i) Prove that $$ \lim_{x\to \infty}\,\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}=\frac{1}{2}$$. ii) Prove that there is a positive constant $c$ such that for every $x\in [1,\infty)$ we have $$\left|\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}-\frac{1}{2} \right| \leq \frac{c}{x}$$

For $n=1,2,\dots$ let \[S_n=\log\left(\sqrt[n^2]{1^1 \cdot 2^2 \cdot \dotsc \cdot n^n}\right)-\log(\sqrt{n}),\] where $\log$ denotes the natural logarithm. Find $\lim_{n \to \infty} S_n$.

Let \((a_n)_{n \geq 1}\) s sequence of reals such that \(\sum_{n \geq 1}{\frac{a_n}{n}}\) converges. Show that \(\lim_{n \rightarrow \infty}{\frac{1}{n} \cdot \sum_{k=1}^{n}{a_k}} = 0\)

Let $ F$ be a closed set in the $ n$-dimensional Euclidean space. Construct a function that is $ 0$ on $ F$, positive outside $ F$ , and whose partial derivatives all exist.

Prove that there exists a topological space $ T$ containing the real line as a subset, such that the Lebesgue-measurable functions, and only those, extend continuously over $ T$. Show that the real line cannot be an everywhere-dense subset of such a space $ T$. [i]A. Csaszar[/i]

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function having the property that $$ f(f(x))=f(x)-\frac{1}{4}x +1, $$ for all real numbers $ x. $ [b]a)[/b] Prove that $ f $ is increasing. [b]b)[/b] Show that the equation $ f(x)=ax $ has at least a real solution in $ x, $ for any real number $ a\ge 1. $ [b]c)[/b] Calculate $ \lim_{x\to\infty } \frac{f(x)}{x} $ supposing that it exists, it's finite, and that $ \lim_{x\to\infty } f(f(x))=\infty . $

Let $f : (a,b) \to \mathbb R$ be a function with the property that for all $x \in (a,b)$ there is a non-degenerated interval $[ a_x,b_x ]$ with $a < a_x \leq x \leq b_x < b$ such that $f$ is constant on $\left[ a_x,b_x \right]$. (a) Prove that $\textrm{Im} \, f$ is finite or numerable. (b) Find all continuous functions which have the property mentioned in the hypothesis.

[b]a)[/b] Give an example of function $ f:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ that admits a primitive $ F:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ having the property that $ F^e $ is a primitive of $ f^e. $ [b]b)[/b] Prove that there is no derivable function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that has a primitive $ G:\mathbb{R}\longrightarrow\mathbb{R} $ such that $ e^G $ is a primitive of $ e^g. $

Let $f: [0,1]\rightarrow\mathbb{R}$ be a continuous function satisfying $xf(y)+yf(x)\le 1$ for every $x,y\in[0,1]$. (a) Show that $\int^1_0 f(x)dx \le \frac{\pi}4$. (b) Find such a funtion for which equality occurs.

Let $a,b\in\mathbb R$, $a\le b$. Assume that $f:[a,b]\to[a,b]$ satisfies $f(x)-f(y)\le|x-y|$ for every $x,y\in[a,b]$. Choose an $x_1\in[a,b]$ and define $$x_{n+1}=\frac{x_n+f(x_n)}2,\qquad n=1,2,3,\ldots.$$Show that $\{x_n\}^\infty_{n=1}$ converges to some fixed point of $f$.

Let $f:\mathbb{Q}\rightarrow \mathbb{Q}$ a monotonic bijective function. a)Prove that there exist a unique continuous function $F:\mathbb{R}\rightarrow \mathbb{R}$ such that $F(x)=f(x),\ (\forall)x\in \mathbb{Q}$. b)Give an example of a non-injective polynomial function $G:\mathbb{R}\rightarrow \mathbb{R}$ such that $G(\mathbb{Q})\subset \mathbb{Q}$ and it's restriction defined on $\mathbb{Q}$ is injective.

Let $ K_n(n=1,2,\ldots)$ be periodical continuous functions of period $ 2 \pi$, and write \[ k_n(f;x)= \int_0^{2\pi}f(t)K_n(x-t)dt .\] Prove that the following statements are equivalent: (i) $ \int_0^{2\pi}|k_n(f;x)-f(x)|dx \rightarrow 0 \;(n \rightarrow \infty)$ for all $ f \in \mathcal{L}_1[0,2 \pi]$. (ii) $ k_n(f;0) \rightarrow f(0)$ for all continuous, $ 2 \pi$-periodic functions $ f$. [i]V. Totik[/i]

Let $ P\subseteq \mathbb{R}^m$ be a non-empty compact convex set and $ f: P\rightarrow \mathbb{R}_{ \plus{} }$ be a concave function. Prove, that for every $ \xi\in \mathbb{R}^m$ \[ \int_{P}\langle \xi,x \rangle f(x)dx\leq \left[\frac {m \plus{} 1}{m \plus{} 2}\sup_{x\in P}{\langle\xi,x\rangle} \plus{} \frac {1}{m \plus{} 2}\inf_{x\in P}{\langle\xi,x\rangle}\right] \cdot\int_{P}f(x)dx.\]

[b]8.[/b] Let $f(x)$ be a convex function defined on the interval $[0, \frac {1}{2}]$ with $f(0)=0$ and $f(\frac{1}{2})=1$; Let further $f(x)$ be differentiable in $(0, \frac {1}{2})$, and differentiable at $0$ and $\frac{1}{2}$ from the right and from the left, respectively. Finally, let $f'(0)>1$. Extend $f(x)$ to $[0.1]$ in the following manner: let $f(x)= f(1-x)$ if $x \in (\frac {1} {2}, 1]$. Show that the set of the points $x$ for shich the terms of the sequence $x_{n+1}=f(x_n)$ ($x_0=x; n = 0, 1, 2, \dots $) are not all different is everywhere dense in $[0,1]$; [b](R. 10)[/b]

$K(x, y), f(x)$ and $g(x)$ are positive and continuous for $x, y \subseteq [0, 1]$. $\int_{0}^{1} f(y) K(x, y) dy = g(x)$ and $\int_{0}^{1} g(y) K(x, y) dy = f(x)$ for all $x \subseteq [0, 1]$. Show that $f = g$ on $[0, 1]$.

Suppose that the function $ g : (0,1) \rightarrow \mathbb{R}$ can be uniformly approximated by polynomials with nonnegative coefficients. Prove that $ g$ must be analytic. Is the statement also true for the interval $ (\minus{}1,0)$ instead of $ (0,1)$? [i]J. Kalina, L. Lempert[/i]

An arithmetic function is a real-valued function whose domain is the set of positive integers. Define the convolution product of two arithmetic functions $ f$ and $ g$ to be the arithmetic function $ f * g$, where \[ (f * g)(n) \equal{} \sum_{ij\equal{}n} f(i) \cdot g(j),\] and $ f^{*k} \equal{} f * f * \ldots * f$ ($ k$ times) We say that two arithmetic functions $ f$ and $ g$ are dependent if there exists a nontrivial polynomial of two variables $ P(x, y) \equal{} \sum_{i,j} a_{ij} x^i y^j$ with real coefficients such that \[ P(f,g) \equal{} \sum_{i,j} a_{ij} f^{*i} * g^{*j} \equal{} 0,\] and say that they are independent if they are not dependent. Let $ p$ and $ q$ be two distinct primes and set \[ f_1(n) \equal{} \begin{cases} 1 & \text{ if } n \equal{} p, \\ 0 & \text{ otherwise}. \end{cases}\] \[ f_2(n) \equal{} \begin{cases} 1 & \text{ if } n \equal{} q, \\ 0 & \text{ otherwise}. \end{cases}\] Prove that $ f_1$ and $ f_2$ are independent.

For a given irrational number $\alpha$ , $y_{1,\alpha} = \alpha$. If $y_{n-1, \alpha}$ is given, let $y_{n, \alpha}$ be the first member of the sequence $\big (\{k \alpha \} \big) ^ \infty_{k = 1}$ to fall in the interval $(0, y_{n-1,\alpha})$ ({ x } denotes the fraction of the number x ). Show that there exists an open set $G\subset (0,1)$ , which has a limit point 0 and for all irrational $\alpha$ , infinitely many members of the $(y_{n,\alpha})$ sequence do not belong to G.

a) Prove that $\{x+y\}-\{y\}$ can only be equal to $\{x\}$ or $\{x\}-1$ for any $x,y\in \mathbb{R}$. b) Let $\alpha\in \mathbb{R}\backslash \mathbb{Q}$. Denote $a_n=\{n\alpha\}$ for all $n\in \mathbb{N}^*$ and define the sequence $(x_n)_{n\ge 1}$ by \[x_n=(a_2-a_1)(a_3-a_2)\cdot \ldots \cdot (a_{n+1}-a_n)\] Prove that the sequence $(x_n)_{n\ge 1}$ is convergent and find it's limit.

Let $n \geq 2$ be an integer. Calculate$$\int \limits_{0}^{\frac{\pi}{2}}\frac{\sin x}{\sin^{2n-1}x+\cos^{2n-1}x}dx$$

Let be two real numbers $ b>a>0, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ with $ x_2>x_1>0 $ and such that $$ ax_{n+2}+bx_n\ge (a+b)x_{n+1} , $$ for any natural numbers $ n. $ Prove that $ \lim_{n\to\infty } x_n=\infty . $ [i]Dan Popescu[/i]

If $ \sum_{m=-\infty}^{+\infty} |a_m| < \infty$, then what can be said about the following expression? \[ \lim_{n \rightarrow \infty} \frac{1}{2n+1} \sum_{m=-\infty}^{+\infty} |a_{m-n}+a_{m-n+1}+...+a_{m+n}|.\] [i]P. Turan[/i]

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative, that satisfies $f\big(x+f'(x)\big)=f(x)$. Let's call this property $(P)$. a) Show that if $f$ is a function with property $(P)$, then there exists a real $x$ such that $f'(x)=0$. b) Give an example of a non-constant function $f$ with property $(P)$. c) Show that if $f$ has property $(P)$ and the equation $f'(x)=0$ has at least two solutions, then $f$ is a constant function.

Show that for every real function f in 1-period $L^2(0, 1)$ there exist three functions $g_1, g_2, g_3$ with the same properties and constants $c_0, c_1, c_2, c_3$ satisfying $$f(x)=c_0+\sum_{i=1}^3(g_i(x+c_i)-g_i(x))$$

[b]2.[/b] Omit the vertices of a closed rectangle; the configuration obtained in such a way will be called a reduced rectangle. Prove tha the set-union of any system of reduced rectangles with parallel sides is equal to the union of countably many elements of the system. [b](St. 3)[/b]