1. A nonzero polynomial $f(x)$ with real coefficients has the property that $f(x)=f^{\prime}(x) f^{\prime \prime}(x)$. What is the leading coefficient of $f(x)$ ?
Answer: $\frac{1}{18}$
Solution: Suppose that the leading term of $f(x)$ is $c x^{n}$, where $c \neq 0$. Then the leading terms of $f^{\prime}(x)$ and of $f^{\prime \prime}(x)$ are $c n x^{n-1}$ and $c n(n-1) x^{n-2}$, respectively, so $c x^{n}=c n x^{n-1} \cdot c n(n-1) x^{n-2}$, which implies that $n=(n-1)+(n-2)$, or $n=3$, and $c=c n \cdot c n(n-1)=18 c^{2}$, or $c=\frac{1}{18}$.