Triangle $ABC$ is given. Let $M$ be a midpoint of segment $AB$ and $T$ be a midpoint of arc $BC$ not containing $A$ of circumcircle of $ABC$. Let $K$ be a point inside triangle $ABC$ such that $MATK$ is an isosceles trapezoid with $AT||MK$. Show that $AK = KC$.
We call a set of squares with sides parallel to the coordinate axes and vertices with integer coordinates friendly if any two of them have exactly two points in common. We consider friendly sets in which each of the squares has sides of length $n$. Determine the largest possible number of squares in such a friendly set.
A rectangle of size $m \times n$ consisting of $m \cdot n$ squares of size $1 \times 1$ is given. Compute the sum of areas of all subrectangles consisting of some of the $m \cdot n$ squares.
Let $ABC$ be a triangle with circumcircle $\omega$. Let $D$, $E$ and $F$ be points on the sides $BC$, $CA$ and $AB$ such that the circumcircle of the triangle $DEF$ touches $\omega$ at $A$. Let $G$ and $H$ be the intersection points of the circumcircles of the triangles $BDE$ and $CDF$ with $\omega$ (different from $B$, $C$), respectively. Prove that the lines $GE$ and $HF$ intersect on $AD$.
Given positive real numbers $a, b, c, d$ that satisfy equalities
$$
a^2 + d^2 - ad = b^2 + c^2 + bc \quad \text{and} \quad a^2 + b^2 = c^2 + d^2,
$$
find all possible values of the expression $\frac{ab+cd}{ad+bc}$.
Let $I$ be the incenter of $\triangle ABC$ and let $H_a$, $H_b$, and $H_c$ be the orthocenters of $\triangle BIC$, $\triangle CIA$, and $\triangle AIB$, respectively. The line $H_aH_b$ meets $AB$ at $X$ and the line $H_aH_c$ meets $AC$ at $Y$. If the midpoint $T$ of the median $AM$ of $\triangle ABC$ lies on $XY$, prove that the line $H_aT$ is perpendicular to $BC$.
Let $a_n$ be the sequence defined by some $a_0$ and the recursion
$$
a_{n+1} = a_n + 2 \cdot 3^n
$$
for $n \ge 0$.
Determine all rational values of $a_0$ such that
$$
\frac{a_k^j}{a_j^k}
$$
is an integer for all integers $j$ and $k$ with $0 \lt j \lt k$.
Two players play a game on an $N \times N$ board. The players alternately mark a cell in such a way that there is never a diagonal on the board containing two marked cells. For which $N \gt 0$ does the starting player have a winning strategy?
The sequence $a_0, a_1, \dots$ is defined by the initial conditions $a_0 = 1$, $a_1 = 6$ and the recursion $a_{n+1} = 4a_n - a_{n-1} + 2$ for $n \gt 1$. Prove that $a_{2^k-1}$ has at least three prime factors for every positive integer $k \gt 3$.
We are to paint $n$ seats in a row, each red or green. We call painting odd, if each monochromatic sequence is of odd length. By monochromatic sequence we mean a sequence of seats in one color, which is bounded by seats of the other color or a wall. Count how many ways of odd painting are there.
Let $q$ be a fixed positive rational number. Call number $x$ *charismatic* if there exist a positive integer $n$ and integers $\alpha_1, \alpha_2, \dots, \alpha_n$ such that
$$
x = (q+1)^{\alpha_1} \cdot (q+2)^{\alpha_2} \cdots (q+n)^{\alpha_n}.
$$
a) Prove that $q$ can be chosen in such a way that every positive rational number turns out to be charismatic.
b) Is it true for every $q$ that, for every charismatic number $x$, the number $x+1$ is charismatic too?
Let $\Gamma$ be the circumcircle of acute triangle $ABC$. The perpendicular to $AB$ from $C$ meets $AB$ at $D$ and $\Gamma$ again at $E$. The bisector of angle $C$ meets $AB$ at $F$ and $\Gamma$ again at $G$. The line $GD$ meets $\Gamma$ again at $H$ and the line $HF$ meets $\Gamma$ again at $I$. Prove that $AI = EB$.
Determine all quadruples $(a, b, c, d)$ of real numbers satisfying the following system of equations.
$$
\begin{aligned}
ab + ac &= 3b + 3c \\
bc + bd &= 5c + 5d \\
ac + cd &= 7a + 7d \\
ad + bd &= 9a + 9b
\end{aligned}
$$
Is it possible to subdivide a convex $2014$-gon, by means of diagonals that do not intersect, into triangles, in such a way that each vertex be incident to an odd number of triangles?
Albert and Betty play the following game. There are two bowls on a table; a red bowl and a blue bowl. At the beginning of the game there are $100$ blue balls in the red bowl and $100$ red balls in the blue bowl. In each turn a player must take one of the following moves:
a) Take two balls of different colors from one bowl and throw the balls away.
b) Take two red balls from the blue bowl and put them in the red bowl.
c) Take two blue balls from the red bowl and put them in the blue bowl.
They take turns alternately, and Albert starts. The player who first takes the last red ball from the blue bowl or the last blue ball from the red bowl wins. Determine who has a winning strategy.
Let $A_0B_0C_0$ be a triangle with area equal to $\sqrt{2}$. We consider the excenters $A_1$, $B_1$ and $C_1$ then we consider the excenters, say $A_2$, $B_2$ and $C_2$, of the triangle $A_1B_1C_1$. By continuing this procedure, examine if it is possible to arrive to a triangle $A_nB_nC_n$ with all coordinates rational.
The graph of the parabola $y = x^2$ is drawn on the Cartesian plane $Oxy$. A triangle $ABC$ is inscribed in the parabola so that its side $AB$ is parallel to the $Ox$ axis and point $C$ lies between the line $AB$ and the $Ox$ axis. It is known that the length of the side $AB$ is 1 less than the length of the altitude $CH$ of the triangle $ABC$.
Find the value of the angle $ACB$.
(a) For which triangles with sides of length $a$, $b$ and $c$ do the inequalities $a^2 + b^2 \gt c^2$, $b^2 + c^2 \gt a^2$ and $a^2 + c^2 \gt b^2$ hold (along with the usual triangle inequalities $a+b\gt c$, $b+c\gt a$ and $c+a\gt b$)?
(b) For which triangles with sides of length $a$, $b$ and $c$ do the inequalities $a^n + b^n \gt c^n$, $b^n + c^n \gt a^n$ and $a^n + c^n \gt b^n$ hold for all positive integers $n$ (along with the usual triangle inequalities $a+b\gt c$, $b+c\gt a$ and $c+a\gt b$)?
