} \text{ such that } y = P_i(z)\}.
$$
Prove that if for any two distinct real numbers $y_1, y_2$ the sets $S_{y_1}, S_{y_2}$ have the same number of elements, then $P_1, \dots, P_n$ all have the same degree.