The vertices A,B,C of an acute-angled triangle ABC lie on the sides B1C1, C1A1, A1B1 respectively of a triangle A1B1C1 similar to the triangle ABC (∠A = ∠A1, etc.). Prove that the orthocenters of triangles ABC and A1B1C1 are equidistant from the circumcenter of △ABC.

An acute, non-isosceles triangle $ABC$ is inscribed in a circle with centre $O$. A line go through $O$ and midpoint $I$ of $BC$ intersects $AB, AC$ at $E, F$ respectively. Let $D, G$ be reflections to $A$ over $O$ and circumcentre of $(AEF)$, respectively. Let $K$ be the reflection of $O$ over circumcentre of $(OBC)$. $a)$ Prove that $D, G, K$ are collinear. $b)$ Let $M, N$ are points on $KB, KC$ that $IM\perp AC$, $IN\perp AB$. The midperpendiculars of $IK$ intersects $MN$ at $H$. Assume that $IH$ intersects $AB, AC$ at $P, Q$ respectively. Prove that the circumcircle of $\triangle APQ$ intersects $(O)$ the second time at a point on $AI$.

An $8\times 8$ chessboard is to be tiled (i.e., completely covered without overlapping) with pieces of the following shapes: [asy] unitsize(.6cm); draw(unitsquare,linewidth(1)); draw(shift(1,0)*unitsquare,linewidth(1)); draw(shift(2,0)*unitsquare,linewidth(1)); label("\footnotesize $1\times 3$ rectangle",(1.5,0),S); draw(shift(8,1)*unitsquare,linewidth(1)); draw(shift(9,1)*unitsquare,linewidth(1)); draw(shift(10,1)*unitsquare,linewidth(1)); draw(shift(9,0)*unitsquare,linewidth(1)); label("\footnotesize T-shaped tetromino",(9.5,0),S); [/asy] The $1\times 3$ rectangle covers exactly three squares of the chessboard, and the T-shaped tetromino covers exactly four squares of the chessboard. [list](a) What is the maximum number of pieces that can be used? (b) How many ways are there to tile the chessboard using this maximum number of pieces?[/list]

$A$ is a compact convex set in plane. Prove that there exists a point $O \in A$, such that for every line $XX'$ passing through $O$, where $X$ and $X'$ are boundary points of $A$, then \[ \frac12 \leq \frac {OX}{OX'} \leq 2.\]

Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points. [i]Proposed by Härmel Nestra, Estonia[/i]

Let $ ABC$ be a triangle, $ B_1$ the midpoint of side $ AB$ and $ C_1$ the midpoint of side $ AC$. Let $ P$ be the point of intersection ($ \neq A$) of the circumcircles of triangles $ ABC_1$ and $ AB_1C$. Let $ Q$ be the point of intersection ($ \neq A$) of the line $ AP$ and the circumcircle of triangle $ AB_1C_1$. Prove that $ \frac{AP}{AQ} \equal{} \frac{3}{2}$.

Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$ such that $|AC|>|BC|>|AB|$ and the incircle touches the sides $BC, CA, AB$ at $D, E, F$ respectively. Let the reflection of $A$ with respect to $F$ and $E$ be $F_1$ and $E_1$ respectively. The circle tangent to $BC$ at $D$ and passing through $F_1$ intersects $AB$ a second time at $F_2$ and the circle tangent to $BC$ at $D$ and passing through $E_1$ intersects $AC$ a second time at $E_2$. The midpoints of the segments $|OE|$ and $|IF|$ are $P$ and $Q$ respectively. Prove that \[|AB| + |AC| = 2|BC| \iff PQ\perp E_2F_2 \].

The graphs of the equations \[ y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k, \] are drawn in the coordinate plane for $k=-10,-9,-8,\ldots,9,10.$ These 63 lines cut part of the plane into equilateral triangles of side $2/\sqrt{3}.$ How many such triangles are formed?

In a triangle $ABC$, points $X$ and $Y$ are on $BC$ and $CA$ respectively such that $CX=CY$,$AX$ is not perpendicular to $BC$ and $BY$ is not perpendicular to $CA$.Let $\Gamma$ be the circle with $C$ as centre and $CX$ as its radius.Find the angles of triangle $ABC$ given that the orthocentres of triangles $AXB$ and $AYB$ lie on $\Gamma$.

A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not inclued the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed $1000x.$

A tetrahedron is called a [i]MEMO-tetrahedron[/i] if all six sidelengths are different positive integers where one of them is $ 2$ and one of them is $ 3$. Let $ l(T)$ be the sum of the sidelengths of the tetrahedron $ T$. (a) Find all positive integers $ n$ so that there exists a MEMO-Tetrahedron $ T$ with $ l(T)\equal{}n$. (b) How many pairwise non-congruent MEMO-tetrahedrons $ T$ satisfying $ l(T)\equal{}2007$ exist? Two tetrahedrons are said to be non-congruent if one cannot be obtained from the other by a composition of reflections in planes, translations and rotations. (It is not neccessary to prove that the tetrahedrons are not degenerate, i.e. that they have a positive volume).

Given is an acute angled triangle $ABC$. A circle going through $B$ and the triangle's circumcenter, $O$, intersects $BC$ and $BA$ at points $P$ and $Q$ respectively. Prove that the intersection of the heights of the triangle $POQ$ lies on line $AC$.

Four segments divide a convex quadrilateral into nine quadrilaterals. The points of intersections of these segments lie on the diagonals of the quadrilateral (see figure). It is known that the quadrilaterals 1, 2, 3, 4 admit inscribed circles. Prove that the quadrilateral 5 also has an inscribed circle. [asy] pair A,B,C,D,E,F,G,H,I,J,K,L; A=(-4.0,4.0);B=(-1.06,4.34);C=(-0.02,4.46);D=(4.14,4.93);E=(3.81,0.85);F=(3.7,-0.42); G=(3.49,-3.05);H=(1.37,-2.88);I=(-1.46,-2.65);J=(-2.91,-2.52);K=(-3.14,-1.03);L=(-3.61,1.64); draw(A--D);draw(D--G);draw(G--J);draw(J--A); draw(A--G);draw(D--J); draw(B--I);draw(C--H);draw(E--L);draw(F--K); pair R,S,T,U,V; R=(-2.52,2.56);S=(1.91,2.58);T=(-0.63,-0.11);U=(-2.37,-1.94);V=(2.38,-2.06); label("1",R,N);label("2",S,N);label("3",T,N);label("4",U,N);label("5",V,N); [/asy] [i]Proposed by Nairi M. Sedrakyan, Armenia[/i]

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].

A circle touches sides $DA$, $AB$, $BC$, $CD$ of a quadrilateral $ABCD$ at points $K$, $L$, $M$, $N$, respectively. Let $S_1$, $S_2$, $S_3$, $S_4$ respectively be the incircles of triangles $AKL$, $BLM$, $CMN$, $DNK$. The external common tangents distinct from the sides of $ABCD$ are drawn to $S_1$ and $S_2$, $S_2$ and $S_3$, $S_3$ and $S_4$, $S_4$ and $S_1$. Prove that these four tangents determine a rhombus.

In an acute angled triangle $ABC$, $\angle A = 30^{\circ}$, $H$ is the orthocenter, and $M$ is the midpoint of $BC$. On the line $HM$, take a point $T$ such that $HM = MT$. Show that $AT = 2 BC$.

Let $ABCD$ be a rectangle and $E$ the reflection of $A$ with respect to the diagonal $BD$. If $EB = EC$, what is the ratio $\frac{AD}{AB}$ ?

Prove that the group of orientation-preserving symmetries of the cube is isomorphic to $S_4$ (the group of permutations of $\{1,2,3,4\}$).(20 points)

On the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1$, $y_2$, $\ldots$, $y_{24}$ on the number line such that for all $k$ such that $1\leq k\leq 12$, we have that $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of \[\textstyle\sum_{n=1}^{24}(|y_n-1|+|y_n+1|).\]

Let $ABC$ be a triangle. The incircle of triangle $ABC$ touches the side $BC$ at $A^{\prime}$, and the line $AA^{\prime}$ meets the incircle again at a point $P$. Let the lines $CP$ and $BP$ meet the incircle of triangle $ABC$ again at $N$ and $M$, respectively. Prove that the lines $AA^{\prime}$, $BN$ and $CM$ are concurrent.

Two equal-radii circles with centres $ O_1$ and $ O_2$ intersect each other at $ P$ and $ Q$, $ O$ is the midpoint of the common chord $ PQ$. Two lines $ AB$ and $ CD$ are drawn through $ P$ ( $ AB$ and $ CD$ are not coincide with $ PQ$ ) such that $ A$ and $ C$ lie on circle $ O_1$ and $ B$ and $ D$ lie on circle $ O_2$. $ M$ and $ N$ are the mipoints of segments $ AD$ and $ BC$ respectively. Knowing that $ O_1$ and $ O_2$ are not in the common part of the two circles, and $ M$, $ N$ are not coincide with $ O$. Prove that $ M$, $ N$, $ O$ are collinear.

Let $ABC$ be a triangle with $AB = 4024$, $AC = 4024$, and $BC=2012$. The reflection of line $AC$ over line $AB$ meets the circumcircle of $\triangle{ABC}$ at a point $D\ne A$. Find the length of segment $CD$. [i]Ray Li.[/i]

Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points. [i]Proposed by Härmel Nestra, Estonia[/i]

How many ways can you mark 8 squares of an $8\times8$ chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different.)

Let $ ABC$ be an equilateral triangle with side length 2, and let $ \Gamma$ be a circle with radius $ \frac {1}{2}$ centered at the center of the equilateral triangle. Determine the length of the shortest path that starts somewhere on $ \Gamma$, visits all three sides of $ ABC$, and ends somewhere on $ \Gamma$ (not necessarily at the starting point). Express your answer in the form of $ \sqrt p \minus{} q$, where $ p$ and $ q$ are rational numbers written as reduced fractions.