Let $ ABCD$ be a cyclic quadrilateral and $ r$ and $ s$ the lines obtained reflecting $ AB$ with respect to the internal bisectors of $ \angle CAD$ and $ \angle CBD$, respectively. If $ P$ is the intersection of $ r$ and $ s$ and $ O$ is the center of the circumscribed circle of $ ABCD$, prove that $ OP$ is perpendicular to $ CD$.

Let $ABC$ be an acute angled triangle. Let $D,E,F$ be points on $BC, CA, AB$ such that $AD$ is the median, $BE$ is the internal bisector and $CF$ is the altitude. Suppose that $\angle FDE=\angle C, \angle DEF=\angle A$ and $\angle EFD=\angle B.$ Show that $ABC$ is equilateral.

At lunch, the seven members of the Kubik family sit down to eat lunch together at a round table. In how many distinct ways can the family sit at the table if Alexis refuses to sit next to Joshua? (Two arrangements are not considered distinct if one is a rotation of the other.)

To [i]clip[/i] a convex $n$-gon means to choose a pair of consecutive sides $AB, BC$ and to replace them by the three segments $AM, MN$, and $NC$, where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, one cuts off the triangle $MBN$ to obtain a convex $(n+1)$-gon. A regular hexagon ${\cal P}_6$ of area 1 is clipped to obtain a heptagon ${\cal P}_7$. Then ${\cal P}_7$ is clipped (in one of the seven possible ways) to obtain an octagon ${\cal P}_8$, and so on. Prove that no matter how the clippings are done, the area of ${\cal P}_n$ is greater than $\frac 13$, for all $n \geq 6$.

Consider a sphere and a plane $\pi$. For a variable point $M \in \pi$, exterior to the sphere, one considers the circular cone with vertex in $M$ and tangent to the sphere. Find the locus of the centers of all circles which appear as tangent points between the sphere and the cone. [i]Octavian Stanasila[/i]

Let squares be constructed on the sides $BC,CA,AB$ of a triangle $ABC$, all to the outside of the triangle, and let $A_1,B_1, C_1$ be their centers. Starting from the triangle $A_1B_1C_1$ one analogously obtains a triangle $A_2B_2C_2$. If $S, S_1, S_2$ denote the areas of triangles$ ABC,A_1B_1C_1,A_2B_2C_2$, respectively, prove that $S = 8S_1 - 4S_2.$

Every brick has $5$ holes in a line. The holes can be filled with bolts (fitting in one hole) and braces (fitting into two neighboring holes). No hole may remain free. One puts $n$ of these bricks in a line to form a pattern from left to right. In this line no two braces and no three bolts may be adjacent. How many different such patterns can be produced with $n$ bricks?

(A.Zaslavsky, 9--10) The reflections of diagonal $ BD$ of a quadrilateral $ ABCD$ in the bisectors of angles $ B$ and $ D$ pass through the midpoint of diagonal $ AC$. Prove that the reflections of diagonal $ AC$ in the bisectors of angles $ A$ and $ C$ pass through the midpoint of diagonal $ BD$ (There was an error in published condition of this problem).

On the real number line, paint red all points that correspond to integers of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer point blue. Find a point $P$ on the line such that, for every integer point $T$, the reflection of $T$ with respect to $P$ is an integer point of a different colour than $T$.

Let $ABCD$ be a cyclic quadrilateral. The lines $BC$ and $AD$ meet at a point $P$. Let $Q$ be the point on the line $BP$, different from $B$, such that $PQ=BP$. Consider the parallelograms $CAQR$ and $DBCS$. Prove that the points $C,Q,R,S$ lie on a circle.

Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.

The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $O$. The circumcircles of triangles $AOB$ and $COD$ intersect again at $K$. Point $L$ is such that the triangles $BLC$ and $AKD$ are similar and equally oriented. Prove that if the quadrilateral $BLCK$ is convex, then it is tangent [has an incircle].

Prove that on a coordinate plane it is impossible to draw a closed broken line such that [list][*] coordinates of each vertex are rational, [*] the length of its every edge is equal to $1$, [*] the line has an odd number of vertices.[/list]

Two three-letter strings, $aaa$ and $bbb$, are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an $a$ when it should have been a $b$, or as a $b$ when it should be an $a$. However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let $S_a$ be the three-letter string received when $aaa$ is transmitted and let $S_b$ be the three-letter string received when $bbb$ is transmitted. Let $p$ be the probability that $S_a$ comes before $S_b$ in alphabetical order. When $p$ is written as a fraction in lowest terms, what is its numerator?

$ABCD$ is a square with centre $O$. Two congruent isosceles triangle $BCJ$ and $CDK$ with base $BC$ and $CD$ respectively are constructed outside the square. let $M$ be the midpoint of $CJ$. Show that $OM$ and $BK$ are perpendicular to each other.

An $n$-stick is a connected figure consisting of $n$ matches of length $1$ which are placed horizontally or vertically and no two touch each other at points other than their ends. Two shapes that can be transformed into each other by moving, rotating or flipping are considered the same. An $n$-mino is a shape which is built by connecting $n$ squares of side length 1 on their sides such that there's a path on the squares between each two squares of the $n$-mino. Let $S_n$ be the number of $n$-sticks and $M_n$ the number of $n$-minos, e.g. $S_3=5$ And $M_3=2$. (a) Prove that for any natural $n$, $S_n \geq M_{n+1}$. (b) Prove that for large enough $n$ we have $(2.4)^n \leq S_n \leq (16)^n$. A [b]grid segment[/b] is a segment on the plane of length 1 which it's both ends are integer points. A polystick is called [b]wise[/b] if using it and it's rotations or flips we can cover all grid segments without overlapping, otherwise it's called [b]unwise[/b]. (c) Prove that there are at least $2^{n-6}$ different unwise $n$-sticks. (d) Prove that any polystick which is in form of a path only going up and right is wise. (e) Extra points: Prove that for large enough $n$ we have $3^n \leq S_n \leq 12^n$ Time allowed for this exam was 2 hours.

The endpoints of a compass are at two lattice points of an infinite unit square grid. It is allowed to rotate the compass around one of its endpoints, not varying its radius, and thus move the other endpoint to another lattice point. Can the endpoints of the compass change places after several such steps?

For $m\ge 3,$ a list of $\binom m3$ real numbers $a_{ijk}$ $(1\le i<j<k\le m)$ is said to be [i]area definite[/i] for $\mathbb{R}^n$ if the inequality \[\sum_{1\le i<j<k\le m}a_{ijk}\cdot\text{Area}(\triangle A_iA_jA_k)\ge0\] holds for every choice of $m$ points $A_1,\dots,A_m$ in $\mathbb{R}^n.$ For example, the list of four numbers $a_{123}=a_{124}=a_{134}=1, a_{234}=-1$ is area definite for $\mathbb{R}^2.$ Prove that if a list of $\binom m3$ numbers is area definite for $\mathbb{R}^2,$ then it is area definite for $\mathbb{R}^3.$

In $\triangle ABC$, $\angle ABC=90^{\circ}$. Points $D,G$ lie on side $AC$. Points $E, F$ lie on segment $BD$, such that $AE \perp BD $ and $GF \perp BD$. Show that if $BE=EF$, then $\angle ABG=\angle DFC$.

Let $I$ be the incentre of a triangle $ABC$ and $\Gamma_a$ be the excircle opposite $A$ touching $BC$ at $D$. If $ID$ meets $\Gamma_a$ again at $S$, prove that $DS$ bisects $\angle BSC$.

In $\triangle ABC$, $AB>AC$. In the circumcircle $(O)$ of $\triangle ABC$, $M$ is the midpoint of arc $BAC$. The incircle $(I)$ of $\triangle ABC$ touches $BC$ at $D$, the line through $D$ parallel to $AI$ intersects $(I)$ again at $P$. Prove that $AP$ and $IM$ intersect at a point on $(O)$.

In triangle $ABC$ it holds $|BC|= \frac{1}{2}(|AB|+|AC|)$. Let $M$ and $N$ be midpoints of $AB$ and $AC$, and let $I$ be the incenter of $ABC$. Prove that $A, M, I, N$ are concyclic.

The bisector of $\angle{BAD}$ in the parallellogram $ABCD$ intersects the lines $BC$ and $CD$ at the points $K$ and $L$ respectively. Prove that the centre of the circle passing through the points $C,\ K$ and $L$ lies on the circle passing through the points $B,\ C$ and $D$.

Let $ABC$ be an acute triangle inscribed in a circle centered at $O$. Let $M$ be a point on the small arc $AB$ of the triangle's circumcircle. The perpendicular dropped from $M$ on the ray $OA$ intersects the sides $AB$ and $AC$ at the points $K$ and $L$, respectively. Similarly, the perpendicular dropped from $M$ on the ray $OB$ intersects the sides $AB$ and $BC$ at $N$ and $P$, respectively. Assume that $KL=MN$. Find the size of the angle $\angle{MLP}$ in terms of the angles of the triangle $ABC$.

Let $AXYZB$ be a regular pentagon with area $5$ inscribed in a circle with center $O$. Let $Y'$ denote the reflection of $Y$ over $\overline{AB}$ and suppose $C$ is the center of a circle passing through $A$, $Y'$ and $B$. Compute the area of triangle $ABC$. [i]Proposed by Evan Chen[/i]