In a triangle $ABC$, with $\widehat{A} > 90^\circ$, let $O$ and $H$ denote its circumcenter and orthocenter, respectively. Let $K$ be the reflection of $H$ with respect to $A$. Prove that $K, O$ and $C$ are collinear if and only if $\widehat{A} - \widehat{B} = 90^\circ$.

How far is it from the point $(9, 17)$ to its reflection across the line \[3x + 4y = 15?\]

Equilateral triangles $AMB,BNC,CKA$ are constructed on the exterior of a triangle $ABC$. The perpendiculars from the midpoints of $MN, NK, KM$ to the respective lines $CA, AB, BC$ are constructed. Prove that these three perpendiculars pass through a single point.

Consider a triangle $ABC$ with incenter $I$. Let $D$ be the intersection of $BI,AC$ and $CI$ intersects the circumcircle of $ABC$ at $M$. Point $K$ lies on the line $MD$ and $\angle KIA=90^\circ$. Let $F$ be the reflection of $B$ about $C$. Prove that $BIKF$ is cyclic.

Let $ABC$ be a triangle with orthocenter $H$. Let $P,Q,R$ be the reflections of $H$ with respect to sides $BC,AC,AB$, respectively. Show that $H$ is incenter of $PQR$.

Let $\omega_P$ and $\omega_Q$ be two circles of radius $1$, intersecting in points $A$ and $B$. Let $P$ and $Q$ be two regular $n$-gons (for some positive integer $n\ge4$) inscribed in $\omega_P$ and $\omega_Q$, respectively, such that $A$ and $B$ are vertices of both $P$ and $Q$. Suppose a third circle $\omega$ of radius $1$ intersects $P$ at two of its vertices $C$, $D$ and intersects $Q$ at two of its vertices $E$, $F$. Further assume that $A$, $B$, $C$, $D$, $E$, $F$ are all distinct points, that $A$ lies outside of $\omega$, and that $B$ lies inside $\omega$. Show that there exists a regular $2n$-gon that contains $C$, $D$, $E$, $F$ as four of its vertices.

Consider a circle $C_1$ and a point $O$ on it. Circle $C_2$ with center $O$, intersects $C_1$ in two points $P$ and $Q$. $C_3$ is a circle which is externally tangent to $C_2$ at $R$ and internally tangent to $C_1$ at $S$ and suppose that $RS$ passes through $Q$. Suppose $X$ and $Y$ are second intersection points of $PR$ and $OR$ with $C_1$. Prove that $QX$ is parallel with $SY$.

Given triangle $ ABC$. Points $ D,E,F$ outside triangle $ ABC$ are chosen such that triangles $ ABD$, $ BCE$, and $ CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.

Consider a charged capacitor made with two square plates of side length $L$, uniformly charged, and separated by a very small distance $d$. The EMF across the capacitor is $\xi$. One of the plates is now rotated by a very small angle $\theta$ to the original axis of the capacitor. Find an expression for the difference in charge between the two plates of the capacitor, in terms of (if necessary) $d$, $\theta$, $\xi$, and $L$. Also, approximate your expression by transforming it to algebraic form: i.e. without any non-algebraic functions. For example, logarithms and trigonometric functions are considered non-algebraic. Assume $ d << L $ and $ \theta \approx 0 $. $\emph{Hint}$: You may assume that $ \frac {\theta L}{d} $ is also very small. [i]Problem proposed by Trung Phan[/i] [hide="Clarification"] There are two possible ways to rotate the capacitor. Both were equally scored but this is what was meant: [asy]size(6cm); real h = 7; real w = 2; draw((-w,0)--(-w,h)); draw((0,0)--(0,h), dashed); draw((0,0)--h*dir(64)); draw(arc((0,0),2,64,90)); label("$\theta$", 2*dir(77), dir(77)); [/asy] [/hide]

In a triangle $ABC,$ incircle touches the sides $BC, CA, AB$ at $D, E, F,$ respectively. A circle $\omega$ passing through $A$ and tangent to line $BC$ at $D$ intersects the line segments $BF$ and $CE$ at $K$ and $L,$ respectively. The line passing through $E$ and parallel to $DL$ intersects the line passing through $F$ and parallel to $DK$ at $P.$ If $R_1, R_2, R_3, R_4$ denotes the circumradius of the triangles $AFD, AED, FPD, EPD,$ respectively, prove that $R_1R_4=R_2R_3.$

Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations: [list=] [*]$L,$ a rotation of $90^{\circ}$ counterclockwise around the origin; [*]$R,$ a rotation of $90^{\circ}$ clockwise around the origin; [*]$H,$ a reflection across the $x$-axis; and [*]$V,$ a reflection across the $y$-axis. [/list] Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.) $\textbf{(A)}\ 2^{37} \qquad\textbf{(B)}\ 3\cdot 2^{36} \qquad\textbf{(C)}\ 2^{38} \qquad\textbf{(D)}\ 3\cdot 2^{37} \qquad\textbf{(E)}\ 2^{39}$

Two triangles are said to be [i]twins [/i] if one of them is an image of the other one under a parallel projection. Prove that two triangles are twins if and only if either at least a side of one of them equals a side of another or both the triangles have equal segments that connect the corresponding vertices with some points on the opposite sides which divide these sides in the same ratio. (E. Barabanov)

Nine distinct positive integers are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of $n$ and the product of any two adjacent numbers in the circle is not a multiple of $n$, where $n$ is a fixed positive integer. Find the smallest possible value for $n$.

Let $O, H$ be circumcenter and orthogonal center of triangle $ABC$. $M,N$ are midpoints of $BH$ and $CH$. $BB'$ is diagonal of circumcircle. If $HONM$ is a cyclic quadrilateral, prove that $B'N=\frac12AC$.

Circles $ \omega_{1}$ and $ \omega_{2}$ intersect at points $ A$ and $ B.$ Points $ C$ and $ D$ are on circles $ \omega_{1}$ and $ \omega_{2},$ respectively, such that lines $ AC$ and $ AD$ are tangent to circles $ \omega_{2}$ and $ \omega_{1},$ respectively. Let $ I_{1}$ and $ I_{2}$ be the incenters of triangles $ ABC$ and $ ABD,$ respectively. Segments $ I_{1}I_{2}$ and $ AB$ intersect at $ E$. Prove that: $ \frac {1}{AE} \equal{} \frac {1}{AC} \plus{} \frac {1}{AD}$

Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.

In the following diagram, let $ABCD$ be a square and let $M,N,P$ and $Q$ be the midpoints of its sides. Prove that \[S_{A'B'C'D'} = \frac 15 S_{ABCD}.\] [asy] import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttzz = rgb(0,0.2,0.6); pen qqzzff = rgb(0,0.6,1); draw((0,4)--(4,4),qqttzz+linewidth(1.6pt)); draw((4,4)--(4,0),qqttzz+linewidth(1.6pt)); draw((4,0)--(0,0),qqttzz+linewidth(1.6pt)); draw((0,0)--(0,4),qqttzz+linewidth(1.6pt)); draw((0,4)--(2,0),qqzzff+linewidth(1.2pt)); draw((2,4)--(4,0),qqzzff+linewidth(1.2pt)); draw((0,2)--(4,4),qqzzff+linewidth(1.2pt)); draw((0,0)--(4,2),qqzzff+linewidth(1.2pt)); dot((0,4),ds); label("$A$", (0.07,4.12), NE*lsf); dot((0,0),ds); label("$D$", (-0.27,-0.37), NE*lsf); dot((4,0),ds); label("$C$", (4.14,-0.39), NE*lsf); dot((4,4),ds); label("$B$", (4.08,4.12), NE*lsf); dot((2,4),ds); label("$M$", (2.08,4.12), NE*lsf); dot((4,2),ds); label("$N$", (4.2,1.98), NE*lsf); dot((2,0),ds); label("$P$", (1.99,-0.49), NE*lsf); dot((0,2),ds); label("$Q$", (-0.48,1.9), NE*lsf); dot((0.8,2.4),ds); label("$A'$", (0.81,2.61), NE*lsf); dot((2.4,3.2),ds); label("$B'$", (2.46,3.47), NE*lsf); dot((3.2,1.6),ds); label("$C'$", (3.22,1.9), NE*lsf); dot((1.6,0.8),ds); label("$D'$", (1.14,0.79), NE*lsf); clip((-4.44,-11.2)--(-4.44,6.41)--(16.48,6.41)--(16.48,-11.2)--cycle); [/asy] [$S_{X}$ denotes area of the $X.$]

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]

Given a convex quadrilateral $ABCD$. The points $P$ and $Q$ are the midpoints of the diagonals $AC$ and $BD$ respectively. The line $PQ$ intersects the lines $AB$ and $CD$ at $N$ and $M$ respectively. Prove that the circumcircles of triangles $NAP$, $NBQ$, $MQD$, and $MPC$ have a common point.

The bisectors of the angles $B,C$ of a triangle $ABC$ intersect the opposite sides in $B', C'$ respectively. Prove that the straight line $B'C'$ intersects the inscribed circle in two different points.

Points $A$, $B$, $C$ and $D$ are chosen on a line in that order, with $AB$ and $CD$ greater than $BC$. Equilateral triangles $APB$, $BCQ$ and $CDR$ are constructed so that $P$, $Q$ and $R$ are on the same side with respect to $AD$. If $\angle PQR=120^\circ$, show that \[\frac{1}{AB}+\frac{1}{CD}=\frac{1}{BC}.\]

Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. Let $l$ be the line tangent to $\Gamma$ at $A$. Let $D$ and $E$ be the intersections of the circumference with center $B$ and radius $AB$ with lines $l$ and $AC$, respectively. Prove the orthocenter of $ABC$ lies on line $DE$.

Let $ABC$ be an acute-angled triangle with $AC<BC.$ A circle passes through $A$ and $B$ and crosses the segments $AC$ and $BC$ again at $A_1$ and $B_1$ respectively. The circumcircles of $A_1B_1C$ and $ABC$ meet each other at points $P$ and $C.$ The segments $AB_1$ and $A_1B$ intersect at $S.$ Let $Q$ and $R$ be the reflections of $S$ in the lines $CA$ and $CB$ respectively. Prove that the points $P,$ $Q,$ $R,$ and $C$ are concyclic.

Points $D,E,F$ are on the sides $BC, CA$ and $AB$, respectively which satisfy $EF || BC$, $D_1$ is a point on $BC,$ Make $D_1E_1 || D_E, D_1F_1 || DF$ which intersect $AC$ and $AB$ at $E_1$ and $F_1$, respectively. Make $\bigtriangleup PBC \sim \bigtriangleup DEF$ such that $P$ and $A$ are on the same side of $BC.$ Prove that $E, E_1F_1, PD_1$ are concurrent. [color=red][Edit by Darij: See my post #4 below for a [b]possible correction[/b] of this problem. However, I am not sure that it is in fact the problem given at the TST... Does anyone have a reliable translation?][/color]

Let $P$ be a regular polygon with $ 4k + 1 $ sides (where $ k $ is a natural) whose vertices are $ A_1, A_2, ..., A_ {4k + 1} $ (in that order ). Each vertex $ A_j $ of $P$ is assigned a natural of the set $ \{1,2, ..., 4k + 1 \} $ such that no two vertices are assigned the same number. On $P$ the following operation is performed: Let $ B_j $ be the midpoint of the side $ A_jA_ {j + 1} $ for $ j = 1,2, ..., 4k + 1 $ (where is consider $ A_ {4k + 2} = A_1 $). If $ a $, $ b $ are the numbers assigned to $ A_ {j} $ and $ A_ {j + 1} $, respectively, the midpoint $ B_j $ is written the number $ 7a-3b $. By doing this with each of the $ 4k + 1 $ sides, the $ 4k + 1 $ vertices initially arranged are erased. We will say that a natural $ m $ is [i]fatal [/i] if for all natural $ k $ , no matter how the vertices of $P$ are initially arranged, it is impossible to obtain $ 4k + 1 $ equal numbers through a finite amount of operations from $ m $. a) Determine if the $ 2010 $ is fatal or not. Justify. b) Prove that there are infinite fatal numbers. [color=#f00]PS. A help in translation of the 2nd paragraph is welcome[/color]. [hide=Original wording]Diremos que un natural $m$ es fatal si no importa cómo se disponen inicialmente los vértices de ${P}$, es imposible obtener mediante una cantidad finita de operaciones $4k+1$ números iguales a $m$.[/hide]