Consider a sequence of positive real numbers $ \left( x_n \right)_{n\ge 1} $ and a primitivable function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ [b]a)[/b] Prove that $ f $ is monotonic and continuous if for any natural numbers $ n $ and real numbers $ x, $ the inequality $$ f\left( x+x_n \right)\geqslant f(x) $$ is true. [b]b)[/b] Show that $ f $ is convex if for any natural numbers $ n $ and real numbers $ x, $ the inequality $$ f\left( x+2x_n \right) +f(x)\geqslant 2f\left( x+x_n \right) $$ is true. [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

Let be two real numbers $ a<b $ and a function $ f:[a,b]\longrightarrow\mathbb{R} $ having 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. [b]a)[/b] Prove that if $ f $ admits antiderivatives, then $ f $ is integrable. [b]b)[/b] Is the converse of [b]a)[/b] true? [i]Marcelina Popa[/i]

$ \lim_{n\to\infty } \frac{1}{n}\sum_{i,j=1}^n \frac{i+j}{i^2+j^2} $

Prove that for any integers $n,p$, $0<n\leq p$, all the roots of the polynomial below are real: \[P_{n,p}(x)=\sum_{j=0}^n {p\choose j}{p\choose {n-j}}x^j\]

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(x)+\sin(f(x)) \ge x,$ for all $x \in \mathbb{R}.$ Prove that $$\int\limits_0^{\pi} f(x) \mathrm{d}x \ge \frac{\pi^2}{2}-2.$$

Find all functions, $ f:\mathbb{R}\longrightarrow\mathbb{R} , $ that have the properties that $ f^2 $ is differentiable and $ f=\left( f^2 \right)' . $

For any measurable set $H \subset R$ , we define the sequence $a_n(H)$ by the formula: $$a_n(H) = \lambda \bigg([0,1] \setminus \bigcup_{k = n}^{2n} (H + \log_2 k) \bigg)$$ where $\lambda$ denotes the Lebesgue measure and $\log_2$ denotes the binary logarithm. Prove that there is a measurable, 1-periodic, positive measure set $H \subset R$ , such that the sequence $a_n( H )$ does not belong to any space $l_p$ ($1 \leq p < \infty$). [hide=not sure about this part]For what numbers $1 \leq p <\infty$ is it true that whenever H is 1-periodic, positive measure, the sequence $a_n( H )$ belongs to the space $l_p$?[/hide]

Consider a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admits primitives. Prove that: $ \text{(i)} $ Every term (function) of the sequence functions $ \left( h_n\right)_{n\ge 2}:\mathbb{R}\longrightarrow\mathbb{R} $ defined, for any natural number $ n $ as $ h_n(x)=x^nf\left( x^3 \right) , $ is primitivable. $ \text{(ii)} $ The function $ \phi :\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ \phi (x) =\left\{ \begin{matrix} e^{-1/x^2} f(x),& \quad x\neq 0 \\ 0,& \quad x=0 \end{matrix} \right. $$ is primitivable. [i]Cristinel Mortici[/i]

Find $$\lim_{N\to\infty}\frac{\ln^2 N}{N} \sum_{k=2}^{N-2} \frac{1}{\ln k \cdot \ln (N-k)}$$

Suppose that $c_n=(-1)^n(n+1)$. While the sum $\sum_{n=0}^\infty c_n$ is divergent, we can still attempt to assign a value to the sum using other methods. The Abel Summation of a sequence, $a_n$, is $\operatorname{Abel}(a_n)=\lim_{x\to1^-}\sum_{n=0}^\infty a_nx^n$. Find $\operatorname{Abel}(c_n)$.

Let $a < b$ be real numbers and let $f : (a, b) \to \mathbb{R}$ be a function such that the functions $g : (a, b) \to \mathbb{R}$, $g(x) = (x - a) f(x)$ and $h : (a, b) \to \mathbb{R}$, $h(x) = (x - b) f(x)$ are increasing. Show that the function $f$ is continuous on $(a, b)$.

Given $a\in (0,1)$ and $C$ the set of increasing functions $f:[0,1]\to [0,\infty )$ such that $\int\limits_{0}^{1}{f(x)}dx=1$ . Determine: $(a)\underset{f\in C}{\mathop{\max }}\,\int\limits_{0}^{a}{f(x)dx}$ $(b)\underset{f\in C}{\mathop{\max }}\,\int\limits_{0}^{a}{{{f}^{2}}(x)dx}$

Given $f,g\in L^1[0,1]$ and $\int_0^1 f = \int_0^1 g=1$, prove that there exists an interval I st $\int_I f = \int_I g=\frac12$.

Find all differentiable functions $ f:(0,\infty )\longrightarrow (-\infty ,\infty ) $ having the property that $$ f'(\sqrt{x}) =\frac{1+x+x^2}{1+x} , $$ for any positive real numbers $ x. $ [i]Ovidiu Țâțan[/i]

Prove that for every $\varepsilon >0$ there exists a positive integer $n$ and there are positive numbers $a_1, \ldots, a_n$ such that for every $\varepsilon < x < 2\pi - \varepsilon$ we have $$\sum_{k=1}^n a_k\cos kx < -\frac{1}{\varepsilon}\left| \sum_{k=1}^n a_k\sin kx\right|$$.

Suppose that $a,b\in\mathbb{R}^+$ which $a+b<1$ and $f:[0,+\infty) \rightarrow [0,+\infty) $ be the increasing function s.t. $\forall x\geq 0 ,\int _0^x f(t)dt=\int _0^{ax} f(t)dt+\int _0^{bx} f(t)dt$. Prove that $\forall x\geq 0 , f(x)=0$

Let be areal number $ r, $ a nonconstant and continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with period $ T $ and $ F $ be its primitive having $ F(0)=0. $ Define the funtion $ g:\mathbb{R}\longrightarrow\mathbb{R} $ as $$ g(x)=\left\{\begin{matrix} f(1/x), & x\neq 0 \\ r, & x=0 \end{matrix}\right. $$ Prove that: [b]a)[/b] the image of $ f $ is closed. [b]b)[/b] $ g $ has the intermediate value property if and only if $ r\in f\left(\mathbb{R}\right) . $ [b]c)[/b] $ g $ is primitivable if and only if $ r=\frac{F(T)}{T} . $

Let $ f(x)$ be a real function such that \[ \lim_{x \rightarrow \plus{}\infty} \frac{f(x)}{e^x}\equal{}1\] and $ |f''(x)|\leq c|f'(x)|$ for all sufficiently large $ x$. Prove that \[ \lim_{x \rightarrow \plus{}\infty} \frac{f'(x)}{e^x}\equal{}1.\] [i]P. Erdos[/i]

Let $c_1,c_2,\ldots,c_{6030}$ be 6030 real numbers. Suppose that for any 6030 real numbers $a_1,a_2,\ldots,a_{6030}$, there exist 6030 real numbers $\{b_1,b_2,\ldots,b_{6030}\}$ such that \[a_n = \sum_{k=1}^{n} b_{\gcd(k,n)}\] and \[b_n = \sum_{d\mid n} c_d a_{n/d}\] for $n=1,2,\ldots,6030$. Find $c_{6030}$. [i]Victor Wang.[/i]

Let $ f$ be a real function defined on the positive half-axis for which $ f(xy)\equal{}xf(y)\plus{}yf(x)$ and $ f(x\plus{}1) \leq f(x)$ hold for every positive $ x$ and $ y$. Show that if $ f(1/2)\equal{}1/2$, then \[ f(x)\plus{}f(1\minus{}x) \geq \minus{}x \log_2 x \minus{}(1\minus{}x) \log_2 (1\minus{}x)\] for every $ x\in (0,1)$. [i]Z. Daroczy, Gy. Maksa[/i]

Let $ \gamma: [0,1]\rightarrow [0,1]\times [0,1]$ be a mapping such that for each $ s,t\in [0,1]$ \[ |\gamma(s) \minus{} \gamma(t)|\leq M|s \minus{} t|^\alpha \] in which $ \alpha,M$ are fixed numbers. Prove that if $ \gamma$ is surjective, then $ \alpha\leq\frac12$

Prove the inequalities $ 0<n\left( \sqrt[n]{2} -1 \right) -\left( \frac{1}{n+1} +\frac{1}{n+2} +\cdots +\frac{1}{n+n}\right) <\frac{1}{2n} , $ where $ n\ge 2. $ [i]Marius Cavachi[/i]

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]

We say that the real-valued function $ f(x)$ defined on the interval $ (0,1)$ is approximately continuous on $ (0,1)$ if for any $ x_0 \in (0,1)$ and $ \varepsilon >0$ the point $ x_0$ is a point of interior density $ 1$ of the set \[ H\equal{} \{x : \;|f(x)\minus{}f(x_0)|< \varepsilon \ \}.\] Let $ F \subset (0,1)$ be a countable closed set, and $ g(x)$ a real-valued function defined on $ F$. Prove the existence of an approximately continuous function $ f(x)$ defined on $ (0,1)$ such that \[ f(x)\equal{}g(x) \;\textrm{for all}\ \;x \in F\ .\] [i]M. Laczkovich, Gy. Petruska[/i]

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 $(0,1)$.