1992 USAMO 第 5 题

Let $\, P(z) \,$ be a polynomial with complex coefficients which is of degree $\, 1992 \,$ and has distinct zeros. Prove that there exist complex numbers $\, a_1, a_2, \ldots, a_{1992} \,$ such that $\, P(z) \,$ divides the polynomial $\left( \cdots \left( (z-a_1)^2 - a_2 \right)^2 \cdots - a_{1991} \right)^2 - a_{1992}$.

令 $\, P(z) \,$ 为具有复系数的多项式，其次数为 $\, 1992 \,$ 并且具有不同的零点。证明存在复数 $\, a_1, a_2, \ldots, a_{1992} \,$ 使得 $\, P(z) \,$ 除多项式 $\left( \cdots \left( (z-a_1)^2 - a_2 \right)^2 \cdots - a_{1991} \right)^2 - a_{1992}$。