Let $ \{ f_n \}$ be a sequence of Lebesgue-integrable functions on $ [0,1]$ such that for any Lebesgue-measurable subset $ E$ of $ [0,1]$ the sequence $ \int_E f_n$ is convergent. Assume also that $ \lim_n f_n\equal{}f$ exists almost everywhere. Prove that $ f$ is integrable and $ \int_E f\equal{}\lim_n \int_E f_n$. Is the assertion also true if $ E$ runs only over intervals but we also assume $ f_n \geq 0 ?$ What happens if $ [0,1]$ is replaced by $ [0,\plus{}\infty) ?$ [i]J. Szucs[/i]

A function $f:\mathbb R\to\mathbb R$ has the property that for every $x,y\in\mathbb R$ there exists a real number $t$ (depending on $x$ and $y$) such that $0<t<1$ and $$f(tx+(1-t)y)=tf(x)+(1-t)f(y).$$ Does it imply that $$f\left(\frac{x+y}2\right)=\frac{f(x)+f(y)}2$$ for every $x,y\in\mathbb R$?

Find the limit $$\lim_{n\to\infty}\sqrt{1+2\sqrt{1+3\sqrt{\ldots+(n-1)\sqrt{1+n}}}}.$$

Let $f:[-\pi/2,\pi/2]\to\mathbb{R}$ be a twice differentiable function which satisfies \[\left(f''(x)-f(x)\right)\cdot\tan(x)+2f'(x)\geqslant 1,\]for all $x\in(-\pi/2,\pi/2)$. Prove that \[\int_{-\pi/2}^{\pi/2}f(x)\cdot \sin(x) \ dx\geqslant \pi-2.\]

Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$. Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$

For each $x>e^e$ define a sequence $S_x=u_0,u_1,\ldots$ recursively as follows: $u_0=e$, and for $n\ge0$, $u_{n+1}=\log_{u_n}x$. Prove that $S_x$ converges to a number $g(x)$ and that the function $g$ defined in this way is continuous for $x>e^e$.

Let $ S \equal{} \{2^0,2^1,2^2,\ldots,2^{10}\}$. Consider all possible positive differences of pairs of elements of $ S$. Let $ N$ be the sum of all of these differences. Find the remainder when $ N$ is divided by $ 1000$.

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

Let $f$ and $g$ be two continuous, distinct functions from $[0,1] \rightarrow (0,+\infty)$ such that $\int_{0}^{1}f(x)dx = \int_{0}^{1}g(x)dx$ Let $y_n=\int_{0}^{1}{\frac{f^{n+1}(x)}{g^{n}(x)}dx}$, for $n\geq 0$, natural. Prove that $(y_n)$ is an increasing and divergent sequence.

Let $(x_i)$ be a decreasing sequence of positive reals, then show that: (a) for every positive integer $n$ we have $\sqrt{\sum^n_{i=1}{x_i^2}} \leq \sum^n_{i=1}\frac{x_i}{\sqrt{i}}$. (b) there is a constant C for which we have $\sum^{\infty}_{k=1}\frac{1}{\sqrt{k}}\sqrt{\sum^{\infty}_{i=k}x_i^2} \le C\sum^{\infty}_{i=1}x_i$.

Prove the following inequality holds if $\{a_n\}$ is a deceasing sequence of positive reals, and $0<\theta<\frac{\pi}{2}$. $$\left|\sum_{n=1}^{2017} a_n \cos n\theta \right| \leq \frac{\pi a_1}{\theta}$$

Consider the set $\mathbb A=\{f\in C^1([-1,1]):f(-1)=-1,f(1)=1\}$. Prove that there is no function in this function space that gives us the minimum of $S=\int_{-1}^1x^2f'(x)^2dx$. What is the infimum of $S$ for the functions of this space?

Let $S$ be the set of points inside and on the boarder of a regular haxagon with side length 1. Find the least constant $r$, such that there exists one way to colour all the points in $S$ with three colous so that the distance between any two points with same colour is less than $r$.

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 $x$.

Prove that, if a continuous function has limits at $ \pm\infty , $ and these are equal, then it touches its maximum or minimum at one point.

Let $\mathcal B$ be a family of open balls in $\mathbb R^n$ and $c<\lambda\left(\bigcup\mathcal B\right)$ where $\lambda$ is the $n$-dimensional Lebesgue measure. Show that there exists a finite family of pairwise disjoint balls $\{U_i\}^k_{i=1}\subseteq\mathcal B$ such that $$\sum_{j=1}^k\lambda(U_j)>\frac c{3^n}.$$

Let two bijective and continuous functions$f,g: \mathbb{R}\to\mathbb{R}$ such that : $\left(f\circ g^{-1}\right)(x)+\left(g\circ f^{-1}\right)(x)=2x$ for any real $x$. Show that If we have a value $x_{0}\in\mathbb{R}$ such that $f(x_{0})=g(x_{0})$, then $f=g$.

let $f:R\to R$ be a continuous function tf(t)>0 for $t\neq 0$. Prove that there exists a non-zero differentiable function $y:[0,\infty)\to R$ such that $y'(t)=f(y(t-1))\,\forall t>1$ and the roots of y are bounded.

Let $a_0=\pi /2$, and let $a_n=\sin (a_{n-1})$ for $n\ge 1$. Determine whether \[ \sum_{n=1}^{\infty}a_n^2 \] converges.

Find $$\lim_{n\to\infty}\frac1{n^3}\left(\sqrt{n^2-1^2}+\sqrt{n^2-2^2}+\ldots+\sqrt{n^2-(n-1)^2}\right).$$

Let be two distinct natural numbers $ k_1 $ and $ k_2 $ and a sequence $ \left( x_n \right)_{n\ge 0} $ which satisfies $$ x_nx_m +k_1k_2\le k_1x_n +k_2x_m,\quad\forall m,n\in\{ 0\}\cup\mathbb{N}. $$ Calculate $ \lim_{n\to\infty}\frac{n!\cdot (-1)^{1+n}\cdot x_n^2}{n^n} . $

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 $P$ be a polynomial of degree $n$ with only real zeros and real coefficients. Prove that for every real $x$ we have $(n-1)(P'(x))^2\ge nP(x)P''(x)$. When does equality occur?

Denote by $ \mathcal{H}$ a set of sequences $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}$ of real numbers having the following properties: (i) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$, then $ S'\equal{}\{s_n\}_{n\equal{}2}^{\infty}\in \mathcal{H}$; (ii) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ T\equal{}\{t_n\}_{n\equal{}1}^{\infty}$, then $ S\plus{}T\equal{}\{s_n\plus{}t_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ ST\equal{}\{s_nt_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$; (iii) $ \{\minus{}1,\minus{}1,\dots,\minus{}1,\dots\}\in \mathcal{H}$. A real valued function $ f(S)$ defined on $ \mathcal{H}$ is called a quasi-limit of $ S$ if it has the following properties: If $ S\equal{}{c,c,\dots,c,\dots}$, then $ f(S)\equal{}c$; If $ s_i\geq 0$, then $ f(S)\geq 0$; $ f(S\plus{}T)\equal{}f(S)\plus{}f(T)$; $ f(ST)\equal{}f(S)f(T)$, $ f(S')\equal{}f(S)$ Prove that for every $ S$, the quasi-limit $ f(S)$ is an accumulation point of $ S$.

Let $ C$ denote the set of functions $ f(x)$, integrable (according to either Riemann or Lebesgue) on $ (a,b)$, with $ 0\le f(x)\le1$. An element $ \phi(x)\in C$ is said to be an "extreme point" of $ C$ if it can not be represented as the arithmetical mean of two different elements of $ C$. Find the extreme points of $ C$ and the functions $ f(x)\in C$ which can be obtained as "weak limits" of extreme points $ \phi_n(x)$ of $ C$. (The latter means that $ \lim_{n\to \infty}\int_a^b \phi_n(x)h(x)\,dx\equal{}\int_a^bf(x)h(x)\,dx$ holds for every integrable function $ h(x)$.)