1993 APMO 第 3 题

Let

$$f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0} \quad \text { and } \quad g(x)=c_{n+1} x^{n+1}+c_{n} x^{n}+\cdots+c_{0}$$

be non-zero polynomials with real coefficients such that $g(x)=(x+r) f(x)$ for some real number $r$. If $a=\max \left(\left|a_{n}\right|, \ldots,\left|a_{0}\right|\right)$ and $c=\max \left(\left|c_{n+1}\right|, \ldots,\left|c_{0}\right|\right)$, prove that $\frac{a}{c} \leq n+1$.

让

$$

f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0} \quad \text { 和 } \quad g(x)=c_{n+1} x^{n+1}+c_{n} x^{n}+\cdots+c_{0}

$$

是具有实数系数的非零多项式，对于某个实数 $r$，$g(x)=(x+r) f(x)$。如果$a=\max \left(\left|a_{n}\right|, \ldots,\left|a_{0}\right|\right)$和$c=\max \left(\left|c_{n+1}\right|, \ldots,\left|c_{0}\right|\right)$，证明$\frac{a}{c} \leq n+1$。