Convex quadrilateral $ABCD$ is described on a circle $\omega$, and is not a trapezius inscribed in a circle. Let the tangency points of $\omega$ and $AB, BC, CD, DA$ be $K, L, M, N$ respectively. A circle with a center $I_K$, different from $\omega$ is tangent to the segement $AB$ and lines $AD, BC$. A circle with center $I_L$, different from $\omega$ is tangent to segment $BC$ and lines $AB, CD$. A circle with center $I_M$, different from $\omega$ is tangent to segment $CD$ and lines $AD, BC$. A circle with center $I_N$, different from $\omega$ is tangent to segment $AD$ and lines $AB, CD$. Prove that the lines $I_KK, I_LL, I_MM, I_NN$ are concurrent.

Let $x$ and $y$ be positive real numbers. 1. Prove: if $x^3 - y^3 \ge 4x$, then $x^2 > 2y$. 2. Prove: if $x^5 - y^3 \ge 2x$, then $x^3 \ge 2y$.

Find all positive integers $d$ with the following property: there exists a polynomial $P$ of degree $d$ with integer coefficients such that $\left|P(m)\right|=1$ for at least $d+1$ different integers $m$.

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

Line $\ell$ intersects sides $BC$ and $AD$ of cyclic quadrilateral $ABCD$ in its interior points $R$ and $S$, respectively, and intersects ray $DC$ beyond point $C$ at $Q$, and ray $BA$ beyond point $A$ at $P$. Circumcircles of the triangles $QCR$ and $QDS$ intersect at $N \neq Q$, while circumcircles of the triangles $PAS$ and $PBR$ intersect at $M\neq P$. Let lines $MP$ and $NQ$ meet at point $X$, lines $AB$ and $CD$ meet at point $K$ and lines $BC$ and $AD$ meet at point $L$. Prove that point $X$ lies on line $KL$.

Police want to arrest on the famous criminals of the country whose name is Kaiser. Kaiser is in one of the streets of a square shaped city with $ n$ vertical streets and $ n$ horizontal streets. In the following cases how many police officers are needed to arrest Kaiser? [img]http://i38.tinypic.com/2i1icec_th.png[/img] [img]http://i34.tinypic.com/28rk4s3_th.png[/img] a) Each police officer has the same speed as Kaiser and every police officer knows the location of Kaiser anytime. b) Kaiser has an infinite speed (finite but with no bound) and police officers can only know where he is only when one of them see Kaiser. Everybody in this problem (including police officers and Kaiser) move continuously and can stop or change his path.

Define a \emph{crossword puzzle} to be a $15 \times 15$ grid of squares, each of which is either black or white. In a crossword puzzle, define a \emph{word} to be a sequence of one or more consecutive white squares in a row or column such that the squares immediately before and after the sequence both are either black or nonexistent. (The latter case would occur if an end of a word coincides with an end of a row or column of the grid.) A crossword puzzle is \emph{tasty} if every word consists of an even number of white squares. Compute the sum of all nonnegative integers $n$ such that there exists a tasty crossword puzzle with exactly $n$ white squares. [i]Proposed by Luke Robitaille[/i]

Let $O$ be the circumcenter of $\triangle ABC$. A circle with center $P$ pass through $A$ and $O$ and $OP$//$BC$. $D$ is a point such that $\angle DBA = \angle DCA = \angle BAC$. Prove that: Circle $(P)$, circle $(BCD)$ and the circle with diameter $(AD)$ share a common point. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMS9jLzlmZjdlN2ExZDJjYjAwYWJlZTQzYWRkYzg3NDlhMTUyZjRlNGJjLmpwZw==&rn=c291dGhlYXN0UDYuanBn[/img]

We are given one red and $k>1$ blue cells, and a pack of $2n$ cards, enumerated by the numbers from $1$ to $2n$. Initially, the pack is situated on the red cell and arranged in an arbitrary order. In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by $1$, or onto an empty cell. Given $k$, what is the maximal $n$ for which it is always possible to move all the cards onto a blue cell?

Let $ABC$ be an acute and scalene triangle. Points $D$ and $E$ are in the interior of the triangle so that $<$ $DAB$ $=$ $<$ $DCB$, $<$ $DAC$ $=$ $<$ $DBC$, $<$ $EAB$ $=$ $<$ $EBC$ and $<$ $EAC$ $=$ $<$ $ECB$. Prove that the triangle $ADE$ is a right triangle.