If $a_0, a_1,\ldots,a_n \in R$ and $\frac{a_0}{1}+\frac{a_1}{2}+\cdots+\frac{a_n}{n+1}=0$, then exists $x\in(0,1)$ s.t. $a_0+a_1x+\cdots+a_nx^n=0$
Prove that $\sum_{k=0}^n a_k x^k = 0$ has at least $1$ real root if $\sum_{k=0}^n \frac{a_k}{k+1} = 0$