Let p be a prime and $f: Z_p \to C$ a complex valued function defined on a cyclic group of order p. Define the Fourier transform of f by the formula: $$\hat f (k) = \sum_{l = 0}^{p-1} f (l) e^{i2\pi kl / p}\qquad(k \in Z_p)$$ Show that if the combined number of zeros of f and $\hat f$ is at least p, then f is identically zero. related: [url]https://artofproblemsolving.com/community/c7h22594[/url]

Show that any positive rational number can be represented as the sum of three positive rational cubes.

Let $\displaystyle{z}$ be a complex number with $\displaystyle{\left| {z + 1} \right| > 2}$. Prove that $\displaystyle{\left| {{z^3} + 1} \right| > 1}$. [i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]

Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.

Let $ f$ be an entire function on $ \mathbb C$ and $ \omega_1,\omega_2$ are complex numbers such that $ \frac {\omega_1}{\omega_2}\in{\mathbb C}\backslash{\mathbb Q}$. Prove that if for each $ z\in \mathbb C$, $ f(z) \equal{} f(z \plus{} \omega_1) \equal{} f(z \plus{} \omega_2)$ then $ f$ is constant.

Prove that for any $0<\delta <2\pi$ there exists a number $m>1$ such that for any positive integer $n$ and unimodular complex numbers $z_1,\ldots, z_n$ with $z_1^v+\dots+z_n^v=0$ for all integer exponents $1\le v\le m$, any arc of length $\delta$ of the unit circle contains at least one of the numbers $z_1,\ldots, z_n$.

Let $ f:\mathbb{C}\longrightarrow\mathbb{C} $ be an holomorphic function which has the property that there exist three positive real numbers $ a,b,c $ such that $ |f(z)|\geqslant a|z|^b , $ for any complex numbers $ z $ with $ |z|\geqslant c. $ Prove that $ f $ is polynomial with degree at least $ \lceil b\rceil . $

Construct a holomorphic function $f(z) = \sum \limits_{n = 0} ^ \infty a_n z^n$ ( | z | <1 ) in the unit circle that can be analytically continued to all points of the unit circle except one point, and for which the sequence $\{a_n\}$ has two limit points, $\infty$ and a finite value.

Let be an holomorphic function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ having the property that $ |f(z)|\le e^{|\text{Im}(z)|} , $ for all complex numbers $ z. $ Prove that the restriction of any of its derivatives (of any order) to the real numbers is everywhere dominated by $ 1. $

Let $k$ and $n$ be positive integers with $n\geq k^2-3k+4$, and let $$f(z)=z^{n-1}+c_{n-2}z^{n-2}+\dots+c_0$$ be a polynomial with complex coefficients such that $$c_0c_{n-2}=c_1c_{n-3}=\dots=c_{n-2}c_0=0$$ Prove that $f(z)$ and $z^n-1$ have at most $n-k$ common roots.

Does there exist an entire function $F \colon \mathbb{C}\to \mathbb{C}$ such that $F$ is not zero everywhere, $|F(z)|\leq e^{|z|}$ for all $z\in \mathbb{C}$, $|F(iy)|\leq 1$ for all $y\in \mathbb{R}$, and $F$ has infinitely many real roots.

$P,Q,R$ are non-zero polynomials that for each $z\in\mathbb C$, $P(z)Q(\bar z)=R(z)$. a) If $P,Q,R\in\mathbb R[x]$, prove that $Q$ is constant polynomial. b) Is the above statement correct for $P,Q,R\in\mathbb C[x]$?

Let $a,\ b>0$ be real numbers, $n\geq 2$ be integers. Evaluate $I_n=\int_{-\infty}^{\infty} \frac{exp(ia(x-ib))}{(x-ib)^n}dx.$

Let $p$ and $q$ be complex polynomials with $\deg p>\deg q$ and let $f(z)=\frac{p(z)}{q(z)}$. Suppose that all roots of $p$ lie inside the unit circle $|z|=1$ and that all roots of $q$ lie outside the unit circle. Prove that $$\max_{|z|=1}|f'(z)|>\frac{\deg p-\deg q}2\max_{|z|=1}|f(z)|.$$

Let $K_1$ be an open disk in the complex plane whose boundary passes through the points -1 and +1, and let $K_2$ be the mirror image of $K_1$ across the real axis. Also, let $D_1 = K_1 \cap K_2$ , and let $D_2$ be the outside of $D_1$ . Suppose that the function $u_1( z )$ is harmonic on $D_1$ and continuous on its closure, $u_2(z)$ harmonic on $D_2$ (including $\infty$) and continuous on its closure, and $u_1(z) = u_2(z)$ at the common boundary of the domains $D_1$ and $D_2$ . Prove that if $u_1( x )\geq 0$ for all $-1 < x <1$, then $u_2 ( x )\geq 0$ for all $x>1$ and $x<-1$.

Let $\Gamma(s)$ denote Euler's gamma function. Construct an even entire function $F(s)$ that does not vanish everywhere, for which the quotient $F(s)/\Gamma(s)$ is bounded on the right halfplane $\{\Re(s)>0\}$.

Let $Z=\{ z_1,\ldots, z_{n-1}\}$, $n\ge 2$, be a set of different complex numbers such that $Z$ contains the conjugate of any its element. a) Show that there exists a constant $C$, depending on $Z$, such that for any $\varepsilon\in (0,1)$ there exists an algebraic integer $x_0$ of degree $n$, whose algebraic conjugates $x_1, x_2, \ldots, x_{n-1}$ satisfy $|x_1-z_1|\le \varepsilon, \ldots, |x_{n-1}-z_{n-1}|\le \varepsilon$ and $|x_0|\le \frac{C}{\varepsilon}$. b) Show that there exists a set $Z=\{ z_1, \ldots, z_{n-1}\}$ and a positive number $c_n$ such that for any algebraic integer $x_0$ of degree $n$, whose algebraic conjugates satisfy $|x_1-z_1|\le \varepsilon,\ldots, |x_{n-1}-z_{n-1}|\le \varepsilon$, it also holds that $|x_0|>\frac{c_n}{\varepsilon}$. (translated by L. Erdős)