2025 USAMO 第 2 题

Let $n$ and $k$ be positive integers with $k<n$. Let $P(x)$ be a polynomial of degree $n$ with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers $a_0, a_1, \dots, a_k$ such that the polynomial $a_kx^k+\cdots+a_1x+a_0$ divides $P(x)$, the product $a_0a_1\cdots a_k$ is zero. Prove that $P(x)$ has a nonreal root.

令 $n$ 和 $k$ 为正整数且 $k<n$。令 $P(x)$ 为具有实数系数、非零常数项且无重根的 $n$ 次多项式。假设对于任何实数 $a_0, a_1, \dots, a_k$，使得多项式 $a_kx^k+\cdots+a_1x+a_0$ 除以 $P(x)$，乘积 $a_0a_1\cdots a_k$ 为零。证明 $P(x)$ 有非实根。