Prove that there is exactly a function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{\ge 0} $ satisfying the following two properties: $ \text{(i)} x\in\mathbb{R}_{> 0}\implies \left( f(x)+f(f(x)) =4018020x \wedge f(x)>0 \right) $ $ \text{(ii)} 0=f(0)+f(f(0)) $

Suppose that ${f: \Bbb{R} \rightarrow \Bbb{R}}$ is an additive function $($that is ${f(x+y) = f(x)+f(y)}$ for all ${x, y \in \Bbb{R}})$ for which ${x \mapsto f(x)f(\sqrt{1-x^2})}$ is bounded of some nonempty subinterval of ${(0,1)}$. Prove that ${f}$ is continuous. [i]Proposed by Zoltán Boros[/i]

Let the Fourier series \[ \frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)\] of a function $ f(x)$ be absolutely convergent, and let \[ a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .\] Show that \[ \frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)\] is uniformly bounded in $ h$. [K. Tandori]

Do there exist continuous functions $f, g: \mathbb{R} \to \mathbb{R}$, both nowhere differentiable, such that $f \circ g$ is differentiable?

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 $C$.

Consider $0<a<T$, $D=\mathbb{R}\backslash \{ kT+a\mid k\in \mathbb{Z}\}$, and let $f:D\to \mathbb{R}$ a $T-$periodic and differentiable function which satisfies $f' > 1$ on $(0, a)$ and $$f(0)=0,\lim_{\substack{x\to a\\x<a}}f(x)=+\infty \text{ and }\lim_{\substack{x\to a\\ x<a}}\frac{f'(x)}{f^2(x)}=1.$$ [list] [*]Prove that for every $n\in \mathbb{N}^*$, the equation $f(x)=x$ has a unique solution in the interval $(nT, nT+a)$ , denoted $x_n$.[/*] [*]Let $y_n=nT+a-x_n$ and $z_n=\int_0^{y_n}f(x)\text{d}x$. Prove that $\lim_{n\to \infty}{y_n}=0$ and study the convergence of the series $\sum_{n=1}^{\infty}{y_n}$ and $\sum_{n=1}^{n}{z_n}$. [/list]

Define a sequence by $a_1=1,a_2=\frac12$, and $a_{n+2}=a_{n+1}-\frac{a_na_{n+1}}2$ for $n$ a positive integer. Find $\lim_{n\to\infty}na_n$.

Let $ a,b,c\in\mathbb{R}_+^*, $ and $ f:[0,a]\longrightarrow [0,b] $ bijective and non-decreasing. Prove that: $$ \frac{1}{b}\int_0^a f^2 (x)dx +\frac{1}{a}\int_0^b \left( f^{-1} (x)\right)^2dx\le ab. $$

For a given irrational number $\alpha$ , $y_{1,\alpha} = \alpha$. If $y_{n-1, \alpha}$ is given, let $y_{n, \alpha}$ be the first member of the sequence $\big (\{k \alpha \} \big) ^ \infty_{k = 1}$ to fall in the interval $(0, y_{n-1,\alpha})$ ({ x } denotes the fraction of the number x ). Show that there exists an open set $G\subset (0,1)$ , which has a limit point 0 and for all irrational $\alpha$ , infinitely many members of the $(y_{n,\alpha})$ sequence do not belong to G.

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 $n$ be a positive integer. We define $f(n)$ as the number of finite sequences $(a_1, a_2, \ldots , a_k)$ of positive integers such that $a_1 < a_2 < a_3 < \cdots < a_k$ and $$a_1+a_2^2+a_3^3+\cdots + a_k^k \leq n.$$ Determine the positive constants $\alpha$ and $C$ such that $$\lim\limits_{n\rightarrow \infty} \frac{f(n)}{n^\alpha}=C.$$

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 \[ f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}. \]

Let the continuous functions $ f_n(x), \; n\equal{}1,2,3,...,$ be defined on the interval $ [a,b]$ such that every point of $ [a,b]$ is a root of $ f_n(x)\equal{}f_m(x)$ for some $ n \not\equal{} m$. Prove that there exists a subinterval of $ [a,b]$ on which two of the functions are equal.

[b]a)[/b] Calculate $ \int_{0}^1 x\sin\left( \pi x^2\right) dx. $ [b]b)[/b] Calculate $ \lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n-1} k\int_{\frac{k}{n}}^{\frac{k+1}{n}} \sin\left(\pi x^2\right) dx. $ [i]Florin Stănescu[/i]

By a partition $\pi$ of an integer $n\ge 1$, we mean here a representation of $n$ as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if $n=4$, then the partitions $\pi$ are $1+1+1+1$, $1+1+2$, $1+3, 2+2$, and $4$). For any partition $\pi$, define $A(\pi)$ to be the number of $1$'s which appear in $\pi$, and define $B(\pi)$ to be the number of distinct integers which appear in $\pi$. (E.g., if $n=13$ and $\pi$ is the partition $1+1+2+2+2+5$, then $A(\pi)=2$ and $B(\pi) = 3$). Prove that, for any fixed $n$, the sum of $A(\pi)$ over all partitions of $\pi$ of $n$ is equal to the sum of $B(\pi)$ over all partitions of $\pi$ of $n$.

Let $ f:[0,\infty )\longrightarrow [0,\infty ) $ a nonincreasing function that satisfies the inequality: $$ \int_0^x f(t)dt <1,\quad\forall x\ge 0. $$ Prove the following affirmations: [b]a)[/b] $ \exists \lim_{x\to\infty} \int_0^x f(t)dt \in\mathbb{R} . $ [b]b)[/b] $ \lim_{x\to\infty} xf(x) =0. $

Consider $f(x)=x^4+ax^3+bx^2+cx+d\; (a,b,c,d\in\mathbb{R})$. It is known that $f$ intersects X-axis in at least $3$ (distinct) points. Show either $f$ has $4$ $\mathbf{distinct}$ real roots or it has $3$ $\mathbf{distinct}$ real roots and one of them is a point of local maxima or minima.

Suppose $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function for which the inequality $f'(x) \leq f'(x+\frac{1}{n})$ holds for every $x\in\mathbb{R}$ and every $n\in\mathbb{N}$.Prove that f is continiously differentiable

For a positive integer $n$, define $f(n)$ to be the number of sequences $(a_1,a_2,\dots,a_k)$ such that $a_1a_2\cdots a_k=n$ where $a_i\geq 2$ and $k\ge 0$ is arbitrary. Also we define $f(1)=1$. Now let $\alpha>1$ be the unique real number satisfying $\zeta(\alpha)=2$, i.e $ \sum_{n=1}^{\infty}\frac{1}{n^\alpha}=2 $ Prove that [list] (a) \[ \sum_{j=1}^{n}f(j)=\mathcal{O}(n^\alpha) \] (b) There is no real number $\beta<\alpha$ such that \[ \sum_{j=1}^{n}f(j)=\mathcal{O}(n^\beta) \] [/list]

$\lim_{x\to0}x\left\lfloor\frac1x\right\rfloor=?$

Let $\{x_n\}_{n\geq 0}$ be a sequence of real numbers which satisfy \[ (x_{n+1} - x_n)(x_{n+1}+x_n+1) \leq 0, \quad n\geq 0. \] a) Prove that the sequence is bounded; b) Is it possible that the sequence is not convergent?

An arithmetic function is a real-valued function whose domain is the set of positive integers. Define the convolution product of two arithmetic functions $ f$ and $ g$ to be the arithmetic function $ f * g$, where \[ (f * g)(n) \equal{} \sum_{ij\equal{}n} f(i) \cdot g(j),\] and $ f^{*k} \equal{} f * f * \ldots * f$ ($ k$ times) We say that two arithmetic functions $ f$ and $ g$ are dependent if there exists a nontrivial polynomial of two variables $ P(x, y) \equal{} \sum_{i,j} a_{ij} x^i y^j$ with real coefficients such that \[ P(f,g) \equal{} \sum_{i,j} a_{ij} f^{*i} * g^{*j} \equal{} 0,\] and say that they are independent if they are not dependent. Let $ p$ and $ q$ be two distinct primes and set \[ f_1(n) \equal{} \begin{cases} 1 & \text{ if } n \equal{} p, \\ 0 & \text{ otherwise}. \end{cases}\] \[ f_2(n) \equal{} \begin{cases} 1 & \text{ if } n \equal{} q, \\ 0 & \text{ otherwise}. \end{cases}\] Prove that $ f_1$ and $ f_2$ are independent.

Let $0 < a < b$. Prove that $\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}$.

Let $ \left( c_n \right)_{n\ge 1} $ be a sequence of real numbers. Prove that the sequences $ \left( c_n\sin n \right)_{n\ge 1} ,\left( c_n\cos n \right)_{n\ge 1} $ are both convergent if and only if $ \left( c_n \right)_{n\ge 1} $ converges to $ 0. $ [i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]

Let be two functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ such that $ g $ has infinite limit at $ \infty . $ [b]a)[/b] Prove that if $ g $ continuous then $ \lim_{x\to\infty } f(x) =\lim_{x\to\infty } f(g(x)) $ [b]b)[/b] Provide an example of what $ f,g $ could be if $ f $ has no limit at $ \infty $ and $ \lim_{x\to\infty } f(g(x)) =0. $