This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 67

2003 SNSB Admission, 4

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 . $

2007 ITest, 48

Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the maximum possible value of \[\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007},\] where, for $1\leq i\leq 2007$, $x_i$ is a nonnegative real number, and \[x_1+x_2+x_3+\cdots+x_{2007}=\pi.\] Find the value of $a+b$.

1995 Miklós Schweitzer, 1

Prove that a harmonic function that is not identically zero in the plane cannot vanish on a two-dimensional positive-measure set.

2003 SNSB Admission, 2

Let be a natural number $ n, $ denote with $ C $ the square in the complex plane whose vertices are the affixes of $ 2n\pi\left( \pm 1\pm i \right) , $ and consider the set $$ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} $$ Prove the following implications. [b]a)[/b] $ \exists \alpha\in\mathbb{R}_{>0}\quad \forall z\in\partial C\quad \left| \cos z \right|\ge\alpha e^{|\text{Im}(z)|} $ [b]b)[/b] $ \forall f\in\Lambda\quad\frac{1}{2\pi i}\int_{\partial C} \frac{f(z)}{z^2\cos z} dz=f'(0)+\frac{4}{\pi^2}\sum_{p=-2n}^{2n-1} \frac{(-1)^{p+1} f(z-p)}{(1+2p)^2} $ [b]c)[/b] $ \forall f\in\Lambda\quad \sum_{p\in\mathbb{Z}}\frac{(-1)^pf\left( \frac{(1+2p)\pi}{2} \right)}{(1+2p)^2} =\frac{\pi^2 f'(0)}{4} $

2008 Iran MO (3rd Round), 2

Let $ g,f: \mathbb C\longrightarrow\mathbb C$ be two continuous functions such that for each $ z\neq 0$, $ g(z)\equal{}f(\frac1z)$. Prove that there is a $ z\in\mathbb C$ such that $ f(\frac1z)\equal{}f(\minus{}\bar z)$

1997 VJIMC, Problem 2

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 $z\in\mathbb C$.

2014 Contests, 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]

2000 IMC, 3

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 $p(z)\in \{0,1\}$.

2005 VJIMC, Problem 1

Let $S_0=\{z\in\mathbb C:|z|=1,z\ne-1\}$ and $f(z)=\frac{\operatorname{Im}z}{1+\operatorname{Re}z}$. Prove that $f$ is a bijection between $S_0$ and $\mathbb R$. Find $f^{-1}$.

2021 IMC, 7

Let $D \subseteq \mathbb{C}$ be an open set containing the closed unit disk $\{z : |z| \leq 1\}$. Let $f : D \rightarrow \mathbb{C}$ be a holomorphic function, and let $p(z)$ be a monic polynomial. Prove that $$ |f(0)| \leq \max_{|z|=1} |f(z)p(z)| $$

2003 SNSB Admission, 3

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 $

1995 IMC, 9

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.

PEN Q Problems, 4

A prime $p$ has decimal digits $p_{n}p_{n-1} \cdots p_0$ with $p_{n}>1$. Show that the polynomial $p_{n}x^{n} + p_{n-1}x^{n-1}+\cdots+ p_{1}x + p_0$ cannot be represented as a product of two nonconstant polynomials with integer coefficients

2002 Miklós Schweitzer, 7

Let the complex function $F(z)$ be regular on the punctuated disk $\{ 0<|z| < R\}$. By a [i]level curve[/i] we mean a component of the level set of $\mathrm{Re}F(z)$, that is, a maximal connected set on which $\mathrm{Re}F(z)$ is constant. Denote by $A(r)$ the union of those level curves that are entirely contained in the punctuated disk $\{ 0<|z|<r\}$. Prove that if the number of components of $A(r)$ has an upper bound independent of $r$ then $F(z)$ can only have a pole type singularity at $0$.

2024 China Team Selection Test, 23

$P(z)=a_nz^n+\dots+a_1z+z_0$, with $a_n\neq 0$ is a polynomial with complex coefficients, such that when $|z|=1$, $|P(z)|\leq 1$. Prove that for any $0\leq k\leq n-1$, $|a_k|\leq 1-|a_n|^2$. [i]Proposed by Yijun Yao[/i]

2003 SNSB Admission, 1

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. $

2021 Alibaba Global Math Competition, 8

Let $f(z)$ be a holomorphic function in $\{\vert z\vert \le R\}$ ($0<R<\infty$). Define \[M(r,f)=\max_{\vert z\vert=r} \vert f(z)\vert, \quad A(r,f)=\max_{\vert z\vert=r} \text{Re}\{f(z)\}.\] Show that \[M(r,f) \le \frac{2r}{R-r}A(R,f)+\frac{R+r}{R-r} \vert f(0)\vert, \quad \forall 0 \le r<R.\]

1976 Miklós Schweitzer, 7

Let $ f_1,f_2,\dots,f_n$ be regular functions on a domain of the complex plane, linearly independent over the complex field. Prove that the functions $ f_i\overline{f}_k, \;1 \leq i,k \leq n$, are also linearly independent. [i]L. Lempert[/i]

2015 Miklos Schweitzer, 9

For a function ${u}$ defined on ${G \subset \Bbb{C}}$ let us denote by ${Z(u)}$ the neignborhood of unit raduis of the set of roots of ${u}$. Prove that for any compact set ${K \subset G}$ there exists a constant ${C}$ such that if ${u}$ is an arbitrary real harmonic function on ${G}$ which vanishes in a point of ${K}$ then: \[\displaystyle \sup_{z \in K} |u(z)| \leq C \sup_{Z(u)\cap G}|u(z)|.\]

2008 Iran MO (3rd Round), 3

For each $ c\in\mathbb C$, let $ f_c(z,0)\equal{}z$, and $ f_c(z,n)\equal{}f_c(z,n\minus{}1)^2\plus{}c$ for $ n\geq1$. a) Prove that if $ |c|\leq\frac14$ then there is a neighborhood $ U$ of origin such that for each $ z\in U$ the sequence $ f_c(z,n),n\in\mathbb N$ is bounded. b) Prove that if $ c>\frac14$ is a real number there is a neighborhood $ U$ of origin such that for each $ z\in U$ the sequence $ f_c(z,n),n\in\mathbb N$ is unbounded.

1949 Miklós Schweitzer, 7

Find the complex numbers $ z$ for which the series \[ 1 \plus{} \frac {z}{2!} \plus{} \frac {z(z \plus{} 1)}{3!} \plus{} \frac {z(z \plus{} 1)(z \plus{} 2)}{4!} \plus{} \cdots \plus{} \frac {z(z \plus{} 1)\cdots(z \plus{} n)}{(n \plus{} 2)!} \plus{} \cdots\] converges and find its sum.

2014 IMS, 12

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.

1979 Miklós Schweitzer, 9

Let us assume that the series of holomorphic functions $ \sum_{k=1}^{\infty}f_k(z)$ is absolutely convergent for all $ z \in \mathbb{C}$. Let $ H \subseteq \mathbb{C}$ be the set of those points where the above sum funcion is not regular. Prove that $ H$ is nowhere dense but not necessarily countable. [i]L. Kerchy[/i]

2025 VJIMC, 4

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 $\sum_{n=0}^{\infty} n|a_n|^2$.

2021 Alibaba Global Math Competition, 5

For the complex-valued function $f(x)$ which is continuous and absolutely integrable on $\mathbb{R}$, define the function $(Sf)(x)$ on $\mathbb{R}$: $(Sf)(x)=\int_{-\infty}^{+\infty}e^{2\pi iux}f(u)du$. (a) Find the expression for $S(\frac{1}{1+x^2})$ and $S(\frac{1}{(1+x^2)^2})$. (b) For any integer $k$, let $f_k(x)=(1+x^2)^{-1-k}$. Assume $k\geq 1$, find constant $c_1$, $c_2$ such that the function $y=(Sf_k)(x)$ satisfies the ODE with second order: $xy''+c_1y'+c_2xy=0$.