# 2014-17 Zeros of a polynomial

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 $|z| \le 1+ \sqrt{\max (|a_0|,|a_1|,|a_2|,\ldots,|a_{n-2}|)}.$

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# 2013-05 Zeros of a cosine series

Let $F(x) = \sum_{n=1}^{1000} \cos (n^{3/2} x).$
Prove that $$F$$ has at least $$80$$ zeros in the interval $$(0, 2013)$$.

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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.