a) Evaluate \[\lim_{n\to \infty} \underbrace{\sqrt{a+\sqrt{a+\ldots+\sqrt{a+\sqrt{b}}}}}_{n\ \text{square roots}}\] with $a,b>0$. b)Let $(a_n)_{n\ge 1}$ and $(x_n)_{n\ge 1}$ such that $a_n>0$ and \[x_n=\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}},\ \forall n\in \mathbb{N}^*\] Prove that: 1) $(x_n)_{n\ge 1}$ is bounded if and only if $(a_n)_{n\ge 1}$ is bounded. 2) $(x_n)_{n\ge 1}$ is convergent if and only if $(a_n)_{n\ge 1}$ is convergent. [i]Valentin Matrosenco[/i]

Let be a real number $ a $ and a function $ f:[a,\infty )\longrightarrow\mathbb{R} $ that is continuous at $ a. $ Prove that $ f $ is primitivable on $ (a,\infty ) $ if and only if $ f $ is primitivable on $ [a,\infty ) . $

Let $\clubsuit\left(x\right)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit\left(8\right)=8$ and $\clubsuit\left(123\right)=1+2+3=6$. For how many two-digit values of $x$ is $\clubsuit\left(\clubsuit\left(x\right)\right)=3$?

Prove that if the function $ f : \mathbb{R}^2 \rightarrow [0,1]$ is continuous and its average on every circle of radius $ 1$ equals the function value at the center of the circle, then $ f$ is constant. [i]V. Totik[/i]

A "letter $ T$" erected at point $ A$ of the $ x$-axis in the $ xy$-plane is the union of a segment $ AB$ in the upper half-plane perpendicular to the $ x$-axis and a segment $ CD$ containing $ B$ in its interior and parallel to the $ x$-axis. Show that it is impossible to erect a letter $ T$ at every point of the $ x$-axis so that the union of those erected at rational points is disjoint from the union of those erected at irrational points. [i]A.Csaszar[/i]

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 $a,\ b>0$ be real numbers, $n\geq 2$ be integers. Evaluate $I_n=\int_{-\infty}^{\infty} \frac{exp(ia(x-ib))}{(x-ib)^n}dx.$

Let $f:[0,1]\to\mathbb{R}$ be a continuous function. Prove that \[\lim_{n\to\infty}\int_0^1 f(x^n) \ dx=f(0).\]Furthermore, if $f(0)=0$ and $f$ is right-differentiable in $0{}$, prove that the limits \[\lim_{\varepsilon\to0}\int_\varepsilon^1\frac{f(x)}{x} \ dx\quad\text{and}\quad\lim_{n\to\infty}\left(n\int_0^1f(x^n) \ dx\right)\]exist, are finite and are equal.

Prove or disprove the following statements: (a) There exists a monotone function $f : [0, 1] \rightarrow [0, 1]$ such that for each $y \in [0, 1]$ the equation $f(x) = y$ has uncountably many solutions $x$. (b) There exists a continuously differentiable function $f : [0, 1] \rightarrow [0, 1]$ such that for each $y \in [0, 1]$ the equation $f(x) = y$ has uncountably many solutions $x$.

A function \[\text{f:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] is called contract if, for every numbers $x,y\in \text{(0,}\infty \text{)}$ we have, $\underset{n\to \infty }{\mathop{\lim }}\,\left( {{f}^{n}}\left( x \right)-{{f}^{n}}\left( y \right) \right)=0$ where ${{f}^{n}}=\underbrace{f\circ f\circ ...\circ f}_{n\ f\text{'s}}$ a) Consider \[f:\text{(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] a function contract, continue with the property that has a fixed point, that existing ${{x}_{0}}\in \text{(0,}\infty \text{) }$ there so that $f\left( {{x}_{0}} \right)={{x}_{0}}.$ Show that $f\left( x \right)>x,$ for every $x\in \text{(0,}{{x}_{0}}\text{)}\,$ and $f\left( x \right)<x$, for every $x\in \text{(}{{x}_{0}}\text{,}\infty \text{)}\,$. b) Show that the given function \[f\text{:(0,}\infty \text{) }\to \text{(0,}\infty \text{)}\] given by $f\left( x \right)=x+\frac{1}{x}$ is contracted but has no fix number.

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 $\rho$.

Suppose that $f: \mathbb{R}\rightarrow\mathbb{R}$ fulfils $\left|\sum^n_{k=1}3^k\left(f(x+ky)-f(x-ky)\right)\right|\le1$ for all $n\in\mathbb{N},x,y\in\mathbb{R}$. Prove that $f$ is a constant function.

Let bet a sequence $\left( a_n \right)_{n\ge 1} $ with $ a_1=1 $ and defined as $ a_n=\sqrt[n]{1+na_{n-1}} . $ Show that $ \left( a_n \right)_{n\ge 1} $ is convergent and determine its limit. [i]Florian Dumitrel[/i]

Let $\mathcal{F}$ be the family of all nonempty finite subsets of $\mathbb{N} \cup \{0\}.$ Find all real numbers $a$ for which the series $$\sum_{A \in \mathcal{F}} \frac{1}{\sum_{k \in A}a^k}$$ is convergent.

$ \int_0^{\pi^2/4} \frac{dx}{1+\sin\sqrt x +\cos\sqrt x} $

Let $ I $ be an open real interval, and let be two functions $ f,g:I\longrightarrow\mathbb{R} $ satisfying the identity: $$ x,y\in I\wedge x\neq y\implies\frac{f(x)-g(y)}{x-y} +|x-y|\ge 0. $$ [b]a)[/b] Prove that $ f,g $ are nondecreasing. [b]b)[/b] Give a concrete example for $ f\neq g. $

Find $ \lim_{n\to\infty} \frac{\pi}{n}\left\{\frac{1}{\sin \frac{\pi (n\plus{}1)}{4n}}\plus{}\frac{1}{\sin \frac{\pi (n\plus{}2)}{4n}}\plus{}\cdots \plus{}\frac{1}{\sin \frac{\pi (n\plus{}n)}{4n}}\right\}$

Let $f:[0, \infty) \to (0, \infty)$ be an increasing function and $g:[0, \infty) \to \mathbb{R}$ be a two times differentiable function such that $g''$ is continuous and $g''(x)+f(x)g(x) = 0, \: \forall x \geq 0.$ $\textbf{a)}$ Provide an example of such functions, with $g \neq 0.$ $\textbf{b)}$ Prove that $g$ is bounded.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a continuous function such that for any $x\in \mathbb{R}$, the limit $\lim_{h\to 0} \left|\frac{f(x+h)-f(x)}{h}\right|$ exists and it is finite. Prove that in any real point, $f$ is differentiable or it has finite one-side derivates, of the same modul, but different signs.

Is is true that if the perfect set $F\subseteq [0,1]$ is of zero Lebesgue measure then those functions in $C^1[0,1]$ which are one-to-one on $F$ form a dense subset of $C^1[0,1]$? (We use the metric $$d(f,g)=\sup_{x\in[0,1]} |f(x)-g(x)| + \sup_{x\in[0,1]} |f'(x)-g'(x)|$$ to define the topology in the space $C^1[0,1]$ of continuously differentiable real functions on $[0,1]$.)

Let p be an odd prime, A be a non-empty subset of residue classes modulo p, $f:A\to\mathbb R$. Suppose that f is not constant and satisfies $f(x) \leq \frac{f(x + h) + f(x-h)}{2}$ whenever $x,x+h,x-h\in A$. Prove that $|A| \leq \frac{p + 1}{2}$.

Let $0 < a < b$. Prove that $\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}$.

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 $x.$

Let $(x_n)_{n \in \mathbb{N}}$ be the sequence defined as $x_n = \sin(2 \pi n! e)$ for all $n \in \mathbb{N}$. Compute $\lim_{n \to \infty} x_n$.

For which values of $ z$ does the series $ \sum_{n\equal{}1}^{\infty}c_1c_2\cdots c_n z^n$ converge, provided that $ c_k>0$ and $ \sum_{k\equal{}1}^{\infty} \frac{c_k}{k}<\infty$ ?