(1) For real numbers $x,\ a$ such that $0<x<a,$ prove the following inequality. \[\frac{2x}{a}<\int_{a-x}^{a+x}\frac{1}{t}\ dt<x\left(\frac{1}{a+x}+\frac{1}{a-x}\right). \] (2) Use the result of $(1)$ to prove that $0.68<\ln 2<0.71.$

For each pair of integers \( j, k \geq 2 \), define the function \( f_{jk} : \mathbb{R} \to \mathbb{R} \) given by \[ f_{jk}(x) = 1 - (1 - x^j)^k. \] (a) Prove that for any integers \( j, k \geq 2 \), there exists a unique real number \( p_{jk} \in (0, 1) \) such that \( f_{jk}(p_{jk}) = p_{jk} \). Furthermore, defining \( \lambda_{jk} := f'_{jk}(p_{jk}) \), prove that \( \lambda_{jk} > 1 \). (b) Prove that \( p^j_{jk} = 1 - p_{kj} \) for any integers \( j, k \geq 2 \). (c) Prove that \( \lambda_{jk} = \lambda_{kj} \) for any integers \( j, k \geq 2 \).

Let K and D be the set of convergent and divergent series of positive terms respectively. Does there exist a bijection between K and D such that for all $\sum a_n,\sum b_n\in K$ and $\sum a_n',\sum b_n'\in D$ , $\frac{a_n}{b_n}\to 0\iff \frac{a_n'}{b_n'}\to 0$ ? Under the bijection, $\sum a_n\leftrightarrow\sum a_n'$ and $\sum b_n\leftrightarrow\sum b_n'$.

Define a sequence $(a_n)$ for $n\ge1$ by $a_1=2$ and $a_{n+1}=a_n^{1+n^{-3/2}}$. Is $(a_n)$ convergent (i.e. $\lim_{n\to\infty}a_n<\infty$)?

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 $

$\textbf{(a)}$ Let $a,b \in \mathbb{R}$ and $f,g :\mathbb{R}\rightarrow \mathbb{R}$ be differentiable functions. Consider the function $$h(x)=\begin{vmatrix} a &b &x\\ f(a) &f(b) &f(x)\\ g(a) &g(b) &g(x)\\ \end{vmatrix}$$ Prove that $h$ is differentiable and find $h'$. \\ \\ $\textbf{(b)}$ Let $n\in \mathbb{N}$, $n\geq 3$, take $n-1$ pairwise distinct real numbers $a_1<a_2<\dots <a_{n-1}$ with sum $\sum_{i=1}^{n-1}a_i = 0$, and consider $n-1$ functions $f_1,f_2,...f_{n-1}:\mathbb{R}\rightarrow \mathbb{R}$, each of them $n-2$ times differentiable over $\mathbb{R}$. Prove that there exists $a\in (a_1,a_{n-1})$ and $\theta, \theta_1,...,\theta_{n-1}\in \mathbb{R}$, not all zero, such that $$\sum_{k=1}^{n-1} \theta_k a_k=\theta a$$ and, at the same time, $$\sum_{k=1}^{n-1}\theta_kf_i(a_k)=\theta f_i^{(n-2)}(a)$$ for all $i\in\{1,2...,n-1\} $. \\ \\ [i](Sergiu Novac)[/i]

Let be a sequence of functions $ \left( f_n \right)_{n\ge 2}:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ defined, for each $ n\ge 2, $ as $$ f_n(x)=2nx^{2+n} -2(n+2)x^{1+n} +(2+n)x +1. $$ [b]a)[/b] Prove that $ f_n $ has an unique local maxima $ x_n, $ for any $ n\ge 2. $ [b]b)[/b] Show that $ 1=\lim_{n\to\infty } x_n. $ [i]Cătălin Zîrnă[/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 a function \( f: \mathbb{R} \to \mathbb{R} \) is morally odd if its graph is symmetric with respect to a point, that is, there exists \((x_0, y_0) \in \mathbb{R}^2\) such that if \((u, v) \in \{(x, f(x)) : x \in \mathbb{R}\}\), then \((2x_0 - u, 2y_0 - v) \in \{(x, f(x)) : x \in \mathbb{R}\}\). On the other hand, \( f \) is said to be morally even if its graph \(\{(x, f(x)) : x \in \mathbb{R}\}\) is symmetric with respect to some line (not necessarily vertical or horizontal). If \( f \) is morally even and morally odd, we say that \( f \) is parimpar. (a) Let \( S \subset \mathbb{R} \) be a bounded set and \( f: S \to \mathbb{R} \) be an arbitrary function. Prove that there exists \( g: \mathbb{R} \to \mathbb{R} \) that is parimpar such that \( g(x) = f(x) \) for all \( x \in S \). (b) Find all polynomials \( P \) with real coefficients such that the corresponding polynomial function \( P: \mathbb{R} \to \mathbb{R} \) is parimpar.

For each positive integer \( n \), list in increasing order all irreducible fractions in the interval \([0, 1]\) that have a positive denominator less than or equal to \( n \): \[ 0 = \frac{p_0}{q_0} < \frac{1}{n} = \frac{p_1}{q_1} < \cdots < \frac{1}{1} = \frac{p_{M(n)}}{q_{M(n)}}. \] Let \( k \) be a positive integer. We define, for each \( n \) such that \( M(n) \geq k - 1 \), \[ f_k(n) = \min \left\{ \sum_{s=0}^{k-1} q_{j+s} : 0 \leq j \leq M(n) - k + 1 \right\}. \] Determine, in function of \( k \), \[ \lim_{n \to \infty} \frac{f_k(n)}{n}. \] For example, if \( n = 4 \), the enumeration is \[ \frac{0}{1} < \frac{1}{4} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{1}{1}, \] where \( p_0 = 0, p_1 = 1, p_2 = 1, p_3 = 1, p_4 = 2, p_5 = 3, p_6 = 1 \) and \( q_0 = 1, q_1 = 4, q_2 = 3, q_3 = 2, q_4 = 3, q_5 = 4, q_6 = 1 \). In this case, we have \( f_1(4) = 1, f_2(4) = 5, f_3(4) = 8, f_4(4) = 10, f_5(4) = 13, f_6(4) = 17 \), and \( f_7(4) = 18 \).

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:\mathbb{R}\longrightarrow\mathbb{R} $ be a continuous function having a primitive $ F $ having the property that $ f-F $ is positive globally. Calculate $ \lim_{x\to\infty } f(x) . $ [i]Florian Dumitrel[/i]

If $ \sum_{m=-\infty}^{+\infty} |a_m| < \infty$, then what can be said about the following expression? \[ \lim_{n \rightarrow \infty} \frac{1}{2n+1} \sum_{m=-\infty}^{+\infty} |a_{m-n}+a_{m-n+1}+...+a_{m+n}|.\] [i]P. Turan[/i]

Let $f\colon\mathbb{N}\rightarrow\mathbb{N}^{\ast}$ be a strictly increasing function. Prove that: [list=a] [*]There exists a decreasing sequence of positive real numbers, $(y_{n})_{n\in\mathbb{N}}$, converging to $0$, such that $y_{n}\leq2y_{f(n)}$, for all $n\in\mathbb{N}$. [*]If $(x_{n})_{n\in\mathbb{N}}$ is a decreasing sequence of real numbers, converging to $0$, then there exists a decreasing sequence of real numbers $(y_{n})_{n\in\mathbb{N}}$, converging to $0$, such that $x_{n}\leq y_{n} \leq2y_{f(n)}$, for all $n\in\mathbb{N}$.[/list]

prove that for any sequence of nonnegative numbers $(a_n)$, $$\liminf_{n\to\infty} (n^2(4a_n(1-a_{n-1})-1))\leq\frac{1}{4}$$

We say that a function $f: \mathbb R \to \mathbb R$ has the property $(P)$ if, for any real numbers $x$, \[ \sup_{t\leq x} f(x) = x. \] a) Give an example of a function with property $(P)$ which has a discontinuity in every real point. b) Prove that if $f$ is continuous and satisfies $(P)$ then $f(x) = x$, for all $x\in \mathbb R$.

Let $f\in C^2[0,N]$ and $|f'(x)|<1$, $f''(x)>0$ for every $x\in [0, N]$. Let $0\leq m_0\ <m_1 < \cdots < m_k\leq N$ be integers such that $n_i=f(m_i)$ are also integers for $i=0,1,\ldots, k$. Denote $b_i=n_i-n_{i-1}$ and $a_i=m_i-m_{i-1}$ for $i=1,2,\ldots, k$. a) Prove that $$-1<\frac{b_1}{a_1}<\frac{b_2}{a_2}<\cdots < \frac{b_k}{a_k}<1$$ b) Prove that for every choice of $A>1$ there are no more than $N / A$ indices $j$ such that $a_j>A$. c) Prove that $k\leq 3N^{2/3}$ (i.e. there are no more than $3N^{2/3}$ integer points on the curve $y=f(x)$, $x\in [0,N]$).

Given a real number $x$, define the series \[ S(x) = \sum_{n=1}^{\infty} \{n! \cdot x\}, \] where $\{s\} = s - \lfloor s \rfloor$ is the fractional part of the number $s$. Determine if there exists an irrational number $x$ for which the series $S(x)$ converges.

Let $ (a_n)_{n \in \mathbb{N}^*}$ be a sequence of real positive numbers such that $ a_n>a_0,n\in \mathbb{N}$. Prove that $ \displaystyle\lim_{n\to\infty}\displaystyle\sum_{k\equal{}0}^{n}(\frac{a_k}{a_{n\minus{}k}})^k\equal{}\infty$.

Let $f:[0,1]\rightarrow R$ be a continuous function and n be an integer number,n>0.Prove that $\int_0^1f(x)dx \le (n+1)*\int_0^1 x^n*f(x)dx $

A function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ has the property that $ \lim_{x\to\infty } \frac{1}{x^2}\int_0^x f(t)dt=1. $ [b]a)[/b] Give an example of what $ f $ could be if it's continuous and $ f/\text{id.} $ doesn't have a limit at $ \infty . $ [b]b)[/b] Prove that if $ f $ is nondecreasing then $ f/\text{id.} $ has a limit at $ \infty , $ and determine it.

Suppose that $ f: [a,b]\rightarrow \mathbb{R} $ be a monotonic function and for every $ x_1,x_2\in [a,b] $ that $ x_1<x_2 $ ,there exist $ c\in (a,b) $ such that $ \int _{x_1}^{x_2}f(x)dx=f(c)(x_1-x_2) $ a) Show that $ f $ be the continuous function on interval $ (a,b) $ b) Suppose that $ f $ is integrable function on interval $ [a,b] $ but $ f $ isn't a monotonic function then ,is it the result of part a) right?

Find the twice-differentiable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that have the property that $$ f'(x)+F(x)=2f(x)+x^2/2, $$ for any real numbers $ x; $ where $ F $ is a primitive of $ f. $ [i]Carmen Botea[/i]

Is there any sequence $(a_n)_{n=1}^{\infty}$ of non-negative numbers, for which $\sum_{n=1}^{\infty} a_n^2<\infty$ , but $\sum_{n=1}^{\infty}\left(\sum_{k=1}^{\infty}\frac{a_{kn}}{k} \right)^2=\infty$ ? [hide=Remark]That contest - Miklos Schweitzer 2010- is missing on the contest page here for now being. The statements of all problems that year can be found [url=http://www.math.u-szeged.hu/~mmaroti/schweitzer/]here[/url], but unfortunately only in Hungarian. I tried google translate but it was a mess. So, it would be wonderful if someone knows Hungarian and wish to translate it. [/hide]

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?