Let $ABC$ be a triangle with $\angle{A} = 60^\circ$. The midperpendicular of segment $AB$ meets line $AC$ at point $C_1$. The midperpendicular of segment $AC$ meets line $AB$ at point $B_1$. Prove that line $B_1C_1$ touches the incircle of triangle $ABC$.

A [i]diagonal line[/i] of a (not necessarily convex) polygon with at least four sides is any line through two non-adjacent vertices of that polygon. Determine all polygons with at least four sides satisfying the following condition: The reflexion of each vertex in each diagonal line lies inside or on the boundary of the polygon. [i]The Problem Selection Committee[/i]

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]

Jordi stands 20 m from a wall and Diego stands 10 m from the same wall. Jordi throws a ball at an angle of 30 above the horizontal, and it collides elastically with the wall. How fast does Jordi need to throw the ball so that Diego will catch it? Consider Jordi and Diego to be the same height, and both are on the same perpendicular line from the wall. $\textbf{(A) } 11 \text{ m/s}\\ \textbf{(B) } 15 \text{ m/s}\\ \textbf{(C) } 19 \text{ m/s}\\ \textbf{(D) } 30 \text{ m/s}\\ \textbf{(E) } 35 \text{ m/s}$

Let $ ABC$ be a non-isosceles triangle, in which $ X,Y,$ and $ Z$ are the tangency points of the incircle of center $ I$ with sides $ BC,CA$ and $ AB$ respectively. Denoting by $ O$ the circumcircle of $ \triangle{ABC}$, line $ OI$ meets $ BC$ at a point $ D.$ The perpendicular dropped from $ X$ to $ YZ$ intersects $ AD$ at $ E$. Prove that $ YZ$ is the perpendicular bisector of $ [EX]$.

Triangle $ABC$ is inscribed in circle $\Omega$ with center $O$. The incircle of $ABC$ is tangent to $BC$, $AC$, $AB$ at $D$, $E$, $F$ respectively, and its center is $I$. The reflection of the tangent line to $\Omega$ at $A$ with respect to $EF$ will be denoted $\ell_A$. We similarly define $\ell_B$, $\ell_C$. Show that the orthocenter of the triangle with sides $\ell_A$, $\ell_B$, $\ell_C$ lies on $OI$.

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

A line $l$ is drawn through the intersection point $H$ of altitudes of acute-angle triangles. Prove that symmetric images $l_a, l_b, l_c$ of $l$ with respect to the sides $BC,CA,AB$ have one point in common, which lies on the circumcircle of $ABC.$

Five people are sitting at a round table. Let $f \ge 0$ be the number of people sitting next to at least one female and $m \ge 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11 $

$A,B,C,D,E$ are points on a circle $\omega$, satisfying $AB=BD$, $BC=CE$. $AC$ meets $BE$ at $P$. $Q$ is on $DE$ such that $BE//AQ$. Suppose $\odot(APQ)$ intersects $\omega$ again at $T$. $A'$ is the reflection of $A$ wrt $BC$. Prove that $A'BPT$ lies on the same circle.

Let $\omega_1$ and $\omega_2$ be two different circles that intersect at two different points, $X$ and $Y$. Let lines $l_1$ and $l_2$ be common tangent lines of these circles such that $l_1$ is tangent $\omega_1$ at $A$ and $\omega_2$ at $C$ and $l_2$ is tangent $\omega_1$ at $B$ and $\omega_2$ at $D$. Let $Z$ be the reflection of $Y$ respect to $l_1$ and let $BC$ and $\omega_1$ meet at $K$ for the second time. Let $AD$ and $\omega_2$ meet at $L$ for the second time. Prove that the line tangent to $\omega_1$ and passes through $K$ and the line tangent to $\omega_2$ and passes through $L$ meet on the line $XZ$.

$P$ is a point in the interior of acute triangle $ABC$. $D,E,F$ are the reflections of $P$ across $BC,CA,AB$ respectively. Rays $AP,BP,CP$ meet the circumcircle of $\triangle ABC$ at $L,M,N$ respectively. Prove that the circumcircles of $\triangle PDL,\triangle PEM,\triangle PFN$ meet at a point $T$ different from $P$.

Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$. Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$. [i]Author: Farzan Barekat, Canada[/i]

Let $ P$ and $ Q$ be the common points of two circles. The ray with origin $ Q$ reflects from the first circle in points $ A_1$, $ A_2$,$ \ldots$ according to the rule ''the angle of incidence is equal to the angle of reflection''. Another ray with origin $ Q$ reflects from the second circle in the points $ B_1$, $ B_2$,$ \ldots$ in the same manner. Points $ A_1$, $ B_1$ and $ P$ occurred to be collinear. Prove that all lines $ A_iB_i$ pass through P.

Let $ABC$ be an isosceles triangle with $AB=AC$ . Let also $\omega$ be a circle of center $K$ tangent to the line $AC$ at $C$ which intersects the segment $BC$ again at $H$ . Prove that $HK \bot AB $.

Consider two circles, tangent at $T$, both inscribed in a rectangle of height $2$ and width $4$. A point $E$ moves counterclockwise around the circle on the left, and a point $D$ moves clockwise around the circle on the right. $E$ and $D$ start moving at the same time; $E$ starts at $T$, and $D$ starts at $A$, where $A$ is the point where the circle on the right intersects the top side of the rectangle. Both points move with the same speed. Find the locus of the midpoints of the segments joining $E$ and $D$.

Let $A'\in(BC),$ $B'\in(CA),C'\in(AB)$ be the points of tangency of the excribed circles of triangle $\triangle ABC$ with the sides of $\triangle ABC.$ Let $R'$ be the circumradius of triangle $\triangle A'B'C'.$ Show that \[ R'=\frac{1}{2r}\sqrt{2R\left(2R-h_{a}\right)\left(2R-h_{b}\right)\left(2R-h_{c}\right)}\] where as usual, $R$ is the circumradius of $\triangle ABC,$ r is the inradius of $\triangle ABC,$ and $h_{a},h_{b},h_{c}$ are the lengths of altitudes of $\triangle ABC.$

In triangle $ABC$, if $L,M,N$ are midpoints of $AB,AC,BC$. And $H$ is orthogonal center of triangle $ABC$, then prove that \[LH^{2}+MH^{2}+NH^{2}\leq\frac14(AB^{2}+AC^{2}+BC^{2})\]

Point $S$ is the midpoint of arc $ACB$ of the circumscribed circle $k$ around triangle $ABC$ with $AC>BC$. Let $I$ be the incenter of triangle $ABC$. Line $SI$ intersects $k$ again at point $T$. Let $D$ be the reflection of $I$ across $T$ and $M$ be the midpoint of side $AB$. Line $IM$ intersects the line through $D$, parallel to $AB$, at point $E$. Prove that $AE=BD$.

Let $\Delta ABC$ be a triangle inscribed in the circumference $\omega$ of center $O$. Let $T$ be the symmetric of $C$ respect to $O$ and $T'$ be the reflection of $T$ respect to line $AB$. Line $BT'$ intersects $\omega$ again at $R$. The perpendicular to $CT$ through $O$ intersects line $AC$ at $L$. Let $N$ be the intersection of lines $TR$ and $AC$. Prove that $\overline{CN}=2\overline{AL}$.

A square hole of depth $h$ whose base is of length $a$ is given. A dog is tied to the center of the square at the bottom of the hole by a rope of length $L >\sqrt{2a^2+h^2}$, and walks on the ground around the hole. The edges of the hole are smooth, so that the rope can freely slide along it. Find the shape and area of the territory accessible to the dog (whose size is neglected).

$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.

Let $ ABC$ be a triangle with $ \angle A < 60^\circ$. Let $ X$ and $ Y$ be the points on the sides $ AB$ and $ AC$, respectively, such that $ CA \plus{} AX \equal{} CB \plus{} BX$ and $ BA \plus{} AY \equal{} BC \plus{} CY$ . Let $ P$ be the point in the plane such that the lines $ PX$ and $ PY$ are perpendicular to $ AB$ and $ AC$, respectively. Prove that $ \angle BPC < 120^\circ$.

Isosceles right triangle $ ABC$ encloses a semicircle of area $ 2\pi$. The circle has its center $ O$ on hypotenuse $ \overline{AB}$ and is tangent to sides $ \overline{AC}$ and $ \overline{BC}$. What is the area of triangle $ ABC$? [asy]defaultpen(linewidth(0.8));pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2); draw(circle(o, 2)); clip(a--b--c--cycle); draw(a--b--c--cycle); dot(o); label("$C$", c, NW); label("$A$", a, NE); label("$O$", o, SE); label("$B$", b, SW);[/asy] $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 3\pi\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 4\pi $

Let $H$ and $I$ be the orthocenter and incenter, respectively, of an acute-angled triangle $ABC$. The circumcircle of the triangle $BCI$ intersects the segment $AB$ at the point $P$ different from $B$. Let $K$ be the projection of $H$ onto $AI$ and $Q$ the reflection of $P$ in $K$. Show that $B$, $H$ and $Q$ are collinear. [i]Proposed by Mads Christensen, Denmark[/i]