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]

Let $f:[0,1]\rightarrow [0,1]$ be a continuous and bijective function. Describe the set: \[A=\{f(x)-f(y)\mid x,y\in[0,1]\backslash\mathbb{Q}\}\] [hide="Note"] You are given the result that [i]there is no one-to-one function between the irrational numbers and $\mathbb{Q}$.[/i][/hide]

Given a positive integer $n,$ what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same? [When $n=8,$ the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is [i]at least[/i] 3.]

Let $x,y$ be real numbers such that $y=|x-1|$. What is the smallest value of $(x-1)^2+(y-2)^2$?

Let $r$ be a real such that $0<r\leq 1$. Denote by $V(r)$ the volume of the solid formed by all points of $(x,\ y,\ z)$ satisfying \[x^2+y^2+z^2\leq 1,\ x^2+y^2\leq r^2\] in $xyz$-space. (1) Find $V(r)$. (2) Find $\lim_{r\rightarrow 1-0} \frac{V(1)-V(r)}{(1-r)^{\frac 32}}.$ (3) Find $\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.$

Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $9{:}99$ just before midnight, $0{:}00$ at midnight, $1{:}25$ at the former $3{:}00$ $\textsc{am}$, and $7{:}50$ at the former $6{:}00$ $\textsc{pm}$. After the conversion, a person who wanted to wake up at the equivalent of the former $6{:}36$ $\textsc{am}$ would have to set his new digital alarm clock for $\text{A:BC}$, where $\text{A}$, $\text{B}$, and $\text{C}$ are digits. Find $100\text{A} + 10\text{B} + \text{C}$.

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\Gamma$. Let $M$ be the midpoint of the arc $ABC$. The circle with center $M$ and radius $MA$ meets $AD, AB$ at $X, Y$. The point $Z \in XY$ with $Z \neq Y$ satisfies $BY=BZ$. Show that $\angle BZD=\angle BCD$.

Let $n>1$ be an integer and let $a_0,a_1,\ldots,a_n$ be non-negative real numbers. Definite $S_k=\sum_{i\equal{}0}^k \binom{k}{i}a_i$ for $k=0,1,\ldots,n$. Prove that\[\frac{1}{n} \sum_{k\equal{}0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k\equal{}0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.\]

For each natural number $k$ find the least number $n$ such that in every tournament with $n$ vertices, there exists a vertex with in-degree and out-degree at least $k$. (Tournament is directed complete graph.)

On an $n$ × $n$ board, the set of all squares that are located on or below the main diagonal of the board is called the$n-ladder$. For example, the following figure shows a $3-ladder$: [asy] draw((0,0)--(0,3)); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((1,0)--(1,3)); draw((0,2)--(2,2)); draw((2,0)--(2,2)); draw((0,3)--(1,3)); draw((3,0)--(3,1)); [/asy] In how many ways can a $99-ladder$ be divided into some rectangles, which have their sides on grid lines, in such a way that all the rectangles have distinct areas?