Given is a parallelogram $ABCD$, with $AB <AC <BC$. Points $E$ and $F$ are selected on the circumcircle $\omega$ of $ABC$ so that the tangenst to $\omega$ at these points pass through point $D$ and the segments $AD$ and $CE$ intersect. It turned out that $\angle ABF = \angle DCE$. Find the angle $\angle{ABC}$. A. Yakubov, S. Berlov

Quadrilateral can be cut into two isosceles triangles in two different ways. a) Can this quadrilateral be nonconvex? b) If given quadrilateral is convex, is it necessarily a rhomb?

There are two round tables with $n{}$ dwarves sitting at each table. Each dwarf has only two friends: his neighbours to the left and to the right. A good wizard wants to seat the dwarves at one round table so that each two neighbours are friends. His magic allows him to make any $2n$ pairs of dwarves into pairs of friends (the dwarves in a pair may be from the same or from different tables). However, he knows that an evil sorcerer will break $n{}$ of those new friendships. For which $n{}$ is the good wizard able to achieve his goal no matter what the evil sorcerer does? [i]Mikhail Svyatlovskiy[/i]

There are 52 cards in the deck, 13 of each suit. Vanya draws from the deck one card at a time. Removed cards are not returned to the deck. Every time Before taking out the card, Vanya makes a wish for some suit.Prove that if Vanya makes a wish every time, , the cards of which are in the deck has no less cards left than cards of any other suit, then the hidden suit will fall with the suit of the card drawn at least 13 times.

Find the number of ways to write the number $2013$ as the sum of two integers greater than or equal to zero so that when adding there is no carry over. Clarification: In the sum $2008+5=2013$ there is carry over from the units to the tens

Each term of an infinite sequene $a_1,a_2, \cdots$ is equal to 0 or 1. For each positive integer $n$, [list] [*] $a_n+a_{n+1} \neq a_{n+2} +a_{n+3}$ and [*] $a_n + a_{n+1}+a_{n+2} \neq a_{n+3} +a_{n+4} + a_{n+5}$ Prove that if $a_1~=~0$ , then $a_{2020}~=~1$

Three circle of unit radius passing through the point $P$ and one of the points of $A, B$ and $C$ each. What can be the radius of the circumcircle of the triangle $ABC$?

Given a scalene triangle $\vartriangle ABC$ with orthocenter $H$ and circumcenter $O$. The exterior angle bisector of $\angle BAC$ intersects circumcircle of $\vartriangle ABC$ at $N \ne A$. Let $D$ be another intersection of $HN$ and the circumcircle of $\vartriangle ABC$. The line passing through $O$, which is parallel to $AN$, intersects $AB,AC$ at $E, F$, respectively. Prove that $DH$ bisects the angle $\angle EDF$. [img]https://3.bp.blogspot.com/-F1mFwojG_I0/XnYNR8ofqSI/AAAAAAAALeo/zge24WF0EO8umPAaXprKAeXJHAj7pr6tQCK4BGAYYCw/s1600/imoc2019g5.png[/img]

Let $a$, $b$, $c$, $d$ and $n$ be positive integers such that $7\cdot 4^n = a^2+b^2+c^2+d^2$. Prove that the numbers $a$, $b$, $c$, $d$ are all $\geq 2^{n-1}$.

Four points $ A, B, C, D$ are chosen on a circle in such a way that arcs $ AB, BC,$ and $ CD$ are of the same length and the $ arc DA$ is longer than these three. Line $ AD$ and the line tangent to the circle at $ B$ intersect at $ E$. Let $ F$ be the other endpoint of the diameter starting at $ C$ of the circle. Prove that triangle $ DEF$ is equilateral.

If $x=t^{(1/(t-1))}$ and $x=t^{(t/(t-1))}$, $t>0$, $t\not=1$, a relation between $x$ and $y$ is $\textbf{(A)}\ y^x=x^{1/y}\qquad \textbf{(B)}\ y^{1/x}=x^{y} \qquad \textbf{(C)}\ y^x=x^{y}\qquad \textbf{(D)}\ x^x=y^y\\ \textbf{(E)}\ \text{none of these}$

Let $S$ be a set of size $n$, and $k$ be a positive integer. For each $1 \le i \le kn$, there is a subset $S_i \subset S$ such that $|S_i| = 2$. Furthermore, for each $e \in S$, there are exactly $2k$ values of $i$ such that $e \in S_i$. Show that it is possible to choose one element from $S_i$ for each $1 \le i \le kn$ such that every element of $S$ is chosen exactly $k$ times.

Inside the triangle $ABC$ choose the point $M$, and on the side $BC$ - the point $K$ in such a way that $MK || AB$. The circle passing through the points $M, \, \, K, \, \, C,$ crosses the side $AC$ for the second time at the point $N$, a circle passing through the points $M, \, \, N, \, \, A, $ crosses the side $AB$ for the second time at the point $Q$. Prove that $BM = KQ$. (Nagel Igor)

Let $ \{a_k\}$ be a sequence of integers such that $ a_1 \equal{} 1$ and $ a_{m \plus{} n} \equal{} a_m \plus{} a_n \plus{} mn$, for all positive integers $ m$ and $ n$. Then $ a_{12}$ is $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 56 \qquad \textbf{(C)}\ 67 \qquad \textbf{(D)}\ 78 \qquad \textbf{(E)}\ 89$

Let $P(x)$ and $Q(x)$ be polynomials of degree $p$ and $q$ respectively such that every coefficient is $1$ or $2023$. If $P(x)$ divides $Q(x)$, prove that $p+1$ divides $q+1$.

A unit square $ABCD$ is given in the plane, with $O$ being the intersection of its diagonals. A ray $l$ is drawn from $O$. Let $X$ be the unique point on $l$ such that $AX + CX = 2$, and let $Y$ be the point on $l$ such that $BY + DY = 2$. Let $Z$ be the midpoint of $\overline{XY}$, with $Z = X$ if $X$ and $Y$ coincide. Find, with proof, the minimum value of the length of $OZ$.

Let the operation $ f$ of $ k$ variables defined on the set $ \{ 1,2,\ldots,n \}$ be called $ \textit{friendly}$ toward the binary relation $ \rho$ defined on the same set if \[ f(a_1,a_2,\ldots,a_k) \;\rho\ \;f(b_1,b_2,\ldots,b_k)\] implies $ a_i \; \rho \ b_i$ for at least one $ i,1\leq i \leq k$. Show that if the operation $ f$ is friendly toward the relations "equal to" and "less than," then it is friendly toward all binary relations. [i]B. Csakany[/i]

Let $n$ be a natural number divisible by $3$. We have a $n \times n$ table and each square is colored either black or white. Suppose that for all $m \times m$ sub-tables from the table ($m > 1$), the number of black squares is not more than white squares. Find the maximum number of black squares.

David and Evan are playing a game. Evan thinks of a positive integer $N$ between 1 and 59, inclusive, and David tries to guess it. Each time David makes a guess, Evan will tell him whether the guess is greater than, equal to, or less than $N$. David wants to devise a strategy that will guarantee that he knows $N$ in five guesses. In David's strategy, each guess will be determined only by Evan's responses to any previous guesses (the first guess will always be the same), and David will only guess a number which satisfies each of Evan's responses. How many such strategies are there? Note: David need not guess $N$ within his five guesses; he just needs to know what $N$ is after five guesses.

Georg investigates which integers are expressible in the form $$\pm 1^2 \pm 2^2 \pm 3^2 \pm \dots \pm n^2.$$ For example, the number $3$ can be expressed as $ -1^2 + 2^2$, and the number $-13$ can be expressed as $+1^2 + 2^2 + 3^2 - 4^2 + 5^2 - 6^2$. Are all integers expressible in this form?

Let $ p$ be a prime and let $ d \in \left\{0,\ 1,\ \ldots,\ p\right\}$. Prove that \[ \sum_{k \equal{} 0}^{p \minus{} 1}{\binom{2k}{k \plus{} d}}\equiv r \pmod{p}, \]where $ r \equiv p\minus{}d \pmod 3$, $ r\in\{\minus{}1,0,1\}$.

Let $d(n)=\sum_{0<d|n}{1}$. Show that, for any natural $n>1$, \[ \sum_{2 \leq i \leq n}{\frac{1}{i}} \leq \sum{\frac{d(i)}{n}} \leq \sum_{1 \leq i \leq n}{\frac{1}{i}} \]

Let $\Gamma$ be a circle and $A$ a point outside $\Gamma$. The tangent lines to $\Gamma$ through $A$ touch $\Gamma$ at $B$ and $C$. Let $M$ be the midpoint of $AB$. The segment $MC$ meets $\Gamma$ again at $D$ and the line $AD$ meets $\Gamma$ again at $E$. Given that $AB=a$, $BC=b$, compute $CE$ in terms of $a$ and $b$.

Let $x, y$ be real numbers such that $x-y, x^2-y^2, x^3-y^3$ are all prime numbers. Prove that $x-y=3$. EDIT: Problem submitted by Leonel Castillo, Panama.

Find the maximum value of \[\sin^2 \theta_1+\sin^2 \theta_2+\cdots+\sin^2 \theta_n\] subject to the restrictions $0 \leq \theta_i , \theta_1+\theta_2+\cdots+\theta_n=\pi.$