Let $ABCD$ be a tangential quadrilateral with inscribed circle $k(O,r)$ which is tangent to the sides $BC$ and $AD$ at $K$ and $L$, respectively. Show that the circle with diameter $OC$ passes through the intersection point of $KL$ and $OD$. [i]Proposed by Ilija Jovchevski[/i]

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Let $n \geq 2, n \in \mathbb{N},$ find the least positive real number $\lambda$ such that for arbitrary $a_i \in \mathbb{R}$ with $i = 1, 2, \ldots, n$ and $b_i \in \left[0, \frac{1}{2}\right]$ with $i = 1, 2, \ldots, n$, the following holds: \[\sum^n_{i=1} a_i = \sum^n_{i=1} b_i = 1 \Rightarrow \prod^n_{i=1} a_i \leq \lambda \sum^n_{i=1} a_i b_i.\]

On Mars a basketball team consists of 6 players. The coach of the team Mars can select any line-up of 6 players among 100 candidates. The coach considers some line-ups as [i]appropriate[/i] while the other line-ups are not (there exists at least one appropriate line-up). A set of 5 candidates is called [i]perspective[/i] if one more candidate could be added to it to obtain an appropriate line-up. A candidate is called [i]universal[/i] if he completes each perspective set of 5 candidates (not containing him) upto an appropriate line-up. The coach has selected a line-up of 6 universal candidates. Determine if it follows that this line-up is appropriate.

The arithmetic mean (average) of a set of $50$ numbers is $38$. If two numbers, namely, $45$ and $55$, are discarded, the mean of the remaining set of numbers is: $ \textbf{(A)}\ 36.5 \qquad\textbf{(B)}\ 37\qquad\textbf{(C)}\ 37.2\qquad\textbf{(D)}\ 37.5\qquad\textbf{(E)}\ 37.52 $

Given a triangle $ABC$. The points $A$, $B$, $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$, $CA$, $AB$. Let $X$ be a variable point on the arc $AB$, and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.

A best-of-9 series is to be played between two teams; that is, the first team to win 5 games is the winner. The Mathletes have a chance of $\tfrac{2}{3}$ of winning any given game. What is the probability that exactly 7 games will need to be played to determine a winner?

Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds: \[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\] (Here $\mathbb{Z}$ denotes the set of integers.) [i]Proposed by Liam Baker, South Africa[/i]

Submit a positive integer $N$ between 1 and 20, inclusive. If $C$ is the total number of teams that submit $N$ for this question, your score will be $\lfloor\frac{N}{C}\rfloor$ [i]Lightning 5.4[/i]

Let O be the circumcenter of a triangle ABC. Points M and N are choosen on the sides AB and BC respectively so that the angle AOC is two times greater than angle MON. Prove that the perimeter of triangle MBN is not less than the lenght of side AC

Points $ A$ and $ B$ lie on a circle centered at $ O$, and $ \angle AOB\equal{}60^\circ$. A second circle is internally tangent to the first and tangent to both $ \overline{OA}$ and $ \overline{OB}$. What is the ratio of the area of the smaller circle to that of the larger circle? $ \textbf{(A)}\ \frac{1}{16} \qquad \textbf{(B)}\ \frac{1}{9} \qquad \textbf{(C)}\ \frac{1}{8} \qquad \textbf{(D)}\ \frac{1}{6} \qquad \textbf{(E)}\ \frac{1}{4}$

Find all polynomials $P(x)$ of degree $1$ such that $\underset {a\le x\le b}{max} P(x) - \underset {a\le x\le b}{min} P(x) =b-a$ , $\forall a,b\in R$ where $a < b$

Let $ S\equal{}\{1,2,\ldots,n\}$. Let $ A$ be a subset of $ S$ such that for $ x,y\in A$, we have $ x\plus{}y\in A$ or $ x\plus{}y\minus{}n\in A$. Show that the number of elements of $ A$ divides $ n$.

There are $n$ different integers on the blackboard. Whenever two of these integers are chosen, either their sum or difference (possibly both) will be a positive integral power of $2$. Find the greatest possible value of $n$.

Find the maximum number of increasing arithmetic progressions that can have a finite sequence of real numbers $a_1<a_2<\cdots<a_n$ of $n\ge 3$ real numbers.

Let $ n$ and $ k$ be positive integers with $ k \geq n$ and $ k \minus{} n$ an even number. Let $ 2n$ lamps labelled $ 1$, $ 2$, ..., $ 2n$ be given, each of which can be either [i]on[/i] or [i]off[/i]. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on). Let $ N$ be the number of such sequences consisting of $ k$ steps and resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off. Let $ M$ be number of such sequences consisting of $ k$ steps, resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off, but where none of the lamps $ n \plus{} 1$ through $ 2n$ is ever switched on. Determine $ \frac {N}{M}$. [i]Author: Bruno Le Floch and Ilia Smilga, France[/i]

The numbers $12,14,37,65$ are one of the solutions of the equation $xy-xz+yt=182$. What number corresponds to which letter?

In the trapezoid $ABCD$ , the base $AB$ is smaller than the $CD$ base. The point $K$ is chosen such that $AK$ is parallel to BC and $BK$ is parallel to $AD$. The points $P$ and $Q$ are chosen on the $AK$ and $BK$ rays respectively, such that $\angle ADP = \angle BCK$ and $\angle BCQ = \angle ADK$. (a) Show that the lines $AD, BC$ and $PQ$ go through the same point. (b) Assuming that the circumscribed circumferences of the $APD$ and $BCQ$ triangles intersect at two points, show that one of those points belongs to the line $PQ$.

Let $M$ be an interior point of the triangle $ABC$ such that $AMC = 90^\circ$, $AMB = 150^\circ$, and $BMC = 120^\circ$. The circumcenters of the triangles $AMC$, $AMB$, and $BMC$ are $P$, $Q$, and $R$ respectively. Prove that the area of $\Delta PQR$ is greater than or equal to the area of $\Delta ABC$.

Circles $o_1, o_2$ with equal radii intersect at points $A, B$. Points $C, D, E, F$ lie in this order on one line, with $C, E$ lying on $o_1$ and $D, F$ on $o_2$. Perpendicular bisectors of $CD$ and $EF$ intersect $AB$ at $X, Y$ respectively. Prove that $AX=BY$.

Conside the continuous $ f: \mathbb{R}\to\mathbb{R}$ . It is also know that equation $f(f(f(x)))=x$ has solution in $\mathbb{R}$. Prove that equation $f(x)=x$ has solution in $\mathbb{R}$.

Find all odd numbers $n\in \mathbb{N}$, for which the number of all natural numbers, that are no bigger than $n$ and coprime with $n$, divides $n^2+3$.

For which positive integer couples $(k,n)$, the equality $\Bigg|\Bigg\{{a \in \mathbb{Z}^+: 1\leq a\leq(nk)!, gcd \left(\binom{a}{k},n\right)=1}\Bigg\}\Bigg|=\frac{(nk)!}{6}$ holds?

Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$, $PB=6$, and $PC=10$. To the nearest integer the area of triangle $ABC$ is: $\textbf{(A)}\ 159\qquad \textbf{(B)}\ 131\qquad \textbf{(C)}\ 95\qquad \textbf{(D)}\ 79\qquad \textbf{(E)}\ 50$

Find all pairs $(a,b)$ of positive integers such that $a^3$ is multiple of $b^2$ and $b-1$ is multiple of $a-1$.