Do numbers $x_1, x_2, \ldots, x_{99}$ exist, where each of them is equal to $\sqrt{2}+1$ or $\sqrt{2}-1$, and satisfy the equation \[x_1x_2+x_2x_3+x_3x_4+\ldots+x_{98}x_{99}+x_{99}x_1=199\,?\] Justify your answer.

Yesterday, $n\ge 4$ people sat around a round table. Each participant remembers only who his two neighbours were, but not necessarily which one sat on his left and which one sat on his right. Today, you would like the same people to sit around the same round table so that each participant has the same two neighbours as yesterday (it is possible that yesterday’s left-hand side neighbour is today’s right-hand side neighbour). You are allowed to query some of the participants: if anyone is asked, he will answer by pointing at his two neighbours from yesterday. a) Determine the minimal number $f(n)$ of participants you have to query in order to be certain to succeed, if later questions must not depend on the outcome of the previous questions. That is, you have to choose in advance the list of people you are going to query, before effectively asking any question. b) Determine the minimal number $g(n)$ of participants you have to query in order to be certain to succeed, if later questions may depend on the outcome of previous questions. That is, you can wait until you get the first answer to choose whom to ask the second question, and so on.

a) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $x_n-2x_{n+1}+x_{n+2} \le 0$ for any $n$ . Moreover $x_o=1,x_{20}=9,x_{200}=6$. What is the maximal value of $x_{2009}$ can be? b) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $2x_n-3x_{n+1}+x_{n+2} \le 0$ for any $n$. Moreover $x_o=1,x_1=2,x_3=1$. Can $x_{2009}$ be greater then $0,678$ ?

A bee travels in a series of steps of length $1$: north, west, north, west, up, south, east, south, east, down. (The bee can move in three dimensions, so north is distinct from up.) There exists a plane $P$ that passes through the midpoints of each step. Suppose we orthogonally project the bee’s path onto the plane $P$, and let $A$ be the area of the resulting figure. What is $A^2$?

Prova that for any real $a$, expresssion $A=(a-1)(a-3)(a-4)(a-6)+10$ is always positive. What is the minimum value that expression $A$ can take and for which values of $a$?

[b]5.[/b] Prove that neither the closed nor the open interval can be decomposed into finitely many mutually disjoint proper subsets which are all congruent by translation. [b](St. 2)[/b]

Let $A=\{1,2,\ldots,n\}$. For a permutation $P=(P(1), P(2), \ldots, P(n))$ of the elements of $A$, let $P(1)$ denote the first element of $P$. Find the number of all such permutations $P$ so that for all $i,j \in A$: (a) if $i < j<P(1)$, then $j$ appears before $i$ in $P$; and (b) if $P(1)<i<j$, then $i$ appears before $j$ in $P$.

Which of the following is not equal to $\dfrac{5}{4}$? $\text{(A)}\ \dfrac{10}{8} \qquad \text{(B)}\ 1\dfrac{1}{4} \qquad \text{(C)}\ 1\dfrac{3}{12} \qquad \text{(D)}\ 1\dfrac{1}{5} \qquad \text{(E)}\ 1\dfrac{10}{40}$

How many solutions are there to the equation \[x^2+2y^2+z^2=xyz\] where $1\le x,y,z\le200$ are positive even integers?

Show that if equiangular hexagon has sides $a, b, c, d, e, f$ in order then $a - d = e - b = c - f$.

Let $ABC$ be a triangle, $P$ the midpoint of $BC$, and $Q$ a point on segment $CA$ such that $|CQ| = 2|QA|$. Let $S$ be the intersection of $BQ$ and $AP$. Prove that $|AS| = |SP|$.

Let $a_1,a_2,a_3,\ldots$ be a sequence of all composite positive integers in increasing order. A sequence $b_1,b_2,b_3,\ldots$ is given for all positive integers $i$ by equation $$b_i=ia_1^2+(i-1)a_2^2+\ldots+2a_{i-1}^2+a_i^2$$ What is the maximum amount of consecutive elements of sequence $b_1,b_2,b_3,\ldots$ which can be divisible by $3$? [i]M. Shutro[/i]

A $2 \times 4 \times 8$ rectangular prism and a cube have the same volume. What is the difference between their surface areas?

([b]3[/b]) Let $ \ell$ be the line through $ (0,0)$ and tangent to the curve $ y \equal{} x^3 \plus{} x \plus{} 16$. Find the slope of $ \ell$.

The quadrilateral $ABCD$ is inscribed in a fixed circle. It has $AB$ parallel to $CD$ and the length $AC$ is fixed, but it is otherwise allowed to vary. If $h$ is the distance between the midpoints of $AC$ and $BD$ and $k$ is the distance between the midpoints of $AB$ and $CD$, show that the ratio $h/k$ remains constant.

We say a number is irie if it can be written in the form $1+\dfrac{1}{k}$ for some positive integer $k$. Prove that every integer $n \geq 2$ can be written as the product of $r$ distinct irie numbers for every integer $r \geq n-1$.

A rectangular grazing area is to be fenced off on three sides using part of a $ 100$ meter rock wall as the fourth side. Fence posts are to be placed every $ 12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $ 36$ m by $ 60$ m? \[ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16 \]

Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $2023$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $2023$ loses. Who wins if every player wants to win? [i]Proposed by Mykhailo Shtandenko[/i]

Let $ABC$ be an acute triangle. Let $P$ be a point such that $PB$ and $PC$ are tangent to circumcircle of $ABC$. Let $X$ and $Y$ be variable points on $AB$ and $AC$, respectively, such that $\angle XPY = 2\angle BAC$ and $P$ lies in the interior of triangle $AXY$. Let $Z$ be the reflection of $A$ across $XY$. Prove that the circumcircle of $XYZ$ passes through a fixed point. (Dominik Burek, Poland)

In triangle $ABC$, let $I$ be the incentre, $O$ be the circumcentre, and $H$ be the orthocentre. It is given that $IO = IH$. Show that one of the angles of triangle $ABC$ must be equal to $60$ degrees.

Find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for all positive integers $m,n$ with $m\ge n$, $$f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3).$$ Here $\varphi(n)$ denotes the number of positive integers coprime to $n$ and not exceeding $n$.

Find all pairs of non-negative integers $(m,n)$ satisfying $\frac{n(n+2)}{4}=m^4+m^2-m+1$

Let $0 = a_1 < a_2 < a_3 < \cdots < a_n < 1$ and $0 = b_1 < b_2 < b_3 \cdots < b_m < 1$ be real numbers such that for no $a_j$ and $b_k$ the relation $a_j + b_k = 1$ is satisfied. Prove that if the $mn$ numbers ${\ a_j + b_k : 1 \leq j \leq n , 1 \leq k \leq m \}}$ are reduced modulo $1$, then at least $m+n -1$ residues will be distinct.

a) Do there exist 5 circles in the plane such that each circle passes through exactly 3 centers of other circles? b) Do there exist 6 circles in the plane such that each circle passes through exactly 3 centers of other circles?

Let $n >3$ be an odd positive integer and $n=\prod_{i=1}^k p_i^{\alpha_i}$ where $p_i$ are primes and $\alpha_i$ are positive integers. We know that \[m=n(1-\frac{1}{p_1})(1-\frac{1}{p_2})(1-\frac{1}{p_3}) \cdots (1-\frac{1}{p_n}).\] Prove that there exists a prime $P$ such that $P|2^m -1$ but $P \nmid n.$