Question 132
Let $P$ and $Q$ be fixed polynomials with real coefficients, let the degree of $Q$ be $2021$, and let $a_1, a_2, \dots, a_{2022}, b_1, b_2, \dots, b_{2022}$ be real numbers, such that $a_1 a_2 \dots a_{2022} \neq 0$. If
$$
P(a_1Q(x) + b_1) + \dots + P(a_{2021}Q(x) + b_{2021}) = P(a_{2022}Q(x) + b_{2022}), \quad \forall x \in \mathbb{R},
$$
prove that $P(x)$ possesses at least one real root.
$$
P(a_1Q(x) + b_1) + \dots + P(a_{2021}Q(x) + b_{2021}) = P(a_{2022}Q(x) + b_{2022}), \quad \forall x \in \mathbb{R},
$$
prove that $P(x)$ possesses at least one real root.