Consider the sequence $(a_n)_{n\ge 1}$ defined by $a_n=n$ for $n\in \{1,2,3.4,5,6\}$, and for $n \ge 7$: $$a_n={\lfloor}\frac{a_1+a_2+...+a_{n-1}}{2}{\rfloor}$$ where ${\lfloor}x{\rfloor}$ is the greatest integer less than or equal to $x$. For example : ${\lfloor}2.4{\rfloor} = 2, {\lfloor}3{\rfloor} = 3$ and ${\lfloor}\pi {\rfloor}= 3$. For all integers $n \ge 2$, let $S_n = \{a_1,a_1,...,a_n\}- \{r_n\}$ where $r_n$ is the remainder when $a_1 + a_2 + ... + a_n$ is divided by $3$. The minus $-$ denotes the ''[i]remove it if it is there[/i]'' notation. For example : $S_4 = {2,3,4}$ because $r_4= 1$ so $1$ is removed from $\{1,2,3,4\}$. However $S_5= \{1,2,3,4,5\}$ betawe $r_5 = 0$ and $0$ is not in the set $\{1,2,3,4,5\}$. 1. Determine $S_7,S_8,S_9$ and $S_{10}$. 2. We say that a set $S_n$ for $n\ge 6$ is well-balanced if it can be partitioned into three pairwise disjoint subsets with equal sum. For example : $S_6 = \{1,2,3,4,5,6\} =\{1,6\}\cup \{2,5\}\cup \{3,4\}$ and $1 +6 = 2 + 5 = 3 + 4$. Prove that $S_7,S_8,S_9$ and $S_{10}$ are well-balanced . 3. Is the set $S_{2019}$ well-balanced? Justify your answer.

Let $A$ be a real number and $(a_{n})$ be a sequence of real numbers such that $a_{1}=1$ and \[1<\frac{a_{n+1}}{a_{n}}\leq A \mbox{ for all }n\in\mathbb{N}.\] $(a)$ Show that there is a unique non-decreasing surjective function $f: \mathbb{N}\rightarrow \mathbb{N}$ such that $1<A^{k(n)}/a_{n}\leq A$ for all $n\in \mathbb{N}$. $(b)$ If $k$ takes every value at most $m$ times, show that there is a real number $C>1$ such that $Aa_{n}\geq C^{n}$ for all $n\in \mathbb{N}$.

For any natural number $ m, \quad\lim_{n\to\infty } n^{1+m} \int_{0}^1 e^{-nx}\ln \left( 1+x^m \right) dx =m! . $ [i]Gheorghe Iurea[/i]

Let $k$ be a fixed positive integer. The $n$th derivative of $\tfrac{1}{x^k-1}$ has the form $\tfrac{P_n(x)}{(x^k-1)^{n+1}}$, where $P_n(x)$ is a polynomial. Find $P_n(1)$.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function defined as the maximum of a finite number of functions $g:\mathbb{R}\rightarrow\mathbb{R}$ of the form $g(x)=C\cdot10^{-|x-d|}$ (with different values of parameters $d{}$ and $C>0$). For real numbers $a<b$ we have $f(a)=f(b)$. Prove that on the segment $[a;b]$ the sum of legnths of segments on which $f$ is increasing is equal to the sum of legnths of segments on which $f$ is decreasing.

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 $n \geq 3$ be an integer. Let $f$ be a function from the set of all integers to itself with the following property: If the integers $a_1,a_2,\ldots,a_n$ form an arithmetic progression, then the numbers $$f(a_1),f(a_2),\ldots,f(a_n)$$ form an arithmetic progression (possibly constant) in some order. Find all values for $n$ such that the only functions $f$ with this property are the functions of the form $f(x)=cx+d$, where $c$ and $d$ are integers.

Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfy the following property for all real numbers $x$ and all polynomials $P$ with real coefficients: If $P(f(x)) = 0$, then $f(P(x)) = 0$.

[color=darkred]Let $\mathcal{C}$ be the set of integrable functions $f\colon [0,1]\to\mathbb{R}$ such that $0\le f(x)\le x$ for any $x\in [0,1]$ . Define the function $V\colon\mathcal{C}\to\mathbb{R}$ by \[V(f)=\int_0^1f^2(x)\ \text{d}x-\left(\int_0^1f(x)\ \text{d}x\right)^2\ ,\ f\in\mathcal{C}\ .\] Determine the following two sets: [list][b]a)[/b] $\{V(f_a)\, |\, 0\le a\le 1\}$ , where $f_a(x)=0$ , if $0\le x\le a$ and $f_a(x)=x$ , if $a<x\le 1\, ;$ [b]b)[/b] $\{V(f)\, |\, f\in\mathcal{C}\}\ .$[/list] [/color]

Let $n$  be a positive integer. A row of $n+ 1$ squares is written from left to right, numbered $0, 1, 2, \cdots, n$ Two frogs, named Alphonse and Beryl, begin a race starting at square 0. For each second that passes, Alphonse and Beryl make a jump to the right according to the following rules: if there are at least eight squares to the right of Alphonse, then Alphonse jumps eight squares to the right. Otherwise, Alphonse jumps one square to the right. If there are at least seven squares to the right of Beryl, then Beryl jumps seven squares to the right. Otherwise, Beryl jumps one square to the right. Let A(n) and B(n) respectively denote the number of seconds for Alphonse and Beryl to reach square n. For example, A(40) = 5 and B(40) = 10. (a) Determine an integer n>200 for which $B(n) <A(n)$. (b) Determine the largest integer n for which$ B(n) \le A(n)$.

[b]a)[/b] Prove that not all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality $$ f(x-1)+f(x+1) =\sqrt 5f(x) ,\quad\forall x\in\mathbb{R} , $$ are periodic. [b]b)[/b] Prove that that all functions $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality $$ g(x-1)+g(x+1)=\sqrt 3g(x) ,\quad\forall x\in\mathbb{R} , $$ are periodic.

Study the real function $f(x) = \left(1 +\frac{1}{x}\right)^x$ defined for $ x \in R - \{-1, 0\}$ . Graphic representation.

Let $ f:(0,1)\longrightarrow \mathbb{R} $ be a bounded function having the property of Darboux. Then: [b]a)[/b] There exists $ g:[0,1)\longrightarrow\mathbb{R} $ with Darboux’s property such that $ g\bigg|_{(0,1)} =f\bigg|_{(0,1)} . $ [b]b)[/b] The function above is uniquely determined if and only if $ f $ has limit at $ 0. $

Functions $ f,g,h$ are differentiable on some open interval around $ 0$ and satisfy the equations and initial conditions \begin{align*}f'&=2f^2gh+\frac1{gh},\ f(0)=1,\\ g'&=fg^2h+\frac4{fh},\ g(0)=1,\\ h'&=3fgh^2+\frac1{fg},\ h(0)=1.\end{align*} Find an explicit formula for $ f(x),$ valid in some open interval around $ 0.$

Prove that \[\arctan \frac 12 +\arctan \frac 13 = \frac{\pi}{4}.\]

Find all function $f:\mathbb{R}^+$ $\rightarrow \mathbb{R}^+$ such that: $f(f(x) + y)f(x) = f(xy + 1) \forall x, y \in \mathbb{R}^+$

The graph of the equation $9x+223y=2007$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous and bounded function such that \[x\int_{x}^{x+1}f(t)\, \text{d}t=\int_{0}^{x}f(t)\, \text{d}t,\quad\text{for any}\ x\in\mathbb{R}.\] Prove that $f$ is a constant function.

Let $a(x)$ and $b(x)$ be continuous functions on $[0,1]$ and let $0 \leq a(x) \leq a <1$ on that range. Under what other conditions (if any) is the solution of the equation for $u,$ $$ u= \max_{0 \leq x \leq 1} b(x) +a(x)u$$ given by $$u = \max_{0 \leq x \leq 1} \frac{b(x)}{1-a(x)}.$$

If $a>0,a\neq1$, $F(x)$ is an odd function. $G(x)=F(x)\cdot(\frac{1}{a^x-1}+\frac{1}{2})$, then $G(x)$ is $\text{(A)}$ odd function $\text{(B)}$ even function $\text{(C)}$ not odd or even function $\text{(D)}$ not sure

Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a monotonically increasing function over the natural numbers, such that $f(f(n))=n^{2}$. What is the smallest, and what is the largest value that $f(2023)$ can take? [i]Proposed by Ilija Jovcheski[/i]

We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers. Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property: \[f(x)f(y)\equal{}2f(x\plus{}yf(x))\] for all positive real numbers $x$ and $y$. [i]Proposed by Nikolai Nikolov, Bulgaria[/i]

Let $ k \equal{} \sqrt[3]{3}$. a, Find all polynomials $ p(x)$ with rationl coefficients whose degree are as least as possible such that $ p(k \plus{} k^2) \equal{} 3 \plus{} k$. b, Does there exist a polynomial $ p(x)$ with integer coefficients satisfying $ p(k \plus{} k^2) \equal{} 3 \plus{} k$

Is it possible to colour all positive real numbers by 10 colours so that every two numbers with decimal representations differing in one place only are of different colours? (We suppose that there is no place in a decimal representations such that all digits starting from that place are 9's.) [i]Proposed by A. Golovanov[/i]

Let $\rho:\mathbb{R}^n\to \mathbb{R}$, $\rho(\mathbf{x})=e^{-||\mathbf{x}||^2}$, and let $K\subset \mathbb{R}^n$ be a convex body, i.e., a compact convex set with nonempty interior. Define the barycenter $\mathbf{s}_K$ of the body $K$ with respect to the weight function $\rho$ by the usual formula \[\mathbf{s}_K=\frac{\int_K\rho(\mathbf{x})\mathbf{x}d\mathbf{x}}{\int_K\rho(\mathbf{x})d\mathbf{x}}.\] Prove that the translates of the body $K$ have pairwise distinct barycenters with respect to $\rho$.