Suppose that \( a_1 + a_2 + \dots + a_n =0 \) for real numbers \( a_1, a_2, \dots, a_n \) and \( n \geq 2\). Set \( a_{n+i}=a_i \) for \( i=1, 2, \dots \). Prove that
\[
\sum_{i=1}^n \frac{1}{a_i (a_i+a_{i+1}) (a_i+a_{i+1}+a_{i+2}) \dots (a_i+a_{i+1}+\dots+a_{i+n-2})} =0
\]
if the denominators are nonzero.
Let \[p(x)=x^n+x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_1 x_1 + a_0\] be a polynomial. Prove that if \(p(z)=0\) for a complex number \(z\), then \[ |z| \le 1+ \sqrt{\max (|a_0|,|a_1|,|a_2|,\ldots,|a_{n-2}|)}.\]
Let \[ F(x) = \sum_{n=1}^{1000} \cos (n^{3/2} x). \]
Prove that \( F \) has at least \( 80 \) zeros in the interval \( (0, 2013) \).
For a nonnegative integer n, let \(F_n(x)=\sum_{m=0}^n \frac{(-2)^m (2n-m)! \Gamma(x+1)}{m! (n-m)! \Gamma(x-m+1)}\). Find all x such that Fn(x)=0.