Olympiad problems

2014 Iran MO (3rd Round), 3

Let $p,q\in \mathbb{R}[x]$ such that $p(z)q(\overline{z})$ is always a real number for every complex number $z$. Prove that $p(x)=kq(x)$ for some constant $k \in \mathbb{R}$ or $q(x)=0$. [i]Proposed by Mohammad Ahmadi[/i]

2004 Miklós Schweitzer, 8

Tags: complex number , complex analysis , real analysis

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 $z_1,\ldots, z_n$.

1978 Miklós Schweitzer, 5

Tags: complex analysis , function , algebra , polynomial , rational function

Suppose that $ R(z)= \sum_{n=-\infty}^{\infty} a_nz^n$ converges in a neighborhood of the unit circle $ \{ z : \;|z|=1\ \}$ in the complex plane, and $ R(z)=P(z) / Q(z)$ is a rational function in this neighborhood, where $ P$ and $ Q$ are polynomials of degree at most $ k$. Prove that there is a constant $ c$ independent of $ k$ such that \[ \sum_{n=-\infty} ^{\infty} |a_n| \leq ck^2 \max_{|z|=1} |R(z)|.\] [i]H. S. Shapiro, G. Somorjai[/i]

2000 IMC, 3

Tags: algebra , polynomial , complex number , complex analysis

Let $p(z)$ be a polynomial of degree $n>0$ with complex coefficients. Prove that there are at least $n+1$ complex numbers $z$ for which $p(z)\in \{0,1\}$.

1995 IMC, 9

Tags: algebra , polynomial , complex analysis

Let all roots of an $n$-th degree polynomial $P(z)$ with complex coefficients lie on the unit circle in the complex plane. Prove that all roots of the polynomial $$2zP'(z)-nP(z)$$ lie on the same circle.

1996 Miklós Schweitzer, 7

Tags: complex analysis

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.

2019 VJIMC, 4

Tags: complex analysis , real analysis , complex number

Let $D=\{ z \in \mathbb{C} : \operatorname{Im}(z) >0 , \operatorname{Re}(z) >0 \} $. Let $n \geq 1 $ and let $a_1,a_2,\dots a_n \in D$ be distinct complex numbers. Define $$f(z)=z \cdot \prod_{j=1}^{n} \frac{z-a_j}{z-\overline{a_j}}$$ Prove that $f'$ has at least one root in $D$. [i]Proposed by Géza Kós (Lorand Eotvos University, Budapest)[/i]

1953 Miklós Schweitzer, 9

Tags: function , complex analysis

[b]9.[/b] Let $w=f(x)$ be regular in $ \left | z \right |\leq 1$. For $0\leq r \leq 1$, denote by c, the image by $f(z)$ of the circle $\left | z \right | = r$. Show that if the maximal length of the chords of $c_{1}$ is $1$, then for every $r$ such that $0\leq r \leq 1$, the maximal length of the chords of c, is not greater than $r$. [b](F. 1)[/b]

2008 IMS, 2

Tags: function , parallelogram , vector , complex number , complex analysis , geometry

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.

2000 Miklós Schweitzer, 7

Tags: function , complex analysis , topology

Let $H(D)$ denote the space of functions holomorphic on the disc $D=\{ z\colon |z|<1 \}$, endowed with the topology of uniform convergence on each compact subset of $D$. If $f(z)=\sum_{n=0}^{\infty} a_nz^n$, then we shall denote $S_n(f,z)=\sum_{k=0}^n a_kz^k$. A function $f\in H(D)$ is called [i]universal[/i] if, for every continuous function $g\colon\partial D\rightarrow \mathbb{C}$ and for every $\varepsilon >0$, there are partial sums $S_{n(j)}(f,z)$ approximating $g$ uniformly on the arc $\{ e^{it} \colon 0\le t\le 2\pi - \varepsilon\}$. Prove that the set of universal functions contains a dense $G_{\delta}$ subset of $H(D)$.

1997 VJIMC, Problem 2

Tags: complex analysis , calculus

Let $f:\mathbb C\to\mathbb C$ be a holomorphic function with the property that $|f(z)|=1$ for all $z\in\mathbb C$ such that $|z|=1$. Prove that there exists a $\theta\in\mathbb R$ and a $k\in\{0,1,2,\ldots\}$ such that $$f(z)=e^{i\theta}z^k$$for all $z\in\mathbb C$.

1977 Miklós Schweitzer, 7

Tags: complex analysis , function , group theory , galois theory , complex number

Let $ G$ be a locally compact solvable group, let $ c_1,\ldots, c_n$ be complex numbers, and assume that the complex-valued functions $ f$ and $ g$ on $ G$ satisfy \[ \sum_{k=1}^n c_k f(xy^k)=f(x)g(y) \;\textrm{for all} \;x,y \in G \ \ .\] Prove that if $ f$ is a bounded function and \[ \inf_{x \in G} \textrm{Re} f(x) \chi(x) >0\] for some continuous (complex) character $ \chi$ of $ G$, then $ g$ is continuous. [i]L. Szekelyhidi[/i]

2003 SNSB Admission, 1

Tags: complex analysis , function , calculus , derivative , complex number , inequalities

Show that if a holomorphic function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ has the property that the modulus of any of its derivatives (of any order) is everywhere dominated by $ 1, $ then $ |f(z)|\le e^{|\text{Im} (z)|} , $ for all complex numbers $ z. $

2017 IMC, 5

Tags: polynomial , complex analysis

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.

2003 SNSB Admission, 3

Tags: complex analysis , function

Let be the set $ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} . $ Show that: $ \text{(1)}\sin\in\Lambda $ $ \text{(2)}\sum_{p\in\mathbb{Z}}\frac{1}{(1+2p)^2} =\frac{\pi^2}{4} $ $ \text{(3)} f\in\Lambda\implies \left| f'(0) \right|\le 1 $

PEN P Problems, 5

Tags: 3d geometry , complex analysis , additive number theory , geometry

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

2003 SNSB Admission, 4

Tags: complex analysis

Consider $ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} . $ Prove that $ g\in\Lambda $ implies $ g'\in\Lambda . $

2006 Iran MO (3rd Round), 6

Tags: polynomial , function , complex analysis , domain , algebra

$P,Q,R$ 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 $P,Q,R\in\mathbb C[x]$?

2000 IMC, 2

Tags: complex analysis

Let $p(x)=x^5+x$ and $q(x)=x^5+x^2$, Find al pairs $(w,z)\in \mathbb{C}\times\mathbb{C}$, $w\not=z$ for which $p(w)=p(z),q(w)=q(z)$.

1962 Miklós Schweitzer, 8

Tags: function , complex analysis

Denote by $ M(r,f)$ the maximum modulus on the circle $ |z|\equal{}r$ of the transcendent entire function $ f(z)$, and by $ M_n(r,f)$ that of the $ nth$ partial sum of the power series of $ f(z)$. Prove that the existence of an entire function $ f_0(z)$ and a corresponding sequence of positive numbers $ r_1<r_2<...\rightarrow \plus{}\infty$ such that \[ \limsup_{n\rightarrow\infty} \frac{M_n(r_n,f_0)}{M(r_n,f_0)}\equal{}\plus{}\infty\] [P. Turan]

2020 Jozsef Wildt International Math Competition, W22

Tags: calculus , integration , complex analysis , complex number

Prove that $$\operatorname{Re}\left(\operatorname{Li}_2\left(\frac{1-i\sqrt3}2\right)+\operatorname{Li}_2\left(\frac{\sqrt3-i}{2\sqrt3}\right)\right)=\frac{7\pi^2}{72}-\frac{\ln^23}8$$ where as usual $$\operatorname{Li}_2(z)=-\int^z_0\frac{\ln(1-t)}tdt,z\in\mathbb C\setminus[1,\infty)$$ [i]Proposed by Paolo Perfetti[/i]

2020 Miklós Schweitzer, 9

Tags: topology , complex analysis , complex number

Let $D\subseteq \mathbb{C}$ be a compact set with at least two elements and consider the space $\Omega=\bigtimes_{i=1}^{\infty} D$ with the product topology. For any sequence $(d_n)_{n=0}^{\infty} \in \Omega$ let $f_{(d_n)}(z)=\sum_{n=0}^{\infty}d_nz^n$, and for each point $\zeta \in \mathbb{C}$ with $|\zeta|=1$ we define $S=S(\zeta,(d_n))$ to be the set of complex numbers $w$ for which there exists a sequence $(z_k)$ such that $|z_k|<1$, $z_k \to \zeta$, and $f_{d_n}(z_k) \to w$. Prove that on a residual set of $\Omega$, the set $S$ does not depend on the choice of $\zeta$.

1998 Miklós Schweitzer, 3

Tags: number theory , complex analysis

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]

2025 VJIMC, 4

Tags: complex analysis , holomorphic , taylor series , root , function

Let $D = \{z\in \mathbb{C}: |z| < 1\}$ be the open unit disk in the complex plane and let $f : D \to D$ be a holomorphic function such that $\lim_{|z|\to 1}|f(z)| = 1$. Let the Taylor series of $f$ be $f(z) = \sum_{n=0}^{\infty} a_nz^n$. Prove that the number of zeroes of $f$ (counted with multiplicities) equals $\sum_{n=0}^{\infty} n|a_n|^2$.

2014 IMS, 12

Tags: complex analysis

Let $U$ be an open subset of the complex plane $\mathbb{C}$ including $\mathbb{D}=\{z \in \mathbb{C} : |z| \le 1\}$ and $f$ be analytic over $U$. Prove that if for every $z$ with a complex norm equal to $1$($|z|=1$) we have $0<Re(\bar{z}f(z))$, then $f$ has only one root in $\mathbb{D}$ and that's simple.