[list=a] [*]Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $g\colon\mathbb{R}\rightarrow\mathbb{R}$, $g(x)=f(x)+f(2x)$, and $h\colon\mathbb{R}\rightarrow\mathbb{R}$, $h(x)=f(x)+f(4x)$, are continuous functions. Prove that $f$ is also continuous. [*]Give an example of a discontinuous function $f\colon\mathbb{R} \rightarrow\mathbb{R}$, with the following property: there exists an interval $I\subset\mathbb{R}$, such that, for any $a$ in $I$, the function $g_{a} \colon\mathbb{R}\rightarrow\mathbb{R}$, $g_{a}(x)=f(x)+f(ax)$, is continuous.[/list]

Function $f: [a,b]\to\mathbb{R}$, $0<a<b$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Prove that there exists $c\in(a,b)$ such that \[ f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}. \]

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]

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 be three polynomials of degree two $ p_1,p_2,p_3\in\mathbb{R} [X] $ and the function $$ f:\mathbb{R}\longrightarrow\mathbb{R} ,\quad f(x)=\max\left( p_1(x),p_2(x),p_3(x)\right) . $$ Then, $ f $ is differentiable if and only if any of these three polynomials dominates the other two.

Show that if n is an arbitrary natural number and $\sqrt n \leq K \leq \frac{n}{2}$, then there exist n distinct integers, $k_j$ ( j = 1, ..., n ) such that $\bigg | \sum_ {j = 1} ^ ne ^ {ik_jt} \bigg | \geq K$ is satisfied on a subset of the interval $(- \pi, \pi)$ with Lebesgue measure at least $\frac{cn}{K^2}$ , where c is a suitable absolute constant.

Let $ a(x)$ and $ r(x)$ be positive continuous functions defined on the interval $ [0,\infty)$, and let \[ \liminf_{x \rightarrow \infty} (x-r(x)) >0.\] Assume that $ y(x)$ is a continuous function on the whole real line, that it is differentiable on $ [0, \infty)$, and that it satisfies \[ y'(x)=a(x)y(x-r(x))\] on $ [0, \infty)$. Prove that the limit \[ \lim_{x \rightarrow \infty}y(x) \exp \left\{ -%Error. "diaplaymath" is a bad command. \int_0^x a(u)du \right \}\] exists and is finite. [i]I. Gyori[/i]

The continuous function and twice differentiable function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $2007^{2}\cdot f(x)+f''(x)=0$. Prove that there exist two such real numbers $k$ and $l$ such that $f(x)=l\cdot\sin(2007x)+k\cdot\cos(2007x)$.

Given a real number $x>1$, prove that there exists a real number $y >0$ such that \[\lim_{n \to \infty} \underbrace{\sqrt{y+\sqrt {y + \cdots+\sqrt y}}}_{n \text{ roots}}=x.\]

We say that a function \( f: \mathbb{R} \to \mathbb{R} \) is morally odd if its graph is symmetric with respect to a point, that is, there exists \((x_0, y_0) \in \mathbb{R}^2\) such that if \((u, v) \in \{(x, f(x)) : x \in \mathbb{R}\}\), then \((2x_0 - u, 2y_0 - v) \in \{(x, f(x)) : x \in \mathbb{R}\}\). On the other hand, \( f \) is said to be morally even if its graph \(\{(x, f(x)) : x \in \mathbb{R}\}\) is symmetric with respect to some line (not necessarily vertical or horizontal). If \( f \) is morally even and morally odd, we say that \( f \) is parimpar. (a) Let \( S \subset \mathbb{R} \) be a bounded set and \( f: S \to \mathbb{R} \) be an arbitrary function. Prove that there exists \( g: \mathbb{R} \to \mathbb{R} \) that is parimpar such that \( g(x) = f(x) \) for all \( x \in S \). (b) Find all polynomials \( P \) with real coefficients such that the corresponding polynomial function \( P: \mathbb{R} \to \mathbb{R} \) is parimpar.

Let $E \subset [0,1]$ be a Lebesgue measurable set having Lebesgue measure $| E |<\frac{1}{2}$. Let $$h (s) = \int _ {\overline {E}} \frac{dt}{{(s-t)}^2}$$ where $\overline {E} = [0,1] \backslash E$. Prove that there is one $t \in \overline {E}$ for which $$\int_E \frac {ds} {h (s) {(s-t)} ^ 2} \leq c {| E |} ^ 2$$ with some absolute constant c .

[list] $(a)$ A sequence $x_1,x_2,\dots$ of real numbers satisfies \[x_{n+1}=x_n \cos x_n \textrm{ for all } n\geq 1.\] Does it follows that this sequence converges for all initial values $x_1?$ (5 points) $(b)$ A sequence $y_1,y_2,\dots$ of real numbers satisfies \[y_{n+1}=y_n \sin y_n \textrm{ for all } n\geq 1.\] Does it follows that this sequence converges for all initial values $y_1?$ (5 points)[/list]

prove that for any sequence of nonnegative numbers $(a_n)$, $$\liminf_{n\to\infty} (n^2(4a_n(1-a_{n-1})-1))\leq\frac{1}{4}$$

Let $(a_n)_{n \ge 1}$ be a sequence of real numbers satisfying the properties: [list=1] [*] the sequence $x_n=\sum\limits_{k=1}^n a_k^2$ is convergent; [*] the sequence $y_n=\sum\limits_{k=1}^n a_k$ is unbounded. [/list] Prove that the sequence $(b_n)_{n \ge 1}$ given by $b_n=\{y_n\}$ is divergent. Note: $\{ x \}$ denotes the fractional part of $x.$

For each positive real number $r$, define $a_0(r) = 1$ and $a_{n+1}(r) = \lfloor ra_n(r) \rfloor$ for all integers $n \ge 0$. (a) Prove that for each positive real number $r$, the limit \[ L(r) = \lim_{n \to \infty} \frac{a_n(r)}{r^n} \] exists. (b) Determine all possible values of $L(r)$ as $r$ varies over the set of positive real numbers. [i]Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.[/i]

Given a real number $p>1$, a continuous function $h\colon [0,\infty)\to [0,\infty)$, and a smooth vector field $Y\colon \mathbb{R}^n \to \mathbb{R}^n$ with $\mathrm{div}~Y=0$, prove the following inequality \[\int_{\mathbb{R}^n}h(|x|)|x|^{p}\leq \int_{\mathbb{R}^{n}}h(|x|)|x+Y(x)|^{p}.\]

Consider the sequence $ \left( I_n \right)_{n\ge 1} , $ where $ I_n=\int_0^{\pi/4} e^{\sin x\cos x} (\cos x-\sin x)^{2n} (\cos x+\sin x )dx, $ for any natural number $ n. $ [b]a)[/b] Find a relation between any two consecutive terms of $ I_n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } nI_n. $ [i]c)[/i] Show that $ \sum_{i=1}^{\infty }\frac{1}{(2i-1)!!} =\int_0^{\pi/4} e^{\sin x\cos x} (\cos x+\sin x )dx. $ [i]Cătălin Țigăeru[/i]

Let $ f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable, $ 2 \pi$-periodic even function. Prove that if \[ f''(x)\plus{}f(x)\equal{}\frac{1}{f(x\plus{} 3 \pi /2 )}\] holds for every $ x$, then $ f$ is $ \pi /2$-periodic. [i]Z. Szabo, J. Terjeki[/i]

Prove that the infinite series $ 1\minus{}\frac{1}{x(x\plus{}1)}\minus{}\frac{x\minus{}1}{2!x^2(2x\plus{}1)}\minus{}\frac{(x\minus{}1)(2x\minus{}1)}{3!(x^3(3x\plus{}1))}\minus{}\frac{(x\minus{}1)(2x\minus{}1)(3x\minus{}1)}{4!x^4(4x\plus{}1)}\minus{}\cdots$ is convergent for every positive $ x$. Denoting its sum by $ F(x)$, find $ \lim_{x\to \plus{}0}F(x)$ and $ \lim_{x\to \infty}F(x)$.

Let be a sequence $ \left( s_n\right)_{n\geqslant 0} $ of positive real numbers, with $ s_0 $ being the golden ratio, and defined as $$ s_{n+2}=\frac{1+s_{n+1}}{s_n} . $$ Establish the necessary and sufficient condition under which $ \left( s_n\right)_{n\geqslant 0} $ is convergent.

Find all differentiable functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f'(x)=\frac{f(x+n)-f(x)}n\] for all real numbers $x$ and all positive integers $n.$

(a) Let $n$ is a positive integer. Calculate $\displaystyle \int_0^1 x^{n-1}\ln x\,dx$.\\ (b) Calculate $\displaystyle \sum_{n=0}^{\infty}(-1)^n\left(\frac{1}{(n+1)^2}-\frac{1}{(n+2)^2}+\frac{1}{(n+3)^2}-\dots \right).$

Sequence $\{a_n\}$ defined by recurrence relation $a_{n+1} = 1+\frac{n^2}{a_n}$. Given $a_1>1$, find the value of $\lim\limits_{n\to\infty} \frac{a_n}{n}$ with proof.

Find the limit of the sequence $x_n$ defined by recurrence relation $$x_{n+2}=\frac{1}{12}x_{n+1}+\frac{1}{2}x_{n}+1$$ where $n=0,1,2,...$ for any initial values $x_2,x_1$.

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