Let $a\leqslant b\leqslant c$ be non-negative integers. A triangle on a checkered plane with vertices in the nodes of the grid is called an $(a,b,c)$[i]-triangle[/i] if there are exactly $a{}$ nodes on one side of it (not counting vertices), exactly $b{}$ nodes on the second side, and exactly $c{}$ nodes on the third side. [list] [*]Does there exist a $(9,10,11)$-triangle? [*]Find all triples of non-negative integers $a\leqslant b\leqslant c$ for which there exists an $(a,b,c)$-triangle. [*]For each such triple, find the minimum possible area of the $(a,b,c)$-triangle. [/list] [i]Proposed by P. Kozhevnikov[/i]

Let $I$ be the incenter of triangle $ABC$, $O_1$ a circle through $B$ tangent to $CI$, and $O_2$ a circle through $C$ tangent to $BI$. Prove that $O_1$,$O_2$ and the circumcircle of $ABC$ have a common point.

For each positive integer $k$, denote by $\tau(k)$ the number of all positive divisors of $k$, including $1$ and $k$. Let $a$ and $b$ be positive integers such that $\tau(\tau(an)) = \tau(\tau(bn))$ for all positive integers $n$. Prove that $a=b$.

We call a positive integer $n\ge 4$[i] beautiful[/i] if there exists some permutation $$\{x_1,x_2,\dots ,x_{n-1}\}$$ of $\{1,2,\dots ,n-1\}$ such that $\{x^1_1,\ x^2_2,\ \dots,x^{n-1}_{n-1}\}$ gives all the residues $\{1,2,\dots, n-1\}$ modulo $n$. Prove that if $n$ is beautiful then $n=2p,$ for some prime number $p.$

Using each of the eight digits $1,3,4,5,6,7,8$ and $9$ exactly once, a three-digit number $A$, two two-digit numbers $B$ and $C$, $B<C$, and a one digit number $D$ are formed. The numbers are such that $A+D=B+C=143$. In how many ways can this be done?

A set of segments with the total length less than $1$ is given on a line. Prove that every set of $n$ points on the line can be translated by a vector of length not exceeding $n/2$, so that all the obtained points are away from the given segments.

Prove that a circle constructed with the side $AB$ of a triangle $ABC$ as a diameter touches the inscribed circle of the triangle $ABC$ if and only if the side $AB$ is equal to the radius of the exircle on that side.

Point $ I_1$ is the reflection of incentre $ I$ of triangle $ ABC$ across the side $ BC$. The circumcircle of $ BCI_1$ intersects the line $ II_1$ again at point $ P$. It is known that $ P$ lies outside the incircle of the triangle $ ABC$. Two tangents drawn from $ P$ to the latter circle touch it at points $ X$ and $ Y$. Prove that the line $ XY$ contains a medial line of the triangle $ ABC$. [i]Author: L. Emelyanov[/i]

Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ respectively intersect each other at points $A$ and $B$, and point $O_1$ lies on $\omega_2$. Let $P$ be an arbitrary point lying on $\omega_1$. Lines $BP, AP$ and $O_1O_2$ cut $\omega_2$ for the second time at points $X$, $Y$ and $C$, respectively. Prove that quadrilateral $XPYC$ is a parallelogram. [i]Proposed by Iman Maghsoudi[/i]

in a quadrilateral $ABCD$ digonals are perpendicular to each other. let $S$ be the intersection of digonals. $K$,$L$,$M$ and $N$ are reflections of $S$ to $AB$,$BC$,$CD$ and $DA$. $BN$ cuts the circumcircle of $SKN$ in $E$ and $BM$ cuts the circumcircle of $SLM$ in $F$. prove that $EFLK$ is concyclic.(20 points)

In a country there are $n$ major cities, $n \ge 4$, connected by railroads, such that each city is directly connected to each other city. Each railroad company in that country has but only one train, which connects a series of cities, at least two, such that the train doesn’t pass through the same city twice in one shift. The companies divided the market such that any two cities are directly$^1$ connected only by one company. Prove that among any $n+1$ companies, there are two which have no common train station or there are two cities that are connected by two trains belonging to two of these $n+1$ companies. $^1$ directly connected means that they are connected by a railroad, without no other station between them

Consider a circle $C$ of center $O$ and its inner point $Q$, different from $O$. Where we must place a point $P$ on the circle $C$ so that the angle $\angle OPQ$ is the largest possible?

Let $\triangle ABC$ be an acute triangle, and let $D$ be the projection of $A$ on $BC$. Let $M,N$ be the midpoints of $AB$ and $AC$ respectively. Let $\Gamma_1$ and $\Gamma_2$ be the circumcircles of $\triangle BDM$ and $\triangle CDN$ respectively, and let $K$ be the other intersection point of $\Gamma_1$ and $\Gamma_2$. Let $P$ be an arbitrary point on $BC$ and $E,F$ are on $AC$ and $AB$ respectively such that $PEAF$ is a parallelogram. Prove that if $MN$ is a common tangent line of $\Gamma_1$ and $\Gamma_2$, then $K,E,A,F$ are concyclic.

Let $a$ and $b$ be real numbers randomly (and independently) chosen from the range $[0,1]$. Find the probability that $a, b$ and $1$ form the side lengths of an obtuse triangle.

In acute-angled triangle $ABC$, let $H$ be the foot of perpendicular from $A$ to $BC$ and also suppose that $J$ and $I$ are excenters oposite to the side $AH$ in triangles $ABH$ and $ACH$. If $P$ is the point that incircle touches $BC$, prove that $I,J,P,H$ are concyclic.

The edges of $K_{2017}$ are each labeled with $1,2,$ or $3$ such that any triangle has sum of labels at least $5.$ Determine the minimum possible average of all $\dbinom{2017}{2}$ labels. (Here $K_{2017}$ is defined as the complete graph on 2017 vertices, with an edge between every pair of vertices.) [i]Proposed by Michael Ma[/i]

Let $A$ be a set of positive integers having the following property: for each positive integer $n$ exactly one of the three numbers $n, 2n$ and $3n$ is an element of $A$. Furthermore, it is given that $2 \in A$. Prove that $13824 \notin A$.

Find all triples $(a, b, p)$ of positive integers, where $p$ is a prime number, such that $a^p - b^p = 2013$.

Let $f(x)$ be a polynomial of nonzero degree. Can it happen that for any real number $a$, an even number of real numbers satisfy the equation $f(x) = a$?

Factor $a^{10} + a^5 + 1$ into nonconstant polynomials with integer coefficients

Let $m,n$ be positive integers with $1 \le n < m$. A box is locked with several padlocks which must all be opened to open the box, and which all have different keys. The keys are distributed among $m$ people. Suppose that among these people, no $n$ can open the box, but any $n+1$ can open it. Find the smallest possible number $l$ of locks and then the total number of keys for which this is possible.

Let $ABCD$ be an isosceles trapezoid such that $AB=10$, $BC=15$, $CD=28$, and $DA=15$. There is a point $E$ such that $\triangle AED$ and $\triangle AEB$ have the same area and such that $EC$ is minimal. Find $EC$.

We are given some three element subsets of $\{1,2, \dots ,n\}$ for which any two of them have at most one common element. We call a subset of $\{1,2, \dots ,n\}$ [i]nice [/i] if it doesn't include any of the given subsets. If no matter how the three element subsets are selected in the beginning, we can add one more element to every 29-element [i]nice [/i] subset while keeping it nice, find the minimum value of $n$.

Pinocchio claims that he can divide an isoceles triangle into three triangles, any two of which can be put together to form a new isosceles triangle. Is Pinocchio lying? (A Shapovalov)

In $\triangle ABC$, $AB = 40$, $BC = 60$, and $CA = 50$. The angle bisector of $\angle A$ intersects the circumcircle of $\triangle ABC$ at $A$ and $P$. Find $BP$. [i]Proposed by Eugene Chen[/i]