Find all real numbers $x$ such that $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor=88$. The notation $\lfloor x\rfloor$ means the greatest integer less than or equal to $x$.

Two students $ A$ and $ B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $ A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $ B.$ If $ B$ answers “no,” the referee puts the question back to $ A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”

A pedestrian walked for $3.5$ hours. In every period of one hour’s duration he walked $5$ kilometres. Is it true that his average speed was $5$ kilometres per hour? (NN Konstantinov, Moscow)

Determine all solutions of the system $$\begin{cases} 2x - 5y + 11z - 6 = 0 \\ -x + 3y - 16z + 8 = 0 \\ 4x - 5y - 83z + 38 = 0 \\ 3x + 11y - z + 9 > 0 \end{cases}$$ in which the first three are equations and the last one is a linear inequality.

For any real number $a$, find all real numbers $x$ that satisfy the following equation. $$(2x + 1)^4 + ax(x + 1) - \frac{x}{2}= 0$$

A loader has two carts. One of them can carry up to 8 kg, and another can carry up to 9 kg. A finite number of sacks with sand lie in a storehouse. It is known that their total weight is more than 17 kg, while each sack weighs not more than 1 kg. What maximum weight of sand can the loader carry on his two carts, regardless of particular weights of sacks? [i]Author: M.Ivanov, D.Rostovsky, V.Frank[/i]

Let $f:] 0. \infty [ \to R$ be a strictly increasing function, such that $$f (x) f\left(f (x) +\frac{1}{x} \right)= 1.$$ Find $f (1)$.

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

For the sequence of real numbers $a_1,a_2,\dots ,a_k$ we say it is [i]invested[/i] on the interval $[b,c]$ if there exists numbers $x_0,x_1,\dots ,x_k$ in the interval $[b,c]$ such that $|x_i-x_{i-1}|=a_i$ for $i=1,2,3,\dots k$ . A sequence is [i]normed[/i] if all its members are not greater than $1$ . For a given natural $n$ , prove : a)Every [i]normed[/i] sequence of length $2n+1$ is [i]invested[/i] in the interval $\left[ 0, 2-\frac{1}{2^n} \right ]$. b) there exists [i]normed[/i] sequence of length $4n+3$ wich is not [i]invested[/i] on $\left[ 0, 2-\frac{1}{2^n} \right ]$.

For each positive integer $ n$, evaluate the sum \[ \sum_{k\equal{}0}^{2n}(\minus{}1)^{k}\frac{\binom{4n}{2k}}{\binom{2n}{k}}\]

Find all polynomials $P(x)$ with real coefficients such that $P(2014) = 1$ and, for some integer $c$: $xP(x-c) = (x - 2014)P(x)$

Let $T$ be the answer to question $16$. Compute the number of distinct real roots of the polynomial $x^4 + 6x^3 +\frac{T}{2}x^2 + 6x + 1$.

The numerical sequence $\{x_n\}$ satisfies the condition $$x_{n+1}=|x_n|-x_{n-1}$$ for all $n > 1$. Prove that the sequence is periodic with period $9$, i.e. for any $n > 1$ we have $x_n = x_{n+9}$. (M Kontsevich, Moscow)

The senior coefficient $a$ in the square polynomial $$P(x) = ax^2 + bx + c$$ is more than $100$. What is the maximal number of integer values of $x$, such that $|P(x)|<50$.

Deos there exist a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$, $y \in \mathbb{R}$, $f(x^2y+f(x+y^2))=x^3+y^3+f(xy)$

Let $A_n = \{1,2,...,n\}$. Prove or disprove: For all integers $n \ge 2$ there exist functions $f,g : A_n \to A_n$ which satisfy $f(f(k)) = g(g(k)) = k$ for $1 \le k \le n$, and $g(f(k)) = k +1$ for $1 \le k \le n -1$.

Find all possible real values for $a, b$ and $c$ such that (a) $a + b + c = 51$, (b) $abc = 4000$, (c) $0 < a \le 10$ and $c \ge 25$.

$x_1,...,x_n$ are reals and $x_1^2+...+x_n^2=1$ Prove, that exists such $y_1,...,y_n$ and $z_1,...,z_n$ such that $|y_1|+...+|y_n| \leq 1$; $max(|z_1|,...,|z_n|) \leq 1$ and $2x_i=y_i+z_i$ for every $i$

Suppose $a_1, a_2, ... , a_{100}$ are positive real numbers such that $$a_k =\frac{ka_{k-1}}{a_{k-1} - (k - 1)}$$ for $k = 2, 3, ... , 100$. Given that $a_{20} = a_{23}$, compute $a_{100}$.

Find all $\{a_n\}_{n\ge 0}$ that satisfies the following conditions. (1) $a_n\in \mathbb{Z}$ (2) $a_0=0, a_1=1$ (3) For infinitly many $m$, $a_m=m$ (4) For every $n\ge2$, $\{2a_i-a_{i-1} | i=1, 2, 3, \cdots , n\}\equiv \{0, 1, 2, \cdots , n-1\}$ $\mod n$

Solve equation : $\sqrt{x+\sqrt{4x+\sqrt{16x}+..+\sqrt{4^nx+3}}}-\sqrt{x}=1$

The numbers $S_1=2^2, S_2=2^4,\ldots, S_n=2^{2n}$ are given. A rectangle $OABC$ is constructed on the Cartesian plane according to these numbers. For this, starting from the point $O$ the points $A_1,A_2,\ldots,A_n$ are consistently marked along the axis $Ox$, and the points $C_1,C_2,\ldots,C_n$ are consistently marked along the axis $Oy$ in such a way that for all $k$ from $1$ to $n$ the lengths of the segments $A_{k-1}A_k=x_k$ and $C_{k-1}C_k=y_k$ are positive integers (let $A_0=C_0=O$, $A_n=A$, and $C_n=C$) and $x_k\cdot y_k=S_k$. [b]a)[/b] Find the maximal possible value of the area of the rctangle $OABC$ and all pairs of sets $(x_1,x_2,\ldots,x_k)$ and $(y_1,y_2,\ldots,y_k)$ at which this maximal area is achieved. [b]b)[/b] Find the minimal possible value of the area of the rctangle $OABC$ and all pairs of sets $(x_1,x_2,\ldots,x_k)$ and $(y_1,y_2,\ldots,y_k)$ at which this minimal area is achieved. [i](E. Manzhulina, B. Rublyov)[/i]

With expressions containing the symbol $*$, the following transformations can be performed: 1) rewrite the expression in the form $x * (y * z) as ((1 * x) * y) * z$; 2) rewrite the expression in the form $x * 1$ as $x$. Conversions can only be performed with an integer expression, but not with its parts. For example, $(1 *1) * (1 *1)$ can be rewritten according to the first rule as $((1 * (1 * 1)) * 1) * 1$ (taking $x = 1 * 1$, $y = 1$ and $z = 1$), but not as $1 * (1 * 1)$ or $(1* 1) * 1$ (in the last two cases, the second rule would be applied separately to the left or right side $1 * 1$). Find all positive integers $n$ for which the expression $\underbrace{1 * (1 * (1 * (...* (1 * 1)...))}_{n units}$ it is possible to lead to a form in which there is not a single asterisk. Note. The expressions $(x * y) * $z and $x * (y * z)$ are considered different, also, in the general case, the expressions $x * y$ and $y * x$ are different.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy the following condition for any real numbers $x{}$ and $y$ $$f(x)+f(x+y) \leq f(xy)+f(y).$$

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers for which \[ a_{n+2} = \left[\frac{2a_n}{a_{n+1}}\right]+\left[\frac{2a_{n+1}}{a_n}\right]. \] Prove that there exists a positive integer $m$ such that $a_m=4$ and $a_{m+1} \in\{3,4\}$. [b]Note.[/b] $[x]$ is the greatest integer not exceeding $x$.