Let G be a domain (connected open set) in the $R^2$ plane whose boundary is locally connected. Prove that for every point q of the boundary of G there exists a simple arc $v_q$ in which $q\in v_q$ and $v_q\setminus\{q\}\subset G$. other questions: (i) Show that local connectedness cannot be replaced by connectedness. (ii) Show that if we replace the $R^2$ plane with $R^3$ space, the statement does not hold.

Let $f:\mathbb{R}\rightarrow (0,\infty)$ be continuous function of period $1$. Prove that for any $a\in\mathbb{R}$ $$\int_0^1\frac{f(x)}{f(x+a)}dx\geq 1.$$

Let $f\colon \mathbb{R}^1\rightarrow \mathbb{R}^2$ be a continuous function such that $f(x)=f(x+1)$ for all $x$, and let $t\in [0,\frac14]$. Prove that there exists $x\in\mathbb{R}$ such that the vector from $f(x-t)$ to $f(x+t)$ is perpendicular to the vector from $f(x)$ to $f(x+\frac12)$. (translated by Miklós Maróti)

Let $ I,J $ be two intervals, $ \varphi :J\longrightarrow\mathbb{R} $ be a continuous function whose image doesn't contain $ 0, $ and $ f,g:I\longrightarrow J $ be two differentiable functions such that $ f'=\varphi\circ f,g'=\varphi\circ g $ and such that the image of $ f-g $ contains $ 0. $ Show that $ f $ and $ g $ are the same function.

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

Let $ 0 \leq c \leq 1$, and let $ \eta$ denote the order type of the set of rational numbers. Assume that with every rational number $ r$ we associate a Lebesgue-measurable subset $ H_r$ of measure $ c$ of the interval $ [0,1]$. Prove the existence of a Lebesgue-measurable set $ H \subset [0,1]$ of measure $ c$ such that for every $ x \in H$ the set \[ \{r : \;x \in H_r\ \}\] contains a subset of type $ \eta$. [i]M. Laczkovich[/i]

Denote by $\lambda (H)$ the Lebesgue outer measure of $H\subseteq \left[ 0,1\right]$. The horizontal and vertical sections of the set $A\subseteq [0, 1]\times [ 0, 1]$ are denoted by $A^y$ and $A_x$ respectively; that is, $A^y=\{ x\in [ 0, 1] \colon (x, y) \in A\}$ and $A_x=\{ y\in [ 0, 1]\colon (x,y)\in A\}$ for all $x,y\in [0,1]$. (a) Is there a decomposition $A\cup B$ of the unit square $[0,1]\times [0,1]$ such that $A^y$ is the union of finitely many segments of total length less than $\frac12$ and $\lambda (B_x)\le \frac12$ for all $x, y\in [0,1]$? (b) Is there a decomposition $A\cup B$ of the unit square $[0,1] \times [0,1]$ such that $A^y$ is the union of finitely many segments of total length not greater than $\frac12$ and $\lambda (B_x)<\frac12$ for all $x,y\in [0,1]$?

Let $F:(1,\infty) \rightarrow \mathbb{R}$ be the function defined by $$F(x)=\int_{x}^{x^{2}} \frac{dt}{\ln(t)}.$$ Show that $F$ is injective and find the set of values of $F$.

Let $f:[0,1]\to\mathbb R$ be an increasing function. Prove that if $\int_0^1f(x)dx=\int_0^1\left(\int_0^xf(t)dt\right)dx=0$ then $f(x)=0,\forall x\in(0,1)$. [i]Mihai Piticari[/i]

Let $f:[0,1]\to\mathbb{R}$ be a function for which there exists a constant $K>0$ such that $|f(x)-f(y)|\le K|x-y|$ for all $x,y\in [0,1].$ Suppose also that for each rational number $r\in [0,1],$ there exist integers $a$ and $b$ such that $f(r)=a+br.$ Prove that there exist finitely many intervals $I_1,\dots,I_n$ such that $f$ is a linear function on each $I_i$ and $[0,1]=\bigcup_{i=1}^nI_i.$

Given a positive integer $ m$ and $ 0 < \delta <\pi$, construct a trigonometric polynomial $ f(x)\equal{}a_0\plus{} \sum_{n\equal{}1}^m (a_n \cos nx\plus{}b_n \sin nx)$ of degree $ m$ such that $ f(0)\equal{}1, \int_{ \delta \leq |x| \leq \pi} |f(x)|dx \leq c/m,$ and $ \max_{\minus{}\pi \leq x \leq \pi}|f'(x)| \leq c/{\delta}$, for some universal constant $ c$. [i]G. Halasz[/i]

Find the number of extrema of the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(x)=\prod_{j=1}^n (x-j)^j, $$ where $ n $ is a natural number.

Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]

Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+y)=f(x+f(y))$ for all $x,y\in\mathbb{R}$.

[b]3.[/b] Is there a real-valued function $Af$, defined on the space of the functions, continuous on $[0,1]$, such that $f(x)\leq g(x) $ and$f(x)\not\equiv g(x) $ inply $Af< Ag$? Is this also true if the functions $f(x)$ are required to be monotonically increasing (rather than continuous) on $[0,1]$? [b](R.4)[/b]

Let $f: [0,1]\rightarrow \mathbb{R}$ continuous. We say that $f$ crosses the axis at $x$ if $f(x)=0$ but $\exists y,z \in [x-\epsilon,x+\epsilon]: f(y)<0<f(z)$ for any $\epsilon$. (a) Give an example of a function that crosses the axis infinitely often. (b) Can a continuous function cross the axis uncountably often?

A sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is called "Devin" if for any $f\in C[0,1]$ $$ \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n f(x_i)=\int_0^1 f(x)\,dx $$ Prove that a sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is "Devin" if and only if for any non-negative integer $k$ it holds $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n x_i^k=\frac{1}{k+1}.$$ [b]Remark[/b]. I left intact the text as it was proposed. Devin is a Bulgarian city and SPA resort, where this competition took place.

Let $ \mathbb{Q}$ and $ \mathbb{R}$ be the set of rational numbers and the set of real numbers, respectively, and let $ f : \mathbb{Q} \rightarrow \mathbb{R}$ be a function with the following property. For every $ h \in \mathbb{Q} , \;x_0 \in \mathbb{R}$, \[ f(x\plus{}h)\minus{}f(x) \rightarrow 0\] as $ x \in \mathbb{Q}$ tends to $ x_0$. Does it follow that $ f$ is bounded on some interval? [i]M. Laczkovich[/i]

Prove that there exists an unique sequence $ \left( c_n \right)_{n\ge 1} $ of real numbers from the interval $ (0,1) $ such that$$ \int_0^1 \frac{dx}{1+x^m} =\frac{1}{1+c_m^m } , $$ for all natural numbers $ m, $ and calculate $ \lim_{k\to\infty } kc_k^k. $ [i]Radu Pop[/i]

Evaluate $\lim_{x\to 1^-}\prod_{n=0}^{\infty}\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$.

Let $(a_n)_{n\ge1}$ be a positive real sequence such that $$\lim_{n\to\infty}\frac{a_n}n=a\in\mathbb R^*_+\enspace\text{and}\enspace\lim_{n\to\infty}\left(\frac{a_{n+1}}{a_n}\right)^n=b\in\mathbb R^*_+$$ Compute $$\lim_{n\to\infty}(a_{n+1}-a_n)$$ [i]Proposed by D.M. Bătinețu-Giurgiu and Neculai Stanciu[/i]

Suppose that ${f: \Bbb{R} \rightarrow \Bbb{R}}$ is an additive function $($that is ${f(x+y) = f(x)+f(y)}$ for all ${x, y \in \Bbb{R}})$ for which ${x \mapsto f(x)f(\sqrt{1-x^2})}$ is bounded of some nonempty subinterval of ${(0,1)}$. Prove that ${f}$ is continuous. [i]Proposed by Zoltán Boros[/i]

Let $(X,d)$ be a metric space and $f:X \to X$ be a function such that $\forall x,y\in X : d(f(x),f(y))=d(x,y)$. $\text{a})$ Prove that for all $x \in X$, $\lim_{n \rightarrow +\infty} \frac{d(x,f^n(x))}{n}$ exists, where $f^n(x)$ is $\underbrace{f(f(\cdots f(x)}_{n \text{times}} \cdots ))$. $\text{b})$ Prove that the amount of the limit does [b][u]not[/u][/b] depend on choosing $x$.

Let be two real numbers $ a<b $ and a differentiable function $ f:[a,b]\longrightarrow\mathbb{R} $ that has a bounded derivative. Show that if $ \frac{f(b)-f(a)}{b-a} $ is equal to the global supremum or infimum of $ f', $ then $ f $ is polynomial with degree $ 1. $ [i]Cătălin Țigăeru[/i]

Let $a_{n+1} = a_n + \frac{1}{{a_n}^{2005}}$ and $a_1=1$. Show that $\sum^{\infty}_{n=1}{\frac{1}{n a_n}}$ converge.