Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative, that satisfies $f\big(x+f'(x)\big)=f(x)$. Let's call this property $(P)$. a) Show that if $f$ is a function with property $(P)$, then there exists a real $x$ such that $f'(x)=0$. b) Give an example of a non-constant function $f$ with property $(P)$. c) Show that if $f$ has property $(P)$ and the equation $f'(x)=0$ has at least two solutions, then $f$ is a constant function.

Suppose the real line is decomposed into two uncountable Borel sets. Prove that a suitable translated copy of the first set intersects the second in an uncountable set.

Find $\sum_{k=1}^\infty\frac{k^2-2}{(k+2)!}$.

Find the remainder when \[\sum_{k=1}^{2^{16}}\binom{2k}{k}(3\cdot 2^{14}+1)^k (k-1)^{2^{16}-1}\]is divided by $2^{16}+1$. ([i]Note:[/i] It is well-known that $2^{16}+1=65537$ is prime.) [i]Victor Wang.[/i]

Find the $ \mathcal{C}^1 $ class functions $ f:[0,2]\longrightarrow\mathbb{R} $ having the property that the application $ x\mapsto e^{-x} f(x) $ is nonincreasing on $ [0,1] , $ nondecreasing on $ [1,2] , $ and satisfying $$ \int_0^2 xf(x)dx=f(0)+f(2) . $$ [i]Cristinel Mortici[/i]

Find both extrema of the function $ x\to\frac{\sin x-3}{\cos x +2} .$

Original in Hungarian; translated with Google translate; polished by myself. Let $f: [0, \infty) \to [0, \infty)$ be a function that is linear between adjacent integers, and for $n = 0, 1, \dots$ satisfies $$f(n) = \begin{cases} 0, & \textrm{if }2\mid n,\\4^l + 1, & \textrm{if }2 \nmid n, 4^{l - 1} \leq n < 4^l(l = 1, 2, \dots).\end{cases}$$ Let $f^1(x) = f(x)$, and $f^k(x) = f(f^{k - 1}(x))$ for all integers $k \geq 2$. Determine the values of $\liminf\nolimits_{k\to\infty}f^k(x)$ and $\limsup\nolimits_{k\to\infty}f^k(x)$ for almost all $x \in [0, \infty)$ under Lebesgue measure. (Not sure whether the last sentence translates correctly; the original: Határozzuk meg Lebesgue majdnem minden $x\in [0, \infty)$-re a $\liminf\nolimits_{k\to\infty}f^k(x)$ és $\limsup\nolimits_{k\to\infty}f^k(x)$ értékét.)

a) Prove that $\{x+y\}-\{y\}$ can only be equal to $\{x\}$ or $\{x\}-1$ for any $x,y\in \mathbb{R}$. b) Let $\alpha\in \mathbb{R}\backslash \mathbb{Q}$. Denote $a_n=\{n\alpha\}$ for all $n\in \mathbb{N}^*$ and define the sequence $(x_n)_{n\ge 1}$ by \[x_n=(a_2-a_1)(a_3-a_2)\cdot \ldots \cdot (a_{n+1}-a_n)\] Prove that the sequence $(x_n)_{n\ge 1}$ is convergent and find it's limit.

Let $0 \le a < b$ be real numbers. Prove that there is no continuous function $f : [a, b] \to \mathbb{R}$ such that \[ \int_a^b f(x)x^{2n} \mathrm dx>0 \quad \text{and} \quad \int_a^b f(x)x^{2n+1} \mathrm dx <0 \] for every integer $n \ge 0$.

Let $(b_{n})_{n\in \mathbb{N}}$ be a sequence of positive real numbers such that $b_{0}=1$, $b_{n}=2+\sqrt{b_{n-1}}-2\sqrt{1+\sqrt{b_{n-1}}}$. Calculate $$\sum_{n=1}^{\infty}b_{n}2^{n}.$$

Evaluate the definite integral $$\int_{-\pi}^{\pi}\frac{\sin nx}{(1+2^{x})\sin x} dx,$$ where $n$ is a natural number.

Let $ a>0$ and $ f: [0,\infty) \to [0,a]$ be a continuous function on $ (0,\infty)$ and having Darboux property on $ [0,\infty)$. Prove that if $ f(0)\equal{}0$ and for all nonnegative $ x$ we have \[ xf(x) \geq \int^x_0 f(t) dt ,\] then $ f$ admits primitives on $ [0,\infty)$.

Suppose that $a_{ij}$ are real numbers in such a way that for each $i$, the series $\sum_{j=1}^{\infty}a_{ij}$ is absolutely convergent. In fact we have a series of absolutely convergent serieses. Also we know that for each bounded sequence $\{b_j\}_j$ we have $\lim_{i\to \infty} \sum_{j=1}^{\infty}a_{ij}b_j=0$. Prove that $\lim_{i\to \infty}\sum_{j=1}^{\infty}|a_{ij}|=0$.

Let $ H$ be a set of real numbers that does not consist of $ 0$ alone and is closed under addition. Further, let $ f(x)$ be a real-valued function defined on $ H$ and satisfying the following conditions: \[ \;f(x)\leq f(y)\ \mathrm{if} \;x \leq y\] and \[ f(x\plus{}y)\equal{}f(x)\plus{}f(y) \;(x,y \in H)\ .\] Prove that $ f(x)\equal{}cx$ on $ H$, where $ c$ is a nonnegative number. [M. Hosszu, R. Borges]

We consider a function $f:\mathbb{R} \rightarrow \mathbb{R}$ for which there exist a differentiable function $g : \mathbb{R} \rightarrow \mathbb{R}$ and exist a sequence $(a_n)_{n \geq 1}$ of real positive numbers, convergent to $0,$ such that \[ g'(x) = \lim_{n \to \infty} \frac{f(x + a_n) - f(x)}{a_n}, \forall x \in \mathbb{R}. \] a) Give an example of such a function f that is not differentiable at any point $x \in \mathbb{R}.$ b) Show that if $f$ is continuous on $\mathbb{R}$, then $f$ is differentiable on $\mathbb{R}.$

Let $f:[0,\infty)\to\mathbb{R}$ be a differentiable function, with a continous derivative. Given that $f(0)=0$ and $0\leqslant f'(x)\leqslant 1$ for every $x>0$ prove that\[\frac{1}{n+1}\int_0^af(t)^{2n+1}\mathrm{d}t\leqslant\left(\int_0^af(t)^n\mathrm{d}t\right)^2,\]for any positive integer $n{}$ and real number $a>0.$

Prove that there is a function ${f: (0,\infty) \rightarrow (0,\infty)}$ which is nowhere continuous and for all ${x,y \in (0,\infty)}$ and any rational ${\alpha}$ we have \[ \displaystyle f\left( \left(\frac{x^\alpha+y^\alpha}{2}\right)^{\frac{1}{\alpha}}\right)\leq \left(\frac{f(x)^\alpha +f(y)^\alpha }{2}\right)^{\frac{1}{\alpha}}. \] Is there such a function if instead the above relation holds for every ${x,y \in (0,\infty)}$ and for every irrational ${\alpha}?$ [i]Proposed by Maksa Gyula and Zsolt Páles[/i]

Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$. Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$

Let be the sequence $ \left( x_n \right)_{n\ge 1} $ defined as $$ x_n= \frac{4009}{4018020} x_{n-1} -\frac{1}{4018020} x_{n-2} + \left( 1+\frac{1}{n} \right)^n. $$ Prove that $ \left( x_n \right)_{n\ge 1} $ is convergent and determine its limit.

Let $ f$ be a continuous, nonconstant, real function, and assume the existence of an $ F$ such that $ f(x\plus{}y)\equal{}F[f(x),f(y)]$ for all real $ x$ and $ y$. Prove that $ f$ is strictly monotone.

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 $$\int\limits_0^{\pi} f(x) \mathrm{d}x \ge \frac{\pi^2}{2}-2.$$

Let $K_1,...,K_d$ be convex, compact sets in $R^d$ with non-empty interior. Suppose they are strongly separated, which means for any choice of $x_1 \in K_1, x_2 \in K_2, ...$, their affine hull is a hyperplane in $R^d$. Also let $0< \alpha_i <1$. A half-space H is called an $\alpha$-cut if $vol(K_i \cap H) = \alpha_i\cdot vol(K_i)$ for all i. How many $\alpha$-cuts are there?

Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function with the property that for any $a,b \in \mathbb{R},$ $a<b,$ there are $c_1,c_2 \in [a,b],$ $c_1 \le c_2$ such that $f(c_1)= \min_{x \in [a,b]} f(x)$ and $f(c_2)= \max_{x \in [a,b]} f(x).$ Prove that $f$ is increasing.

Let be an even natural number $ n $ and a function $ f:[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ f(x)=\int_0^x \prod_{k=0}^n (s-k) ds. $$ Show that [b]a)[/b] $ f(n)=0. $ [b]b)[/b] $ f $ is globally nonnegative. [i]Gheorghe Grigore[/i]