Let be a real number $ c $ and a differentiable function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ such that $$ f(c)\neq \frac{1}{b-a}\int_a^b f(x)dx, $$ for any real numbers $ a\neq b. $ Prove that $ f'(c)=0. $ [i]Florin Rotaru[/i]

Let H be a $G_{\delta}$ subset of $\mathbb R$ whose closure has a positive Lebesgue measure. Prove that the set $H + H + H + H = \{ x + y + z + u : x , y , z , u \in H \}$ contains an interval.

Let $ f(x)$ be a nonzero, bounded, real function on an Abelian group $ G$, $ g_1,...,g_k$ are given elements of $ G$ and $ \lambda_1,...,\lambda_k$ are real numbers. Prove that if \[ \sum_{i=1}^k \lambda_i f(g_ix) \geq 0\] holds for all $ x \in G$, then \[ \sum_{i=1}^k \lambda_i \geq 0.\] [i]A. Mate[/i]

Find all differentiable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the conditions: $ \text{(i)}\quad\forall x\in\mathbb{Z} \quad f'(x) =0 $ $ \text{(ii)}\quad\forall x\in\mathbb{R}\quad f'(x)=0\implies f(x)=0 $

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$ ?

Define in the plane the sequence of vectors $v_1, v_2, \ldots$ with initial values $v_1 = (1, 0)$, $v_2 = (-1/\sqrt{2}, 1/\sqrt{2})$ and satisfying the relationship $$v_n=\frac{v_{n-1}+v_{n-2}}{\lVert v_{n-1}+v_{n-2}\rVert},$$ for $n \geq 3$. Show that the sequence is convergent and determine its limit. [b]Note:[/b] The expression $\lVert v \rVert$ denotes the length of the vector $v$.

Let be two real numbers $ a<b, $ a natural number $ n\ge 2, $ and a continuous function $ f:[a,b]\longrightarrow (0,\infty ) $ whose image contains $ 1 $ and that admits a primitive $ F:[a,b]\longrightarrow [a,b] . $ Prove that there is a real number $ c\in (a,b) $ such that $$ (\underbrace{F\circ\cdots\circ F}_{\text{n times}} )(b) -(\underbrace{F\circ\cdots\circ F}_{\text{n times}} )(a) =(f(c))^{n+1} (b-a) $$ [i]Vlad Mihaly[/i]

[b]Problem 4[/b] Find all continuously differentiable functions $ f : \mathbb{R} \rightarrow \mathbb{R} $, such that for every $a \geq 0$ the following relation holds: $$\iiint \limits_{D(a)} xf \left( \frac{ay}{\sqrt{x^2+y^2}} \right) \ dx \ dy\ dz = \frac{\pi a^3}{8} (f(a) + \sin a -1)\ , $$ where $D(a) = \left\{ (x,y,z)\ :\ x^2+y^2+z^2 \leq a^2\ , \ |y|\leq \frac{x}{\sqrt{3}} \right\}\ .$

Evaluate: $\lim_{n\to\infty} \frac{1}{2n} \ln\binom{2n}{n}$

a) Show that for each function $f:\mathbb{Q} \times \mathbb{Q} \rightarrow \mathbb{R}$, there exists a function $g:\mathbb{Q}\rightarrow \mathbb{R}$ with $f(x,y) \leq g(x)+g(y) $ for all $x,y\in \mathbb{Q}$. b) Find a function $f:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$, for which there is no function $g:\mathbb{Q}\rightarrow \mathbb{R}$ such that $f(x,y) \leq g(x)+g(y) $ for all $x,y\in \mathbb{R}$.

Prove that the function \[ f(\nu)= \int_1^{\frac{1}{\nu}} \frac{dx}{\sqrt{(x^2-1)(1-\nu^2x^2)}}\] (where the positive value of the square root is taken) is monotonically decreasing in the interval $ 0<\nu<1$. [P. Turan]

Does the set of real numbers have a well-order $\prec$ such that the intersection of the subset $\{(x,y) : x\prec y\}$ of the plane with every line is Lebesgue measurable on the line?

Let $C$ be a fixed circle, $u > 0$ be a fixed real and let $v_0 , v_1 , v_2 , \ldots$ be a sequence of positive real numbers. Two ants $A$ and $B$ walk around the perimeter of $C$ in opposite directions, starting from the same starting point. Ant $A$ has a constant speed $u$, while ant $B$ has an initial speed $v_0$. For each positive integer $n$, when the two ants collide for the $n$−th time, they change the directions in which they walk around the perimeter of $C$, with ant $A$ remaining at speed $u$ and ant $B$ stops walking at speed $v_{n-1}$ to walk at speed $v_n$. (a) If the sequence $\{v_n\}$ is strictly increasing, with $\lim_{n\rightarrow \infty} v_n = +\infty$, prove that there is exactly one point in $C$ that ant $A$ will pass "infinitely" many times. (b) Prove that there is a sequence $\{v_n\}$ with $\lim_{n\rightarrow\infty} v_n = +\infty$, such that ant $A$ will pass "infinitely" many times through all points on the circle $C$.

Let be two functions $ f,g:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ having that properties that $ f $ is continuous, $ g $ is nondecreasing and unbounded, and for any sequence of rational numbers $ \left( x_n \right)_{n\ge 1} $ that diverges to $ \infty , $ we have $$ 1=\lim_{n\to\infty } f\left( x_n \right) g\left( x_n \right) . $$ Prove that $1=\lim_{x\to\infty } f\left( x \right) g\left( x \right) . $ [i]Radu Gologan[/i]

Consider a function $f:\mathbb{R}\rightarrow \mathbb{R}$. For $x\in \mathbb{R}$ we say that $f$ is [i]increasing in $x$[/i] if there exists $\epsilon_x > 0$ such that $f(x)\geq{f(a)}$, $\forall a\in (x-\epsilon_x,x)$ and $f(x)\leq f(b)$, $\forall b\in (x,x+\epsilon_x)$. $\textbf{(a)}$ Prove that if $f$ is increasing in $x$, $\forall x\in \mathbb{R}$ then $f$ is increasing over $\mathbb{R}$. $\textbf{(b)}$ We say that $f$ is [i]increasing to the left[/i] in $x$ if there exists $\epsilon_x > 0$ such that $f(x)\geq f(a) $, $ \forall a \in (x-\epsilon_x,x)$. Provide an example of a function $f: [0,1]\rightarrow \mathbb{R}$ for which there exists an infinite set $M \subset (0,1)$ such that $f$ is increasing to the left in every point of $M$, yet $f$ is increasing over no proper subinterval of $[0,1]$.

[b]Problem 1[/b] Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be differentiable on $\mathbb{R}$. Prove that there exists $x \in [0, 1]$ such that $$\frac{4}{\pi} ( f(1) - f(0) ) = (1+x^2) f'(x) \ .$$

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.

The definite integrals between $0$ and $1$ of the squares of the continuous real functions $f(x)$ and $g(x)$ are both equal to $1$. Prove that there is a real number $c$ such that \[f(c)+g(c)\leq 2\]

Let be four real numbers. Find the polynom of least degree such that two of these numbers are some locally extreme values, and the other two are the respective points of local extrema.

Let $ f:[0,1]\longrightarrow (0,\infty ) $ be a continuous function and $ \left( b_n \right)_{n\ge 1} $ be a sequence of numbers from the interval $ (0,1) $ that converge to $ 0. $ [b]a)[/b] Demonstrate that for any fixed $ n, $ the equation $ F(x)=b_nF(1)+\left( 1-b_n\right) F(0) $ has an unique solution, namely $ x_n, $ where $ F $ is a primitive of $ f. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } \frac{x_n}{b_n} . $

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 $$\int^\infty_1f(x)\text dx<\infty.$$

Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+y)=f(x+f(y))$ for all $x,y\in\mathbb{R}$.

Find all functions $\mathbb{R}^+\rightarrow\mathbb{R}^+$ for which we have for all $x,y\in \mathbb{R}^+$ that $f(x)f(yf(x))=f(x+y)$.

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]

Prove that the set of rational-valued, multiplicative arithmetical functions and the set of complex rational-valued, multiplicative arithmetical functions form isomorphic groups with the convolution operation $ f \circ g$ defined by \[{ (f \circ g)(n)= %Error. "displatmath" is a bad command. \sum_{d|n} f(d)g(\frac nd}).\] (We call a complex number $ \textit{complex rational}$, if its real and imaginary parts are both rational.) [i]B. Csakany[/i]