RU-ALLRUSSIAN 2021 Problem 112

Let $P(x)$ be a nonzero polynomial of degree $n>1$ with nonnegative coefficients such that function $y=P(x)$ is odd. Is that possible thet for some pairwise distinct points $A_{1}, A_{2}, \dots A_{n}$ on the graph $G: y = P(x)$ the following conditions hold: tangent to $G$ at $A_{1}$ passes through $A_{2}$ , tangent to $G$ at $A_{2}$ passes through $A_{3}$ , $\dots$ , tangent to $G$ at $A_{n}$ passes through $A_{1}$ ?