Find the minimum possible sum of lengths of edges of a prism all of whose edges are tangent of a unit sphere. [Muller-Pfeiffer].

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

Prove that \[ \sum_{n=1}^\infty {1\over n^n} = \int_0^1 x^{-x}\,dx. \]

Let $A$ be an infinite subset of the set of natural numbers, and denote by $\tau_A(n)$ the number of divisors of $n$ in $A$. Construct a set $A$ for which $$\sum_{n\le x}\tau_A(n)=x+O(\log\log x)$$ and show that there is no set for which the error term is $o(\log\log x)$ in the above formula. (translated by Miklós Maróti)

Let $f:[0,1]\to(0,\infty)$ be a continous function on $[0,1]$ and let $A=\int_0^1 f(t)\mathrm{d}t.$[list=a] [*]Consider the function $F:[0,1]\to[0,A]$ defined by \[F(x)=\int_0^xf(t)\mathrm{d}t.\]Prove that $F(x)$ has an inverse function, which is differentiable. [*]Prove that there exists a unique function $g:[0,1]\to[0,1]$ for which\[\int_0^xf(t)\mathrm{d}t=\int_{g(x)}^1f(t)\mathrm{d}t\]is satisfied for every $x\in [0,1].$ [*]Prove that there exists $c\in[0,1]$ for which\[\lim_{x\to c}\frac{g(x)-c}{x-c}=-1,\]whre $g$ is the function uniquely determined at b. [/list]

Let $[0, 1]$ be the set $\{x \in \mathbb{R} : 0 \leq x \leq 1\}$. Does there exist a continuous function $g : [0, 1] \to [0, 1]$ such that no line intersects the graph of $g$ infinitely many times, but for any positive integer $n$ there is a line intersecting $g$ more than $n$ times? [i]Proposed by Ethan Tan[/i]

Let $\alpha>1$ and $f:\left[\frac{1}{\alpha},\alpha\right]\rightarrow \left[\frac{1}{\alpha},\alpha\right]$, a bijective function. If $f^{-1}(x)=\frac{1}{f(x)},\ \forall x\in \left[\frac{1}{\alpha},\alpha\right]$, prove that: a)$f$ has at least one point of discontinuity; b)if $f$ is continuous in $1$, then $f$ has an infinity points of discontinuity; c)there is a function $f$ which satisfies the conditions from the hypothesis and has a finite number of points of dicontinuity. [i]Radu Mortici [/i]

Prove that there exists an ordered set in which every uncountable subset contains an uncountable, well-ordered subset and that cannot be represented as a union of a countable family of well-ordered subsets. [i]A. Hajnal[/i]

$ \lim_{n\to\infty }\left( \frac{1}{e}\sum_{i=0}^n \frac{1}{i!} \right)^{n!} $

Let $(a_n)_{n \in \mathbb{N}}$ be a sequence of integers. Define $a_n^{(0)} = a_n$ for all $n \in \mathbb{N}$. For all $M \geq 0$, we define $(a_n^{(M + 1)})_{n \in \mathbb{N}}:\, a_n^{(M + 1)} = a_{n + 1}^{(M)} - a_n^{(M)}, \forall n \in \mathbb{N}$. We say that $(a_n)_{n \in \mathbb{N}}$ is $\textrm{(M + 1)-self-referencing}$ if there exists $k_1$ and $k_2$ fixed positive integers such that $a_{n + k_1} = a_{n + k_2}^{(M + 1)}, \forall n \in \mathbb{N}$. (a) Does there exist a sequence of integers such that the smallest $M$ such that it is $\textrm{M-self-referencing}$ is $M = 2022$? (a) Does there exist a stricly positive sequence of integers such that the smallest $M$ such that it is $\textrm{M-self-referencing}$ is $M = 2022$?

Let $ 0 \leq c \leq 1$, and let $ \eta$ denote the order type of the set of rational numbers. Assume that with every rational number $ r$ we associate a Lebesgue-measurable subset $ H_r$ of measure $ c$ of the interval $ [0,1]$. Prove the existence of a Lebesgue-measurable set $ H \subset [0,1]$ of measure $ c$ such that for every $ x \in H$ the set \[ \{r : \;x \in H_r\ \}\] contains a subset of type $ \eta$. [i]M. Laczkovich[/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 \[ f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}. \]

Find the differentiable functions $ f:\mathbb{R}\longrightarrow (-\infty ,1) $ with the property $ f(1)=-1 $ and $$ f(x+y)=f(x)+f(y)-f(x)f(y) , $$ for any reals $ x,y. $ [i]Vasile Pop[/i]

Prove that there are real numbers $ a_1, a_2, ..$ such that: i) For all real numbers x, the serie $ f(x) \equal{} \sum_{n \equal{} 1}^\infty a_nx^n$ converge; ii) f is a bijection of R to R; iii) f'(x) >0; iv) f(Q) = A, where Q is the set of rational numbers and A is the set of algebraic numbers.

Prove that the parity of each term of the sequence $ \left( \left\lfloor \left( \lfloor \sqrt q \rfloor +\sqrt{q} \right)^n \right\rfloor \right)_{n\ge 1} $ is opposite to the parity of its index, where $ q $ is a squarefree natural number.

Given a set $S$ of $2n-1$, $n\in \mathbb N$, different irrational numbers. Prove that there are $n$ different elements $x_1, x_2, \ldots, x_n\in S$ such that for all non-negative rational numbers $a_1, a_2, \ldots, a_n$ with $a_1+a_2+\ldots + a_n>0$ we have that $a_1x_1+a_2x_2+\cdots +a_nx_n$ is an irrational number.

A sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is called "Devin" if for any $f\in C[0,1]$ $$ \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n f(x_i)=\int_0^1 f(x)\,dx $$ Prove that a sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is "Devin" if and only if for any non-negative integer $k$ it holds $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n x_i^k=\frac{1}{k+1}.$$ [b]Remark[/b]. I left intact the text as it was proposed. Devin is a Bulgarian city and SPA resort, where this competition took place.

Does there exist a function $ f(x,y)$ of two real variables that takes natural numbers as its values and for which $ f(x,y)\equal{}f(y,z)$ implies $ x\equal{}y\equal{}z?$ [i]A. Hajnal[/i]

Let $p(n)\geq 0$ for all positive integers $n$. Furthermore, $x(0)=0, v(0)=1$, and \[x(n)=x(n-1)+v(n-1), \qquad v(n)=v(n-1)-p(n)x(n) \qquad (n=1,2,\dots).\] Assume that $v(n)\to 0$ in a decreasing manner as $n \to \infty$. Prove that the sequence $x(n)$ is bounded if and only if $\sum_{n=1}^{\infty}n\cdot p(n)<\infty$.

a) Let $a\in \mathbb{R}$ and $f \colon \mathbb{R} \to \mathbb{R}$ be a continuous function for which there exists an antiderivative $F \colon \mathbb{R} \to \mathbb{R} $, such that $F(x)+a\cdot f(x) \geq 0$, for any $x \in \mathbb{R}$, and$ \lim_{|x| \to \infty} \frac{F(x)}{e^{|\alpha \cdot x|}}=0$ holds for any $\alpha \in \mathbb{R}^*$. Prove that $F(x) \geq 0$ for all $x \in \mathbb{R}$. b) Let $n\geq 2$ be a positive integer, $g \in \mathbb{R}[X]$, $g = X^n + a_1X^{n-1}+ \dots + a_{n-1}X+a_n$ be a polynomial with all of its roots being real, and $f \colon \mathbb{R} \to \mathbb{R}$ a polynomial function such that $f(x)+a_1\cdot f'(x)+a_2\cdot f^{(2)}(x)+\dots+a_n\cdot f^{(n)}(x) \geq 0$ for any $x \in \mathbb{R}$. Prove that $f(x) \geq 0$ for all $x \in \mathbb{R}$.

Let $ a\in (1,\infty) $ and a countinuous function $ f:[0,\infty)\longrightarrow\mathbb{R} $ having the property: $$ \lim_{x\to \infty} xf(x)\in\mathbb{R} . $$ [b]a)[/b] Show that the integral $ \int_1^{\infty} \frac{f(x)}{x}dx $ and the limit $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $ both exist, are finite and equal. [b]b)[/b] Calculate $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $

Let $p$ be an even non-negative continous function with $\int _{\mathbb{R}} p(x) dx =1$ and let $n$ be a positive integer. Let $\xi_1,\xi_2,\xi_3 \dots ,\xi_n$ be independent identically distributed random variables with density function $p$ . Define \begin{align*} X_{0} & = 0 \\ X_{1} & = X_0+ \xi_1 \\ & \vdotswithin{ = }\notag \\ X_{n} & = X_{n-1} + \xi_n \end{align*} Prove that the probability that all random variables $X_1,X_2 \dots X_{n-1}$ lie between $X_0$ and $X_n$ is $\frac{1}{n}$. [i]Proposed by Fedor Petrov (St.Petersburg State University).[/i]

$\exists ? f: R\to R$ continuos function that: $\forall x_0\in R \lim\limits_{x \to x_0} \frac{|f(x)-f(x_0)|}{|x-x_0|}=+\infty$

Consider a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admits bounded primitives. Prove that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(x)=\left\{ \begin{matrix} x, & \quad x\le 0 \\ f(1/x)\cdot\ln x ,& \quad x>0 \end{matrix}\right. $$ admits primitives. [i]Florian Dumitrel[/i]

Let $ a $ be a positive real number and $ \left( x_n\right)_{n\ge 1} $ be a sequence of pairwise distinct real numbers satisfying the properties: $ \text{(i) } x_n\in (0,a) , $ for any natural numbers $ n $ $ \text{(ii) } \left| x_n-x_m \right|\geqslant\frac{m+n}{amn} , $ for all pairs $ (m,n) $ of distinct natural numbers Show that $ a\geqslant 2. $