In an $ ABC$ triangle such that $ m(\angle B)>m(\angle C)$, the internal and external bisectors of vertice $ A$ intersects $ BC$ respectively at points $ D$ and $ E$. $ P$ is a variable point on $ EA$ such that $ A$ is on $ [EP]$. $ DP$ intersects $ AC$ at $ M$ and $ ME$ intersects $ AD$ at $ Q$. Prove that all $ PQ$ lines have a common point as $ P$ varies.

Square $600\times 600$ is divided into figures of four types, shown in figure. In the figures of the two types, shown on the left, in painted black, the cells recorded number $2^k$, where $k$ is the number of the column, where is this cell (columns numbered from left to right by numbers from $1$ to $600$). Prove that the sum of all recorded numbers are divisible by $9$. [asy] // Set up the drawing area size(10cm,0); defaultpen(fontsize(10pt)); unitsize(0.8cm); // A helper function to draw a single unit square // c = coordinates of the lower-left corner // p = fill color (default is white) void drawsq(pair c, pen p=white) { fill(shift(c)*unitsquare, p); draw(shift(c)*unitsquare); } // --- Shape 1 (left) --- // 2 columns, 3 rows, black square in the middle-left drawsq((1,1), black); // middle-left black drawsq((2,0)); // bottom-right drawsq((2,1)); // middle-right drawsq((2,2)); // top-right // --- Shape 2 (next to the first) --- // 2 columns, 3 rows, black square in the middle-right drawsq((4,0)); drawsq((4,1)); drawsq((4,2)); drawsq((5,1), black); // middle-right black // --- Shape 3 (the "T" shape, 3 across the bottom + 1 in the middle top) --- drawsq((7,0)); drawsq((8,0)); drawsq((9,0)); drawsq((8,1)); // --- Shape 4 (the "T" shape, 3 across the top + 1 in the middle bottom) --- drawsq((11,1)); drawsq((12,1)); drawsq((13,1)); drawsq((12,0)); [/asy]

A game is played with 16 cards laid out in a row. Each card has a black side and a red side, and initially the face-up sides of the cards alternate black and red with the leftmost card black-side-up. A move consists of taking a consecutive sequence of cards (possibly only containing 1 card) with leftmost card black-side-up and the rest of the cards red-side-up, and flipping all of these cards over. The game ends when a move can no longer be made. What is the maximum possible number of moves that can be made before the game ends? [i]Ray Li.[/i] [size=85][i]See a close variant [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=810&t=500913]here[/url].[/i][/size]

Let $ a,\ b$ be positive real numbers. Prove that $ \int_{a \minus{} 2b}^{2a \minus{} b} \left|\sqrt {3b(2a \minus{} b) \plus{} 2(a \minus{} 2b)x \minus{} x^2} \minus{} \sqrt {3a(2b \minus{} a) \plus{} 2(2a \minus{} b)x \minus{} x^2}\right|dx$ $ \leq \frac {\pi}3 (a^2 \plus{} b^2).$ [color=green]Edited by moderator.[/color]

Let $G$ be the set of points $(x, y)$ such that $x$ and $y$ are positive integers less than or equal to 20. Say that a ray in the coordinate plane is [i]ocular[/i] if it starts at $(0, 0)$ and passes through at least one point in $G$. Let $A$ be the set of angle measures of acute angles formed by two distinct ocular rays. Determine \[ \min_{a \in A} \tan a. \]

In the parliament of Nekurnekadzeme, all activities take place in commissions, which consist of exactly three members. The constitution stipulates that any three commissions must have at least five members. We will call a family of commissions a [i]clique[/i] if every two of them have exactly two members in common, but if any other commission is added to this family, this condition is no longer fulfilled. Prove that two different cliques cannot have more than one commission in common.

The sequence \[\log_{12}{162},\, \log_{12}{x},\, \log_{12}{y},\, \log_{12}{z},\, \log_{12}{1250}\] is an arithmetic progression. What is $x$? $ \textbf{(A)} \ 125\sqrt{3} \qquad \textbf{(B)} \ 270 \qquad \textbf{(C)} \ 162\sqrt{5} \qquad \textbf{(D)} \ 434 \qquad \textbf{(E)} \ 225\sqrt{6}$

On sides $AB, BC$ and $CA$ of triangle $ABC$ are located points $P, M$ and $K$, respectively, so that $AM, BK$ and $CP$ intersect in one point and the sum of the vectors $\overrightarrow{AM}, \overrightarrow{BK}$ and $\overrightarrow{CP}$ equals $ \overrightarrow{0}$. Prove that $K, M$ and $P$ are midpoints of the sides of triangle $ABC$ on which they are located.

Let $n$ be a positive integer. In order to find the integer closest to $\sqrt n$, Mary finds $a^2$, the closest perfect square to $n$. She thinks that a is then the number she is looking for. Is she always correct?

A trapezoid $ABCD$ is given with $BC // AD$. Assume that the bisectors of the angles $BAD$ and $CDA$ intersect on the perpendicular bisector of the line segment $BC$. Prove that $|AB|= |CD|$ or $|AB| +|CD| =|AD|$.

A triangle $ABC$ with circumcenter $U$ is given, so that $\angle CBA = 60^o$ and $\angle CBU = 45^o$ apply. The straight lines $BU$ and $AC$ intersect at point $D$. Prove that $AD = DU$. (Karl Czakler)

The diameter of a circle of radius $R$ is divided into $2n$ equal parts. The point $M$ is taken on the circle. Prove that the sum of the squares of the distances from the point $M$ to the points of division (together with the ends of the diameter) does not depend on the choice of the point $M$. Calculate this sum.

The wizard Albus and Brian are playing a game on a square of side length $2n+1$ meters surrounded by lava. In the centre of the square there sits a toad. In a turn, a wizard chooses a direction parallel to a side of the square and enchants the toad. This will cause the toad to jump $d$ meters in the chosen direction, where $d$ is initially equal to $1$ and increases by $1$ after each jump. The wizard who sends the toad into the lava loses. Albus begins and they take turns. Depending on $n$, determine which wizard has a winning strategy.

Among $2000$ distinct positive integers, there are equally many even and odd ones. The sum of the numbers is less than $3000000.$ Show that at least one of the numbers is divisible by $3.$

Nikolai and Peter are dividing a cake in the shape of a triangle. Firstly, Nikolai chooses one point $P$ inside the triangle and after that Peter cuts the cake by any line he chooses through $P$, then takes one of the pieces and leaves the other one for Nikolai. What’s the greatest portion of the cake Nikolai can be sure he could take, if he chooses $P$ in the best way possible?

Let $\omega$ be the circumcircle of the acute nonisosceles triangle $\Delta ABC$. Point $P$ lies on the altitude from $A$. Let $E$ and $F$ be the feet of the altitudes from P to $CA$, $BA$ respectively. Circumcircle of triangle $\Delta AEF$ intersects the circle $\omega$ in $G$, different from $A$. Prove that the lines $GP$, $BE$ and $CF$ are concurrent.

Mr. Ganzgenau would like to take his tea mug out of the microwave right at the front. But Mr. Ganzgenau's microwave doesn't really want to be very precise play along. To be precise, the two of them play the following game: Let $n$ be a positive integer. The turntable of the microwave makes one in $n$ seconds full turn. Each time the microwave is switched on, an integer number of seconds turned either clockwise or counterclockwise so that there are n possible positions in which the tea mug can remain. One of these positions is right up front. At the beginning, the microwave turns the tea mug to one of the $n$ possible positions. After that Mr. Ganzgenau enters an integer number of seconds in each move, and the microwave decides either clockwise or counterclockwise this number of spin for seconds. For which $n$ can Mr. Ganzgenau force the tea cup after a finite number of puffs to be able to take it out of the microwave right up front? (Birgit Vera Schmidt) [hide=original wording, in case it doesn't make much sense]Herr Ganzgenau möchte sein Teehäferl ganz genau vorne aus der Mikrowelle herausnehmen. Die Mikrowelle von Herrn Ganzgenau möchte da aber so ganz genau gar nicht mitspielen. Ganz genau gesagt spielen die beiden das folgende Spiel: Sei n eine positive ganze Zahl. In n Sekunden macht der Drehteller der Mikrowelle eine vollständige Umdrehung. Bei jedem Einschalten der Mikrowelle wird eine ganzzahlige Anzahl von Sekunden entweder im oder gegen den Uhrzeigersinn gedreht, sodass es n mögliche Positionen gibt, auf denen das Teehäferl stehen bleiben kann. Eine dieser Positionen ist ganz genau vorne. Zu Beginn dreht die Mikrowelle das Teehäferl auf eine der n möglichen Positionen. Danach gibt Herr Ganzgenau in jedem Zug eine ganzzahlige Anzahl von Sekunden ein, und die Mikrowelle entscheidet, entweder im oder gegen den Uhrzeigersinn diese Anzahl von Sekunden lang zu drehen. Für welche n kann Herr Ganzgenau erzwingen, das Teehäferl nach endlich vielen Zügen ganz genau vorne aus der Mikrowelle nehmen zu können? (Birgit Vera Schmidt) [/hide]

A clock with hands of the same length has stopped at a certain time between $00:00$ and $12:00$. Is it possible to determine the correct time when the clock stopped, no matter when it stopped, if it has: a) two hands, showing the hour and the minute? b) three hands, showing the hour, the minute and the second?

Let $f_1,f_2,\ldots $ be a sequence of non-increasing functions from the naturals to the naturals. Show there exists $i < j$ such that $$f_i(n) \leq f_j(n) \text{ for all } n \in \mathbb{N}.$$

Determine all real numbers $a$ such that the two polynomials $x^2+ax+1$ and $x^2+x+a$ have at least one root in common.

Trapezium $ABCD$, where $BC||AD$, is inscribed in a circle, $E$ is midpoint of the arc $AD$ of this circle not containing point $C$ . Let $F$ be the foot of the perpendicular drawn from $E$ on the line tangent to the circle at the point $C$ . Prove that $BC=2CF$.

Let $n$ be the number of ways that $10$ dollars can be changed into dimes and quarters, with at least one of each coin being used. Then $n$ equals: $\textbf{(A)}\ 40 \qquad \textbf{(B)}\ 38 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ 19 $

How many three-digit numbers are divisible by 13? $ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 76\qquad\textbf{(E)}\ 77$

In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam.

Point \( A_1 \) inside the acute-angled triangle \( ABC \) is such that \[ \angle ACB = 2\angle A_1BC \quad \text{and} \quad \angle ABC = 2\angle A_1CB. \] Point \( A_2 \) is chosen so that points \( A \) and \( A_2 \) lie on opposite sides of line \( BC \), \( AA_2 \perp BC \), and the perpendicular bisector of \( AA_2 \) is tangent to the circumcircle of \( \triangle ABC \). Define points \( B_1, B_2, C_1, C_2 \) analogously. Prove that the circumcircles of \( \triangle AA_1A_2 \), \( \triangle BB_1B_2 \), and \( \triangle CC_1C_2 \) intersect at exactly two common points. [i]Proposed by Vadym Solomka[/i]