Problem: If
$C_0+\frac{C_1}{2}+\cdots + \frac{C_{n-1}}{n}+\frac{C_n}{n+1} =0,$
where $C_0,...,C_n$ are real constants, prove that the equation
$C_0+C_1x+\cdots +C_{n-1}x^{n-1}+C_nx^n=0$
has at least one real root between $0$ and $1$.
Source: W. Rudin, Principles of Mathematical Analysis, Chapter 5, exercise 4.