In the following exercise, $ C_G (e) $ denotes the centralizer of the element $ e $ in the group $ G. $ [b]a)[/b] Prove that $ \max_{\sigma\in S_n\setminus\{1\}} \left| C_{S_n} (\sigma ) \right| <\frac{n!}{2} , $ for any natural number $ n\ge 4. $ [b]b)[/b] Show that $ \lim_{n\to\infty} \left(\frac{1}{n!}\cdot\max_{\sigma\in S_n\setminus\{1\}} \left| C_{S_n} (\sigma ) \right|\right) =0. $ [i]Alexandru Cioba[/i]

Show that one cannot find compact sets $A_1, A_2, A_3, \ldots$ in $\mathbb{R}$ such that (1) All elements of $A_n$ are rational. (2) Any compact set $K\subset \mathbb{R}$ which only contains rational numbers is contained in some $A_{m}$.

Let be two nonnegative real numbers $ a,b $ with $ b>a, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ of real numbers such that the sequence $ \left( \frac{x_1+x_2+\cdots +x_n}{n^a} \right)_{n\ge 1} $ is bounded. Show that the sequence $ \left( x_1+\frac{x_2}{2^b} +\frac{x_3}{3^b} +\cdots +\frac{x_n}{n^b} \right)_{n\ge 1} $ is convergent.

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]

For any natural number $ m, \quad\lim_{n\to\infty } n^{1+m} \int_{0}^1 e^{-nx}\ln \left( 1+x^m \right) dx =m! . $ [i]Gheorghe Iurea[/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 every positive $ \alpha$, natural number $ n$, and at most $ \alpha n$ points $ x_i$, construct a trigonometric polynomial $ P(x)$ of degree at most $ n$ for which \[ P(x_i) \leq 1, \; \int_0^{2 \pi} P(x)dx=0,\ \; \textrm{and}\ \; \max P(x) > cn\ ,\] where the constant $ c$ depends only on $ \alpha$. [i]G. Halasz[/i]

[color=darkred]Let $\mathcal{C}$ be the set of integrable functions $f\colon [0,1]\to\mathbb{R}$ such that $0\le f(x)\le x$ for any $x\in [0,1]$ . Define the function $V\colon\mathcal{C}\to\mathbb{R}$ by \[V(f)=\int_0^1f^2(x)\ \text{d}x-\left(\int_0^1f(x)\ \text{d}x\right)^2\ ,\ f\in\mathcal{C}\ .\] Determine the following two sets: [list][b]a)[/b] $\{V(f_a)\, |\, 0\le a\le 1\}$ , where $f_a(x)=0$ , if $0\le x\le a$ and $f_a(x)=x$ , if $a<x\le 1\, ;$ [b]b)[/b] $\{V(f)\, |\, f\in\mathcal{C}\}\ .$[/list] [/color]

Given a set $S$ of $2n-1$, $n\in \mathbb N$, different irrational numbers. Prove that there are $n$ different elements $x_1, x_2, \ldots, x_n\in S$ such that for all non-negative rational numbers $a_1, a_2, \ldots, a_n$ with $a_1+a_2+\ldots + a_n>0$ we have that $a_1x_1+a_2x_2+\cdots +a_nx_n$ is an irrational number.

Let $ \pi_n(x)$ be a polynomial of degree not exceeding $ n$ with real coefficients such that \[ |\pi_n(x)| \leq \sqrt{1\minus{}x^2} \;\textrm{for}\ \;\minus{}1\leq x \leq 1 \ .\] Then \[ |\pi'_n(x)| \leq 2(n\minus{}1).\] [i]P. Turan[/i]

Let $ f:(0,1)\longrightarrow \mathbb{R} $ be a bounded function having the property of Darboux. Then: [b]a)[/b] There exists $ g:[0,1)\longrightarrow\mathbb{R} $ with Darboux’s property such that $ g\bigg|_{(0,1)} =f\bigg|_{(0,1)} . $ [b]b)[/b] The function above is uniquely determined if and only if $ f $ has limit at $ 0. $

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous and bounded function such that \[x\int_{x}^{x+1}f(t)\, \text{d}t=\int_{0}^{x}f(t)\, \text{d}t,\quad\text{for any}\ x\in\mathbb{R}.\] Prove that $f$ is a constant function.

Let $\rho:\mathbb{R}^n\to \mathbb{R}$, $\rho(\mathbf{x})=e^{-||\mathbf{x}||^2}$, and let $K\subset \mathbb{R}^n$ be a convex body, i.e., a compact convex set with nonempty interior. Define the barycenter $\mathbf{s}_K$ of the body $K$ with respect to the weight function $\rho$ by the usual formula \[\mathbf{s}_K=\frac{\int_K\rho(\mathbf{x})\mathbf{x}d\mathbf{x}}{\int_K\rho(\mathbf{x})d\mathbf{x}}.\] Prove that the translates of the body $K$ have pairwise distinct barycenters with respect to $\rho$.

Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admit a primitive $ F $ defined as $ F(x)=\left\{\begin{matrix} f(x)/x, & x\neq 0 \\ 2007, & x=0 \end{matrix}\right. . $

Consider the functions $f,g,h:\mathbb{R}_{\geqslant 0}\to\mathbb{R}_{\geqslant 0}$ and the binary operation $*:\mathbb{R}_{\geqslant 0}\times \mathbb{R}_{\geqslant 0}\to \mathbb{R}_{\geqslant 0}$ defined as \[x*y=f(x)+g(y)+h(x)\cdot|x-y|,\]for all $x,y\in\mathbb{R}_{\geqslant 0}$. Suppose that $(\mathbb{R}_{\geqslant 0},*)$ is a commutative monoid. Determine the functions $f,g,h$.

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.

Find the pairs of functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ with $ f $ continuous, $ g $ differentiable and satisfying: $$ -\sin g(x) + \int \cos f(x)dx =\cos g(x) +\int \sin f(x)dx $$

Find $ \lim_{n\to\infty}a_n$ where $ (a_n)_{n\ge1}$ is defined by $ a_n\equal{}\frac1{\sqrt{n^2\plus{}8n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}16n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}24n\minus{}1}}\plus{}\ldots\plus{}\frac1{\sqrt{9n^2\minus{}1}}$.

Let $ m:[0,1]\longrightarrow\mathbb{R} $ be a metric map. [b]a)[/b] Prove that $ -\text{identity} +m $ is continuous and nonincreasing. [b]b)[/b] Show that $ \int_0^1\int_0^x (-t+m(t))dtdx=\int_0^1 (x-1)(x-m(x))dx. $ [b]c)[/b] Demonstrate that $ \int_0^1\int_0^x m(t)dtdx -\frac{1}{2}\int_0^1 m(x)dx\ge -\frac{1}{12} . $ [i]Gabriela Constantinescu[/i] and [i]Nelu Chichirim[/i]

Let $K_1,...,K_d$ be convex, compact sets in $R^d$ with non-empty interior. Suppose they are strongly separated, which means for any choice of $x_1 \in K_1, x_2 \in K_2, ...$, their affine hull is a hyperplane in $R^d$. Also let $0< \alpha_i <1$. A half-space H is called an $\alpha$-cut if $vol(K_i \cap H) = \alpha_i\cdot vol(K_i)$ for all i. How many $\alpha$-cuts are there?

How many zeroes does the function $f(x)=2^x -1 -x^2 $ have on the real line?

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} ,f(x)=\left\{\begin{matrix} \sin x , & x\not\in\mathbb{Q} \\ 0, & x\in\mathbb{Q}\end{matrix}\right. . $ [b]a)[/b] Determine the maximum length of an interval $ I\subset\mathbb{R} $ such that $ f|_I $ is discontinuous everywhere, yet has the intermediate value property. [b]b)[/b] Study the convergence of the sequence $ \left( x_n\right)_{n\in\mathbb{N}\cup\{ 0\}} $ defined by $ x_0\in (0,\pi /2),x_{n+1}=f\left( x_n\right),\forall n\ge 0. $

Let $p\in(0,1)$ and $a>0$ be real numbers. Determine the asymptotic behavior of the sequence $\{a_n\}_{n=1}^\infty$ defined recursively by $$a_1=a,a_{n+1}=\frac{a_n}{1+a_n^p},n\in\mathbb N$$ [i]Proposed by Arkady Alt[/i]

Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge (z-y)\int_{x}^{z}f(u)\,\mathrm{du} \end{align*}

Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p\equal{}\displaystyle\frac{\binom p0}{2\cdot 4}\minus{}\frac{\binom p1}{3\cdot5}\plus{}\frac{\binom p2}{4\cdot6}\minus{}\ldots\plus{}(\minus{}1)^p\cdot\frac{\binom pp}{(p\plus{}2)(p\plus{}4)}$. Find $ \lim_{n\to\infty}(a_0\plus{}a_1\plus{}\ldots\plus{}a_n)$.