Let be a real number $ a\in (0,1) $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with the property that: $$ \lim_{x\to 0} f(x) =0= \lim_{x\to 0} \frac{f(x)-f(ax)}{x} $$ Prove that $ \lim_{x\to\infty } \frac{f(x)}{x} =0. $

Let $p$ be the [i]potentioral[/i] function, from positive integers to positive integers, defined by $p(1) = 1$ and $p(n + 1) = p(n)$, if $n + 1$ is not a perfect power and $p(n + 1) = (n + 1) \cdot p(n)$, otherwise. Is there a positive integer $N$ such that, for all $n > N,$ $p(n) > 2^n$?

[color=darkred]Let $f\colon [0,\infty)\to\mathbb{R}$ be a continuous function such that $\int_0^nf(x)f(n-x)\ \text{d}x=\int_0^nf^2(x)\ \text{d}x$ , for any natural number $n\ge 1$ . Prove that $f$ is a periodic function.[/color]

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 $f:\mathbb{R}\rightarrow \mathbb{R}$ 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.

Let $ a>0$, $ d>0$ and put $ f(x)\equal{}\frac{1}{a}\plus{}\frac{x}{a(a\plus{}d)}\plus{}\cdots\plus{}\frac{x^n}{a(a\plus{}d)\cdots(a\plus{}nd)}\plus{}\cdots$ Give a closed form for $ f(x)$.

Let $ f:[-1,1]\longrightarrow\mathbb{R} $ a derivable function and a non-negative integer $ n. $ Show that there is a $ c\in [-1,1] $ so that: $$ \int_{-1}^1 x^{2n+1} f(x)dx =\frac{2}{2n+3}f'(c). $$

A function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has property $ \mathcal{F} , $ if for any real number $ a, $ there exists a $ b<a $ such that $ f(x)\le f(a), $ for all $ x\in (b,a) . $ [b]a)[/b] Give an example of a function with property $ \mathcal{F} $ that is not monotone on $ \mathbb{R} . $ [b]b)[/b] Prove that a continuous function that has property $ \mathcal{F} $ is nondecreasing.

Compute the sum of the series $\sum_{k=0}^{\infty} \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)} = \frac{1}{1\cdot2\cdot3\cdot4} + \frac{1}{5\cdot6\cdot7\cdot8} + ...$

Let $ f: [0,1]\rightarrow \mathbb{R}$ a derivable function such that $ f(0)\equal{}f(1)$, $ \int_0^1f(x)dx\equal{}0$ and $ f^{\prime}(x) \neq 1\ ,\ (\forall)x\in [0,1]$. i)Prove that the function $ g: [0,1]\rightarrow \mathbb{R}\ ,\ g(x)\equal{}f(x)\minus{}x$ is strictly decreasing. ii)Prove that for each integer number $ n\ge 1$, we have: $ \left|\sum_{k\equal{}0}^{n\minus{}1}f\left(\frac{k}{n}\right)\right|<\frac{1}{2}$

Let U be the set formed as the union of three open intervals, $U = (-100, -10) \cup (1/101, 1/11) \cup (101/100, 11/10)$. Show that $\int_{U} \frac{(x^2-x)^2}{(x^3-3x+1)^2} dx$ is rational.

Let $ F$ be a closed set in the $ n$-dimensional Euclidean space. Construct a function that is $ 0$ on $ F$, positive outside $ F$ , and whose partial derivatives all exist.

Let $f:[0,\infty)\to(0,\infty)$ a continous function such that $\lim_{n\to\infty} \int^x_0 f(t)dt$ exists and it is finite. Prove that \[ \lim_{x\to\infty} \frac 1{\sqrt x} \int^x_0 \sqrt {f(t)} dt = 0. \] [i]Radu Miculescu[/i]

Function $f: [a,b]\to\mathbb{R}$, $0<a<b$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Prove that there exists $c\in(a,b)$ such that \[ f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}. \]

A real valued function $f$ is defined on the interval $(-1,2)$. A point $x_0$ is said to be a fixed point of $f$ if $f(x_0)=x_0$. Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$. Show that if $f'(1)>1$, then $f$ has a fixed point in the interval $(0,1)$.

[b]7.[/b] Define the generalized derivative at $x_0$ of the function $f(x)$ by $\lim_{h \to 0} 2 \frac{ \frac{1}{h} \int_{x_0}^{x_0+h} f(t) dt - f(x_0)}{h}$ Show that there exists a function, continuous everywhere, which is nowhere differentiable in this general sense [b]( R. 8)[/b]

Let $I \subset \mathbb{R}$ be an open interval and $f:I \to \mathbb{R}$ a twice differentiable function such that $f(x)f''(x)=0,$ for any $x \in I.$ Prove that $f''(x)=0,$ for any $x \in I.$

Let $f:[-1,1]\rightarrow\mathbb{R}$ be a continuous function. Show that: a) if $\int_0^1 f(\sin (x+\alpha ))\, dx=0$, for every $\alpha\in\mathbb{R}$, then $f(x)=0,\ \forall x\in [-1,1]$. b) if $\int_0^1 f(\sin (nx))\, dx=0$, for every $n\in\mathbb{Z}$, then $f(x)=0,\ \forall x\in [-1,1]$.

Define $\mathcal{A}=\{(x,y)|x=u+v,y=v, u^2+v^2\le 1\}$. Find the length of the longest segment that is contained in $\mathcal{A}$.

Show that if n is an arbitrary natural number and $\sqrt n \leq K \leq \frac{n}{2}$, then there exist n distinct integers, $k_j$ ( j = 1, ..., n ) such that $\bigg | \sum_ {j = 1} ^ ne ^ {ik_jt} \bigg | \geq K$ is satisfied on a subset of the interval $(- \pi, \pi)$ with Lebesgue measure at least $\frac{cn}{K^2}$ , where c is a suitable absolute constant.

Find a perfect set $ H \subset [0,1]$ of positive measure and a continuous function $ f$ defined on $ [0,1]$ such that for any twice differentiable function $ g$ defined on $ [0,1]$, the set $ \{ x \in H : \;f(x)\equal{}g(x)\ \}$ is finite. [i]M. Laczkovich[/i]

Let $ f$ be a differentiable real function and let $ M$ be a positive real number. Prove that if \[ |f(x\plus{}t)\minus{}2f(x)\plus{}f(x\minus{}t)| \leq Mt^2 \; \textrm{for all}\ \;x\ \; \textrm{and}\ \;t\ , \] then \[ |f'(x\plus{}t)\minus{}f'(x)| \leq M|t|.\] [i]J. Szabados[/i]

Let be a number $ a\in \left[ 1,\infty \right) $ and a function $ f\in\mathcal{C}^2(-a,a) . $ Show that the sequence $$ \left( \sum_{k=1}^n f\left( \frac{k}{n^2} \right) \right)_{n\ge 1} $$ is convergent, and determine its limit.

Let $ 0<a_k<1$ for $ k=1,2,... .$ Give a necessary and sufficient condition for the existence, for every $ 0<x<1$, of a permutation $ \pi_x$ of the positive integers such that \[ x= \sum_{k=1}^{\infty} \frac{a_{\pi_x}(k)}{2^k}.\] [i]P. Erdos[/i]

Consider the sequence of real numbers $(a_n)_{n\ge1}$ such that $$\lim_{n\to\infty}\frac1{n^r}\sum_{k=1}^n\frac{a_k}k=l\in\mathbb R,r\in\mathbb N^*$$ Show that: $$\lim_{n\to\infty}\left(\dfrac{\displaystyle\sum_{p=n+1}^{2n}\sum_{k=1}^p\sum_{i=1}^k\frac{a_i}{p\cdot i}}{n^{r+1}}\right)=l\left(\frac{2^{r+1}}{r(r+1)}-\frac{2^r}{(r+1)^2}\right)$$ [i]Proposed by Florin Stănescu and Şerban Cioculescu[/i]