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]

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?

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).

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

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 $\frac{AD}{AB}$ ?

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 \[\textstyle\sum_{n=1}^{24}(|y_n-1|+|y_n+1|).\]

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]

The entries in a $ 3\times3$ array include all the digits from 1 through 9, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? $ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 42\qquad\textbf{(E)}\ 60$

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.

In quadrilateral $ABCD$, $\angle DAC = 98^{\circ}$, $\angle DBC = 82^\circ$, $\angle BCD = 70^\circ$, and $BC = AD$. Find $\angle ACD.$

Two circles $C_1$ and $C_2$ with the respective radii $r_1$ and $r_2$ intersect in $A$ and $B.$ A variable line $r$ through $B$ meets $C_1$ and $C_2$ again at $P_r$ and $Q_r$ respectively. Prove that there exists a point $M,$ depending only on $C_1$ and $C_2,$ such that the perpendicular bisector of each segment $P_rQ_r$ passes through $M.$

Let $ABCD$ be a quadrilateral inscribed in circle $\omega$. Define $E = AA \cap CD$, $F = AA \cap BC$, $G = BE \cap \omega$, $H = BE \cap AD$, $I = DF \cap \omega$, and $J = DF \cap AB$. Prove that $GI$, $HJ$, and the $B$-symmedian are concurrent. [i]Proposed by Robin Park[/i]

$ ABCD$ is a $ 4\times 4$ square. $ E$ is the midpoint of $ \left[AB\right]$. $ M$ is an arbitrary point on $ \left[AC\right]$. How many different points $ M$ are there such that $ \left|EM\right|\plus{}\left|MB\right|$ is an integer? $\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$

Let $ABC$ be a triangle,$O$ its circumcenter and $R$ the radius of its circumcircle.Denote by $O_{1}$ the symmetric of $O$ with respect to $BC$,$O_{2}$ the symmetric of $O$ with respect to $AC$ and by $O_{3}$ the symmetric of $O$ with respect to $AB$. (a)Prove that the circles $C_{1}(O_{1},R)$, $C_{2}(O_{2},R)$, $C_{3}(O_{3},R)$ have a common point. (b)Denote by $T$ this point.Let $l$ be an arbitary line passing through $T$ which intersects $C_{1}$ at $L$, $C_{2}$ at $M$ and $C_{3}$ at $K$.From $K,L,M$ drop perpendiculars to $AB,BC,AC$ respectively.Prove that these perpendiculars pass through a point.

A function $f$ is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real $x$. If $x = 0$ is a root of $f(x) = 0$, what is the least number of roots $f(x) = 0$ must have in the interval $-1000 \le x \le 1000$?

Given a triangle $ \triangle{ABC} $ with circumcircle $ \Omega $. Denote its incenter and $ A $-excenter by $ I, J $, respectively. Let $ T $ be the reflection of $ J $ w.r.t $ BC $ and $ P $ is the intersection of $ BC $ and $ AT $. If the circumcircle of $ \triangle{AIP} $ intersects $ BC $ at $ X \neq P $ and there is a point $ Y \neq A $ on $ \Omega $ such that $ IA = IY $. Show that $ \odot\left(IXY\right) $ tangents to the line $ AI $.

The eleven members of a cricket team are numbered $1,2,...,11$. In how many ways can the entire cricket team sit on the eleven chairs arranged around a circular table so that the numbers of any two adjacent players differ by one or two ?