Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having a positive period $ T. $ Prove that: $$ \lim_{n\to\infty } e^{-nT}\int_0^{nT} e^tf(t)dt=\frac{1}{e^T-1}\int_0^T e^tf(t)dt $$

Let $(X,d)$ be a nonempty connected metric space such that the limit of every convergent sequence, is a term of that sequence. Prove that $X$ has exactly one element.

Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$.

Let $ \left( c_n \right)_{n\ge 1} $ be a sequence of real numbers. Prove that the sequences $ \left( c_n\sin n \right)_{n\ge 1} ,\left( c_n\cos n \right)_{n\ge 1} $ are both convergent if and only if $ \left( c_n \right)_{n\ge 1} $ converges to $ 0. $ [i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]

[b]a)[/b] Let $ \left( x_n \right)_{n\ge 1} $ be a sequence of real numbers having the property that $ \left| x_{n+1} -x_n \right|\leqslant 1/2^n, $ for any $ n\geqslant 1. $ Show that $ \left( x_n \right)_{n\ge 1} $ is convergent. [b]b)[/b] Create a sequence $ \left( y_n \right)_{n\ge 1} $ of real numbers that has the following properties: $ \text{(i) } \lim_{n\to\infty } \left( y_{n+1} -y_n \right) = 0 $ $ \text{(ii) } $ is bounded $ \text{(iii) } $ is divergent [i]Eugen Popa[/i]

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 $n.$

Let $f:[-1,1]\to\mathbb{R}$ be a continuous function having finite derivative at $0$, and \[I(h)=\int^h_{-h}f(x)\text{ d}x,\ h\in [0,1].\] Prove that a) there exists $M>0$ such that $|I(h)-2f(0)h|\le Mh^2$, for any $h\in [0,1]$. b) the sequence $(a_n)_{n\ge 1}$, defined by $a_n=\sum_{k=1}^n\sqrt{k}|I(1/k)|$, is convergent if and only if $f(0)=0$. [i]Calin Popescu[/i]

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

Let $N{}$ be a fixed positive integer, $S{}$ be the set $\{1, 2,\ldots , N\}$ and $\mathcal{F}$ be the set of functions $f:S\to S$ such that $f(i)\geqslant i$ for all $i\in S.$ For each $f\in\mathcal{F}$ let $P_f$ be the unique polynomial of degree less than $N{}$ satisfying $P_f(i) = f(i)$ for all $i\in S.$ If $f{}$ is chosen uniformly at random from $\mathcal{F}$ determine the expected value of $P_f'(0)$ where\[P_f'(0)=\frac{\mathrm{d}P_f(x)}{\mathrm{d}x}\bigg\vert_{x=0}.\][i]Proposed by Ishan Nath[/i]

Let $\alpha$ be a real number, $1<\alpha<2$. (a) Show that $\alpha$ can uniquely be represented as the infinte product \[ \alpha = \left(1+\dfrac1{n_1}\right)\left(1+\dfrac1{n_2}\right)\cdots \] with $n_i$ positive integers satisfying $n_i^2\le n_{i+1}$. (b) Show that $\alpha\in\mathbb{Q}$ iff from some $k$ onwards we have $n_{k+1}=n_k^2$.

[b]8.[/b] Let $(a_n)_{n=1}^{\infty}$ be a sequence of positive numbers and suppose that $\sum_{n=1}^{\infty} a_n^2$ is divergent. Let further $0<\epsilon<\frac{1}{2}$. Show that there exists a sequence $(b_n)_{n=1}^{\infty}$ of positive numbers such that $\sum_{n=1}^{\infty}b_n^2$ is convergent and $\sum_{n=1}^{N}a_n b_n >(\sum_{n=1}^{N}a_n^2)^{\frac{1}{2}-\epsilon}$ for every positive integer $N$. [b](S. 8)[/b]

Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R}_{\ge 0} $ that admits primitives and such that $ \lim_{x\to 0 } \frac{f(x)}{x} =0. $ Prove that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} , $ defined as $$ g(x)=\left\{ \begin{matrix} f(x)/x ,&\quad x\neq 0\\ 0,& \quad x=0 \end{matrix} \right. , $$ is primitivable.

$f: (0,1) \rightarrow [0, \infty)$ is zero except at a countable set of points $a_{1}, a_2, a_3, ... $ . Let $b_n = f(a_n)$. Show that if $\sum b_{n}$ converges, then $f$ is differentiable at at least one point. Show that for any sequence $b_{n}$ of non-negative reals with $\sum b_{n} =\infty$ , we can find a sequence $a_{n}$ such that the function $f$ defined as above is nowhere differentiable.

Let be a real function that has the intermediate value property and is monotone on the irrationals. Show that it's continuous. [i]Mihai Piticari[/i]

For each $n\geq 1$ let $$a_{n}=\sum_{k=0}^{\infty}\frac{k^{n}}{k!}, \;\; b_{n}=\sum_{k=0}^{\infty}(-1)^{k}\frac{k^{n}}{k!}.$$ Show that $a_{n}\cdot b_{n}$ is an integer.

A sequence of real numbers $x_1,x_2,\ldots ,x_n$ is given such that $x_{i+1}=x_i+\frac{1}{30000}\sqrt{1-x_i^2},\ i=1,2,\ldots ,$ and $x_1=0$. Can $n$ be equal to $50000$ if $x_n<1$?

[b]a)[/b] Let $ h:\mathbb{R}\longrightarrow\mathbb{R} $ he a function that admits a primitive $ H $ such that the function $ h/H $ is constant. Prove that there is a real number $ \gamma $ such that $ h(x)=\gamma\cdot\exp \left( x\cdot\frac{h}{H} (x) \right) , $ for any real number $ x. $ [b]b)[/b] Find the functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ that admit the primitives $ F,G, $ respectively, that satisfy $ f=\frac{G+g}{2},g=\frac{F+f}{2} $ and $ f(0)=g(0)=0. $

Let $e_1,\ldots, e_n$ be semilines on the plane starting from a common point. Prove that if there is no $u\not\equiv 0$ harmonic function on the whole plane that vanishes on the set $e_1\cup \cdots \cup e_n$, then there exists a pair $i,j$ of indices such that no $u\not\equiv 0$ harmonic function on the whole plane exists that vanishes on $e_i\cup e_j$.

Let $ g : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $ x+g(x)$ is strictly monotone (increasing or decreasing), and let $ u : [0,\infty) \rightarrow \mathbb{R}$ be a bounded and continuous function such that \[ u(t)+ \int_{t-1}^tg(u(s))ds\] is constant on $ [1,\infty)$. Prove that the limit $ \lim_{t\rightarrow \infty} u(t)$ exists. [i]T. Krisztin[/i]

Let be a real number $ r $ and two functions $ f:[r,\infty )\longrightarrow\mathbb{R} , F_1:(r,\infty )\longrightarrow\mathbb{R} $ satisfying the following two properties. $ \text{(i)} f $ has Darboux's intermediate value property. $ \text{(ii)} F_1$ is differentiable and $ F'_1=f\bigg|_{(r,\infty )} $ [b]1)[/b] Provide an example of what $ f,F_1 $ could be if $ f $ hasn't a lateral limit at $ r, $ and $ F_1 $ has lateral limit at $ r. $ Moreover, if $ f $ has lateral limit at $ r, $ show that [b]2)[/b] $ F_1 $ has a finite lateral limit at $ r. $ [b]3)[/b] the function $ F:[r,\infty )\longrightarrow\mathbb{R} $ defined as $$ F(x)=\left\{ \begin{matrix} F_1(x) ,& \quad x\in (r,\infty ) \\ \lim_{\stackrel{x\to r}{x>r}} F_1(x), & \quad x=r \end{matrix} \right. $$ is a primitive of $ f. $

Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of real numbers of the interval $ [1,\infty) . $ Suppose that the sequence $ \left( \left[ x_n^k\right]\right)_{n\ge 1} $ is convergent for all natural numbers $ k. $ Prove that $ \left( x_n\right)_{n\ge 1} $ is convergent. Here, $ [\beta ] $ means the greatest integer smaller than $ \beta . $

Let $\{x_n\}$ be a sequence of real numbers in the interval $[0,1)$. Prove that there exists a sequence $1 < n_1 < n_2 < n_3 < \cdots$ of positive integers such that the following limit exists $$\lim_{i,j \to \infty} x_{n_i+n_j}. $$ That is, there exists a real number $L$ such that for every $\epsilon > 0,$ there exists a positive integer $N$ such that if $i,j > N$, then $|x_{n_i+n_j}-L| < \epsilon.$

Define $E\subseteq \{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$ such that $E$ posseses the following properties:\\ $\textbf{(i)}$ If $\int_0^1 f(x)g(x) dx = 0$ for $f\in E$ with $\int_0^1f^2(t)dt \neq 0$, then $g\in E$; \\ $\textbf{(ii)}$ There exists $h\in E$ with $\int_0^1 h^2(t)dt\neq 0$.\\ Prove that $E=\{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$. \\ [i](Andrei Bâra)[/i]

Let $ F $ be the primitive of a continuous function $ f:\mathbb{R}\longrightarrow (0,\infty ), $ with $ F(0)=0. $ Determine for which values of $ \lambda \in (0,1) $ the function $ \left( F^{-1}\circ \lambda F \right)/\text{id.} $ has limit at $ 0, $ and calculate it.

Let be a natural number $ k, $ and a matrix $ M\in\mathcal{M}_k(\mathbb{R}) $ having the property that $$ \det\left( I-\frac{1}{n^2}\cdot A^2 \right) +1\ge\det \left( I -\frac{1}{n}\cdot A \right) +\det \left( I +\frac{1}{n}\cdot A \right) , $$ for all natural numbers $ n. $ Prove that the trace of $ A $ is $ 0. $ [i]Nelu Chichirim[/i]