A point $O$ is chosen inside the square $ABCD$. The square $A'B'C'D'$ is the image of the square $ABCD$ under the homothety with center at point $O$ and coefficient $k> 1$ (points $A', B', C', D' $ are images of points $A, B, C, D$ respectively). Prove that the sum of the areas of the quadrilaterals $A'ABB'$ and $C'CDD'$ is equal to the sum of the areas quadrilaterals $B'BCC'$ and $D'DAA'$.

Let $n$ be a positive integer and let $a, b, c, d$ be integers such that $n | a + b + c + d$ and $n | a^2 + b^2 + c^2 + d^2. $ Show that $$n | a^4 + b^4 + c^4 + d^4 + 4abcd.$$

Viorel is $10$ years old, and his mother is $31$ years old. In how many years will Viorel be half his mother's age?

On a dance evening, each of the gentlemen present has sex with at least one of the ladies present danced and each of the ladies present danced with at least one of the gentlemen present. No gentleman has sex with every lady present and no lady has sex with every gentleman present danced. It must be proven that there were two such ladies and two such gentlemen among those present has that that evening each of the two ladies with exactly one of the two men, and each of the both men danced with exactly one of the two women. It is assumed that the dance evening did not take place without ladies and gentlemen, i.e. the crowd, which consists of all the ladies and gentlemen present, is not empty. [hide=original wording]An einem Tanzabend hat jeder der anwesenden Herren mit mindestens einer der anwesenden Damen getanzt und jede der anwesenden Damen mit mindestens einem der anwesenden Herren. Kein Herr hat mit jeder der anwesenden Damen und keine Dame mit jedem der anwesenden Herren getanzt. Es ist zu beweisen, dass es unter den Anwesenden zwei solche Damen und zwei solche Herren gegeben hat, dass an dem Abend jede der beiden Damen mit genau einem der beiden Herren, und jeder der beiden Herren mit genau einer der beiden Damen getanzt hat. Es wird vorausgesetzt, dass der Tanzabend nicht ohne Damen und Herren stattgefunden hat, d.h., die Menge, die aus allen anwesenden Damen und Herren besteht, ist nicht leer.[/hide]

[b]Problem 1. [/b]In the cells of square table are written the numbers $1$, $0$ or $-1$ so that in every line there is exactly one $1$, amd exactly one $-1$. Each turn we change the places of two columns or two rows. Is it possible, from any such table, after finite number of turns to obtain its opposite table (two tables are opposite if the sum of the numbers written in any two corresponding squares is zero)? [i] Emil Kolev[/i]

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

Two circles of the same radii intersect in two distinct points $P$ and $Q$. A line passing through $P$, not touching any of the circles, intersects the circles again at $A$ and $B$. Prove that $Q$ lies on the perpendicular bisector of $AB$.

For positive real numbers $x,y,z$ with $xy+yz+zx=1$, prove that $$\frac{2}{xyz}+9xyz \geq 7(x+y+z)$$

We consider the $n$ vertices of a regular polygon with $n$ sides. There is a set of triangles with vertices at these $n$ points with the property that for each triangle in the set, the sides of at least one are not the side of any other triangle in the set. What is the largest amount of triangles that can have the set? [hide=original wording]Consideramos los n vértices de un polígono regular de n lados. Se tiene un conjunto de triángulos con vértices en estos n puntos con la propiedad que para cada triángulo del conjunto, al menos uno de sus lados no es lado de ningún otro triángulo del conjunto. ¿Cuál es la mayor cantidad de triángulos que puede tener el conjunto?[/hide]

A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen? $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 16$

Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ in points $A$, $B$. The intersection of tangents to $G_2$ in points $A$, $B$ is point $C$. If $C \in G_1$, find all possible values of $p$.

About pentagon $ABCDE$ is known that angle $A$ and angle $C$ are right and that the sides $| AB | = 4$, $| BC | = 5$, $| CD | = 10$, $| DE | = 6$. Furthermore, the point $C'$ that appears by mirroring $C$ in the line $BD$, lies on the line segment $AE$. Find angle $E$.

Let $$N = \sum^{512}_{i=0}i {512 \choose i}.$$ What is the greatest integer $a$ such that $2^a$ is a divisor of $N$?

Simplify: $ \frac{a^3}{(a\minus{}b)(a\minus{}c)}\plus{}\frac{b^3}{(b\minus{}a)(b\minus{}c)}\plus{}\frac{c^3}{(c\minus{}a)(c\minus{}b)}$

let $ H $ the orthocenter of the triangle $ ABC $ pass two lines $ l_1 $ and $ l_2 $ through $ H $ such that $ l_1 \bot l_2 $ we have $ l_1 \cap BC = D $ and $ l_1 \cap AB = Z $ also $ l_2 \cap BC = E $ and $ l_2 \cap AC = X $ like this picture pass a line $ d_1$ through $ D $ parallel to $ AC $ and another line $ d_2 $ through $ E $ parallel to $ AB $ let $ d_1 \cap d_2 = Y $ prove $ X $ $ , $ $ Y $ and $ Z $ are on a same line

Let $ABC$ be a triangle such that $\angle CAB = 60^o$. Consider $D, E$ points on sides $AC$ and $AB$ respectively such that $BD$ bisects angle $\angle ABC$ , $CE$ bisects angle $\angle BCA$ and let $I$ be the intersection of them. Prove that $|ID| =|IE|$.

Prove that every infinite sequence $S$ of distinct positive integers contains either an infinite subsequence such that for every pair of terms, neither term ever divides the other, or an infinite subsequence such that in every pair of terms, one always divides the other.

On a circle, numbers from $1$ to $100$ are arranged in some order. We call a pair of numbers [i]good [/i] if these two numbers do not stand side by side, and at least on one of the two arcs into which they break a circle, all the numbers are less than each of them. What can be the total number of [i]good [/i] pairs?

In the United States, coins have the following thicknesses: penny, $ 1.55$ mm; nickel, $ 1.95$ mm; dime, $ 1.35$ mm; quarter, $ 1.75$ mm. If a stack of these coins is exactly $ 14$ mm high, how many coins are in the stack? $ \textbf{(A)}\ 7\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 9\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 11$

Mr. Rose gave a test to his two calculus classes. His first period class has $20$ students, and their average score on the test was $80$. His second period class has $30$ students, and their average score was $90$. What was the average score of all $50$ of his calculus students?

Let $ ABC$ be a triangle, and $ I$ its incenter. Let the incircle of $ ABC$ touch side $ BC$ at $ D$, and let lines $ BI$ and $ CI$ meet the circle with diameter $ AI$ at points $ P$ and $ Q$, respectively. Given $ BI \equal{} 6, CI \equal{} 5, DI \equal{} 3$, determine the value of $ \left( DP / DQ \right)^2$.

A family of sets $F$ is called perfect if the following condition holds: For every triple of sets $X_1, X_2, X_3\in F$, at least one of the sets $$ (X_1\setminus X_2)\cap X_3,$$ $$(X_2\setminus X_1)\cap X_3$$ is empty. Show that if $F$ is a perfect family consisting of some subsets of a given finite set $U$, then $\left\lvert F\right\rvert\le\left\lvert U\right\rvert+1$. [i]Proposed by Michał Pilipczuk[/i]

Let $D$ be a point on the side $AC$ of a triangle $ABC$. Suppose that the incircle of triangle $BCD$ intersects $BD$ and $CD$ at $X$, $Y$, respectively. Show that $XY$ passes through a fixed point when $D$ is moving on the side $AC$.

The numbers $1, 2, 3, . . . , 101$ are written in a row in some order. Prove that it is always possible to erase $90 $ of the numbers so that the remaining $11$ numbers remain arranged in either increasing or decreasing order.

In circle with radius $10$, point $M$ is on chord $PQ$ such that $PM=5$ and $MQ=10$. Through point $M$ we draw chords $AB$ and $CD$, and points $X$ and $Y$ are intersection points of chords $AD$ and $BC$ with chord $PQ$ (see picture), respectively. If $XM=3$ find $MY$ [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYy9kLzBiMmFmM2ViOGVmOTlmZDA5NGY2ZWY4MjM1YWI0ZDZjNjJlNzA1LnBuZw==&rn=Z2VvbWV0cmlqYS5wbmc=[/img]