Let $ABC$ be a triangle and $AD,BE,CF$ be cevians concurrent at a point $P$. Suppose each of the quadrilaterals $PDCE,PEAF$ and $PFBD$ has both circumcircle and incircle. Prove that $ABC$ is equilateral and $P$ coincides with the center of the triangle.

There is a row that consists of digits from $ 0$ to $ 9$ and Ukrainian letters (there are $ 33$ of them) with following properties: there aren’t two distinct digits or letters $ a_i$, $ a_j$ such that $ a_i > a_j$ and $ i < j$ (if $ a_i$, $ a_j$ are letters $ a_i > a_j$ means that $ a_i$ has greater then $ a_j$ position in alphabet) and there aren’t two equal consecutive symbols or two equal symbols having exactly one symbol between them. Find the greatest possible number of symbols in such row.

Let $M$ be a point which has coordinates $(p\times 1994,7p\times 1994)$ in the Cartesian plane ($p$ is a prime). Find the number of right-triangles satisfying the following conditions: (1) all vertexes of the triangle are lattice points, moreover $M$ is on the right-angled corner of the triangle; (2) the origin ($0,0$) is the incenter of the triangle.

Find all triples $(a, b, n)$ of positive integers such that $a$ and $b$ are both divisors of $n$, and $a+b = \frac{n}{2}$ .

The vault of a bank has $N$ locks. To open it, they must be operated simultaneously. Five executives have some of the keys, so any trio can open the vault, but no pair can do it. Determine $N$.

On the plane are given $ k\plus{}n$ distinct lines , where $ k>1$ is integer and $ n$ is integer as well.Any three of these lines do not pass through the same point . Among these lines exactly $ k$ are parallel and all the other $ n$ lines intersect each other.All $ k\plus{}n$ lines define on the plane a partition of triangular , polygonic or not bounded regions. Two regions are colled different, if the have not common points or if they have common points only on their boundary.A regions is called ''good'' if it contained in a zone between two parallel lines . If in a such given configuration the minimum number of ''good'' regionrs is $ 176$ and the maximum number of these regions is $ 221$, find $ k$ and $ n$. Babis

Two ants crawl along the perimeter of a polygonal table, so that the distance between them is always $10$ cm. Each side of the table is more than $1$ meter long. At the initial moment both ants are on the same side of the table. (a) [i](2 points)[/i] Suppose that the table is a convex polygon. Is it always true that both ants can visit each point on the perimeter? (b) [i](4 points)[/i] Is it always true (this time without assumption of convexity) that each point on the perimeter can be visited by at least one ant?

Let $ ABCD$ be a convex quadrilateral with $ BA\neq BC$. Denote the incircles of triangles $ ABC$ and $ ADC$ by $ \omega_{1}$ and $ \omega_{2}$ respectively. Suppose that there exists a circle $ \omega$ tangent to ray $ BA$ beyond $ A$ and to the ray $ BC$ beyond $ C$, which is also tangent to the lines $ AD$ and $ CD$. Prove that the common external tangents to $ \omega_{1}$ and $\omega_{2}$ intersect on $ \omega$. [i]Author: Vladimir Shmarov, Russia[/i]

Find all natural numbers $n\geq 3$ satisfying one can cut a convex $n$-gon into different triangles along some of the diagonals (None of these diagonals intersects others at any point other than vertices) and the number of diagonals are used at each vertex is even.

A triangle with vertices $A(0, 2)$, $B(-3, 2)$, and $C(-3, 0)$ is reflected about the $x$-axis, then the image $\triangle A'B'C'$ is rotated counterclockwise about the origin by $90^{\circ}$ to produce $\triangle A''B''C''$. Which of the following transformations will return $\triangle A''B''C''$ to $\triangle ABC$? $\textbf{(A)}$ counterclockwise rotation about the origin by $90^{\circ}$. $\textbf{(B)}$ clockwise rotation about the origin by $90^{\circ}$. $\textbf{(C)}$ reflection about the $x$-axis $\textbf{(D)}$ reflection about the line $y = x$ $\textbf{(E)}$ reflection about the $y$-axis.

If positive integers $a$ and $b$ have 99 and 101 different positive divisors respectively (including 1 and the number itself), can the product $ab$ have exactly 150 positive divisors?

101 numbers are written in a circle. Near the first number the statement "This number is bigger than the next one" is written, near the second "This number is bigger that the next two" and etc, near the 100th "This number is bigger than the next 100 numbers". What is the maximum possible amount of the statements that can be true? [i]M. Karpuk[/i]

Determine the smallest constant $N$ so that the following may hold true: Geostan has deployed secret agents in Combostan. All pairs of agents can communicate, either directly or through other agents. The distance between two agents is the smallest number of agents in a communication chain between the two agents. Andreas and Edvard are among these agents, and Combostan has given Noah the task of determining the distance between Andreas and Edvard. Noah has a list of numbers, one for each agent. The number of an agent describes the longest of the two distances from the agent to Andreas and Edvard. However, Noah does not know which number corresponds to which agent, or which agents have direct contact. Given this information, he can write down $N$ numbers and prove that the distance between Andreas and Edvard is one of these $N$ numbers. The number $N$ is independent of the agents’ communication network.

Find all prime numbers $p$ for which $p^2 +11$ has exactly six positive divisors.

$g(x) \colon \int_{10}^{x} \log_{10}(\log_{10}(t^2-1000t+10^{1000})) dt$ (a) Find the domain of $g(x)$ (b) Approximate the value of $g(1000)$ (c) Find $x \in [10, 1000]$ to maximize the slope of $g(x)$ (d) Find $x \in [10, 1000]$ to minimize the slope of $g(x)$ (e) Determine, if it exists, $\lim_{x \to \infty} \frac{\ln(x)}{g(x)}$

Given $4$ lines in Euclidean $3-$space: $L_1: x = 1, y = 0;$ $L_2: y = 1, z = 0;$ $L_3: x = 0, z = 1;$ $L_4: x = y, y = -6z.$ Find the equations of the two lines which both meet all of the $L_i.$

For $R>1$ let $\mathcal{D}_R =\{ (a,b)\in \mathbb{Z}^2: 0<a^2+b^2<R\}$. Compute $$\lim_{R\rightarrow \infty}{\sum_{(a,b)\in \mathcal{D}_R}{\frac{(-1)^{a+b}}{a^2+b^2}}}.$$ [i]Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro[/i]

Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

Let $A$ and $A' $ be fixed points on two equal circles in the plane and let $AB$ and $A' B'$ be arcs of these circles of the same length $x$. Find the locus of the midpoint of segment $BB'$ when $x$ varies: (a) if the arcs have the same direction, (b) if the arcs have opposite directions.

Mark three nodes on a cellular paper so that the semiperimeter of the obtained triangle would be equal to the sum of its two smallest medians. [i](Proposed by L.Emelyanov)[/i]

If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]

Let a,b,c,d be real numbers satisfying $|a|,|b|,|c|,|d|>1$ and $abc+abd+acd+bcd+a+b+c+d=0$. Prove that $\frac {1} {a-1}+\frac {1} {b-1}+ \frac {1} {c-1}+ \frac {1} {d-1} >0$

In the center of each cell of a checkered rectangle $M{}$ there is a point-like light bulb. All the light bulbs are initially switched off. In one turn it is allowed to choose a straight line not intersecting any light bulbs such that on one side of it all the bulbs are switched off, and to switch all of them on. In each turn at least one bulb should be switched on. The task is to switch on all the light bulbs using the largest possible number of turns. What is the maximum number of turns if: [list=a] [*]$M$ is a square of size $21 \times 21$; [*]$M$ is a rectangle of size $20 \times 21$? [/list] [i]Alexandr Shapovalov[/i]

Let $ABC$ be a triangle with circumcircle $\omega$ and $\ell$ a line without common points with $\omega$. Denote by $P$ the foot of the perpendicular from the center of $\omega$ to $\ell$. The side-lines $BC,CA,AB$ intersect $\ell$ at the points $X,Y,Z$ different from $P$. Prove that the circumcircles of the triangles $AXP$, $BYP$ and $CZP$ have a common point different from $P$ or are mutually tangent at $P$. [i]Proposed by Cosmin Pohoata, Romania[/i]

Given a quadrilateral $ ABCD$ with $ \angle B\equal{}\angle D\equal{}90^{\circ}$. Point $ M$ is chosen on segment $ AB$ so taht $ AD\equal{}AM$. Rays $ DM$ and $ CB$ intersect at point $ N$. Points $ H$ and $ K$ are feet of perpendiculars from points $ D$ and $ C$ to lines $ AC$ and $ AN$, respectively. Prove that $ \angle MHN\equal{}\angle MCK$.