Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function. Define $\tilde{f}:[0,1]\rightarrow\mathbb{R}$ as $$\tilde{f}(x)=\begin{dcases} \frac{1}{x}\cdot\int_0^x f(t)dt &\text{ for } x>0;\\ f(0) &\text{ for }x=0.\end{dcases}$$ Show that: [list=a] [*] $\tilde{f}$ is continuous in $0$ and differentiable on $(0,1]$. [*] the following equality takes place: $$\int_0^1 f^2(x)dx = \left(\int_0^1 f(x)dx\right)^2 + \int_0^1 \left(f(x)-\tilde{f}(x)\right)^2dx.$$

Find $$ \inf_{\substack{ n\ge 1 \\ a_1,\ldots ,a_n >0 \\ a_1+\cdots +a_n <\pi }} \left( \sum_{j=1}^n a_j\cos \left( a_1+a_2+\cdots +a_j \right)\right) . $$

Let $f:[1,\infty)\to(0,\infty)$ be a non-increasing function such that $$\limsup_{n\to\infty}\frac{f(2^{n+1})}{f(2^n)}<\frac12.$$Prove that $$\int^\infty_1f(x)\text dx<\infty.$$

Let $ I $ be a real interval, and $ f:I\longrightarrow\mathbb{R} $ be a continuous function. Prove that $ f $ is monotone if and only if $ \min(\left( f(a),f(b)\right) \le\frac{1}{b-a}\int_a^b f(x)dx \le\max\left( f(a),f(b) \right) , $ for any distinct $ a,b\in I. $

Let $ a < c < b$ be three real numbers and let $ f: [a,b]\rightarrow \mathbb{R}$ be a continuos function in $ c$. If $ f$ has primitives on each of the intervals $ [a,c)$ and $ (c,b]$, then prove that it has primitives on the interval $ [a,b]$.

Let $ \{f_n \}_{n=0}^{\infty}$ be a uniformly bounded sequence of real-valued measurable functions defined on $ [0,1]$ satisfying \[ \int_0^1 f_n^2=1.\] Further, let $ \{ c_n \}$ be a sequence of real numbers with \[ \sum_{n=0}^{\infty} c_n^2= +\infty.\] Prove that some re-arrangement of the series $ \sum_{n=0}^{\infty} c_nf_n$ is divergent on a set of positive measure. [i]J. Komlos[/i]

Let $ f: \mathbb R \to \mathbb R$ be a function, two times derivable on $ \mathbb R$ for which there exist $ c\in\mathbb R$ such that \[ \frac { f(b)\minus{}f(a) }{b\minus{}a} \neq f'(c) ,\] for all $ a\neq b \in \mathbb R$. Prove that $ f''(c)\equal{}0$.

Let us call a continuous function $ f : [a,b] \rightarrow \mathbb{R}^2 \;\textit{reducible}$ if it has a double arc (that is, if there are $ a \leq \alpha < \beta \leq \gamma < \delta \leq b$ such that there exists a strictly monotone and continuous $ h : [\alpha,\beta] \rightarrow [\gamma,\delta]$ for which $ f(t)\equal{}f(h(t))$ is satisfied for every $ \alpha \leq t \leq \beta$); otherwise $ f$ is irreducible. Construct irreducible $ f : [a,b] \rightarrow \mathbb{R}^2$ and $ g : [c,d] \rightarrow \mathbb{R}^2$ such that $ f([a,b])\equal{}g([c,d])$ and (a) both $ f$ and $ g$ are rectifiable but their lengths are different; (b) $ f$ is rectifiable but $ g$ is not. [i]A. Csaszar[/i]

Study the convergence of a sequence $ \left( x_n\right)_{n\ge 0} $ for which $ x_0\in\mathbb{R}\setminus\mathbb{Q} , $ and $ x_{n+1}\in \left\{ \frac{x_n+1}{x_n} , \frac{x_n+2}{2x_n-1}\right\} , $ for all $ n\ge 1. $

Let $\sum_{n=1}^\infty a_n$ be a convergent series of positive terms (so $a_i>0$ for all $i$) and set $b_n=\frac1{na_n^2}$ for $n\ge1$. Prove that $\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n}$ is convergent.

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]

Determine all continuous, strictly monotone functions $\phi : \mathbb{R}^+\to\mathbb{R}$ such that $$F(x,y)=\phi^{-1} \left(\frac{x\phi(x)+y\phi(y)}{x+y}\right) + \phi^{-1} \left(\frac{y\phi(x)+x\phi(y)}{x+y}\right) $$ is homogeneous of degree 1, ie $F(tx,ty)=tF(x,y) , \forall x,y,t\in\mathbb{R}^+$ [hide=Note]F(x,y)=F(y,x) and F(x,x)=2x[/hide]

Let $ f(x)\equal{}\sum_{n\equal{}1}^{\infty} a_n/(x\plus{}n^2), \;(x \geq 0)\ ,$ where $ \sum_{n\equal{}1}^{\infty} |a_n|n^{\minus{} \alpha} < \infty$ for some $ \alpha > 2$. Let us assume that for some $ \beta > 1/{\alpha}$, we have $ f(x)\equal{}O(e^{\minus{}x^{\beta}})$ as $ x \rightarrow \infty$. Prove that $ a_n$ is identically $ 0$. [i]G. Halasz[/i]

Show that for all $n \in \mathbb{N}$, \[n = \sum^{}_{d \vert n}\phi(d).\]

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function that admits a primitive $ F. $ [b]a)[/b] Show that there exists a real number $ c $ such that $ f(c)-F(c)>1 $ if $ \lim_{x\to\infty } \frac{1+F(x)}{e^x} =-\infty . $ [b]b)[/b] Prove that there exists a real number $ c' $ such that $ f(c') -(F(c'))^2<1. $ [i]Cristinel Mortici[/i]

Prove that for any interval $[a,b]$ of real numbers and any positive integer $n$ there exists a positive integer $k$ and a partition of the given interval \[a = x (0) < x (1) < x (2) < \cdots < x (k-1) < x (k) = b\] such that \[\int_{x(0)}^{x(1)}f(x)dx+\int_{x(2)}^{x(3)}f(x)dx+\cdots=\int_{x(1)}^{x(2)}f(x)dx+\int_{x(3)}^{x(4)}f(x)dx+\cdots\] for all polynomials $f$ with real coefficients and degree less than $n$.

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers such that $a_0=0$ and $$a_{n+1}^3=a_n^2-8\quad \text{for} \quad n=0,1,2,…$$ Prove that the following series is convergent: $$\sum_{n=0}^{\infty}{|a_{n+1}-a_n|}.$$ [i]Proposed by Orif Ibrogimov, National University of Uzbekistan[/i]

Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.

Let be a surjective function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that if the sequence $ \left( f\left( x_n \right) \right)_{n\ge 1} $ is convergent, then the sequence $ \left( x_n \right)_{n\ge 1} $ is convergent. Prove that it is continuous.

Let $f:[0,\infty)\to\mathbb R$ be a continuous function s.t. $\lim_{x\to\infty}\frac {f(x)}x=0$. Let $(x_n)_n$ be a sequence of positive real numbers s.t. $\left(\frac{x_n}n\right)_n$ is bounded. Prove that $\lim_{n\to\infty}\frac{f(x_n)}n=0$. [i]Dorin Andrica, Eugen Paltanea[/i]

(a) Let $p>1$ a real number. Find a real constant $c_p$ for which the following statement holds: If $f: [-1,1]\rightarrow\mathbb{R}$ is a continuously differentiable function with $f(1)>f(-1)$ and $|f'(y)|\le1 \forall y\in[-1,1]$, then $\exists x\in[-1,1]: f'(x)>0$ so that $\forall y\in[-1,1]: |f(y)-f(x)|\le c_p\sqrt[p]{f'(x)}|y-x|$. (b) What if $p=1$?

Let $ p \geq 1$ be a real number and $ \mathbb{R}_\plus{}\equal{}(0, \infty)$. For which continuous functions $ g : \mathbb{R}_\plus{} \rightarrow \mathbb{R}_\plus{}$ are following functions all convex? \[ M_n(x)\equal{}\left[ \frac{\sum_{i\equal{}1}^n g(\frac{x_i}{x_{i\plus{}1}}) x_{i\plus{}1}^p}{\sum_{i\equal{}1}^n g(\frac{x_i}{x_{i\plus{}1}})} \right ]^\frac 1p ,\] \[ x\equal{}(x_1,\ldots, x_{n\plus{}1}) \in \mathbb{R}_\plus{} ^ {n\plus{}1} , \; n\equal{}1,2,\ldots\] [i]L. Losonczi[/i]

Let $D=\{ (x,y) \mid x>0, y\neq 0\}$ and let $u\in C^1(\overline {D})$ be a bounded function that is harmonic on $D$ and for which $u=0$ on the $y$-axis. Prove that $u$ is identically zero. (translated by Miklós Maróti)

Let $ 0 \leq c \leq 1$, and let $ \eta$ denote the order type of the set of rational numbers. Assume that with every rational number $ r$ we associate a Lebesgue-measurable subset $ H_r$ of measure $ c$ of the interval $ [0,1]$. Prove the existence of a Lebesgue-measurable set $ H \subset [0,1]$ of measure $ c$ such that for every $ x \in H$ the set \[ \{r : \;x \in H_r\ \}\] contains a subset of type $ \eta$. [i]M. Laczkovich[/i]

[b]3.[/b] Is there a real-valued function $Af$, defined on the space of the functions, continuous on $[0,1]$, such that $f(x)\leq g(x) $ and$f(x)\not\equiv g(x) $ inply $Af< Ag$? Is this also true if the functions $f(x)$ are required to be monotonically increasing (rather than continuous) on $[0,1]$? [b](R.4)[/b]