$ABC$ is an obtuse triangle. (angle $B$ is obtuse) Its circumcircle is $O$. A tangent line for $O$ passing $C$ meets with $AB$ at $B_1$. Let $O_1$ be a circumcenter of triangle $AB_1C$. $B_2$ is a point on the segment $BB_1$. Let $C_1$ be a contact point of the tangent line for $O$ passing $B_2$, which is more closer to $C$. Let $O_2$ be a circumcenter of triangle $AB_2C_1$. Prove that if $OO_2$ and $AO_1$ is perpendicular, then five points $O,O_2,O_1,C_1,C$ are concyclic.

Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A',B',C'$. Let $L,M,N$ be the midpoints of segments $AA', BB', CC'$; Let $G$ be the centroid of the triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L,M,N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\epsilon$?

Evaluate \[\int_0^a \sqrt{2ax-x^2}\ dx \ (a>0)\]

Let $z_1, z_2, z_3$ be pairwise distinct complex numbers satisfying $|z_1| = |z_2| = |z_3| = 1$ and \[\frac{1}{2 + |z_1 + z_2|}+\frac{1}{2 + |z_2 + z_3|}+\frac{1}{2 + |z_3 + z_1|} =1.\] If the points $A(z_1),B(z_2),C(z_3)$ are vertices of an acute-angled triangle, prove that this triangle is equilateral.

Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent. [b]Note by Darij:[/b] A [i]cyclic quadrilateral [/i]is a quadrilateral inscribed in a circle.

Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that \[\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),\] where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than $4.$

Consider the triangle $ABC{}$ and let $I_A{}$ be its $A{}$-excenter. Let $M,N$ and $P{}$ be the projections of $I_A{}$ onto the lines $AC,BC{}$ and $AB{}$ respectively. Prove that if $\overrightarrow{I_AM}+\overrightarrow{I_AP}=\overrightarrow{I_AN}$ then $ABC{}$ is an equilateral triangle.

Given two non-intersecting chords $AB$ and $CD$ of a circle $k$ and a length $a <CD$. Determine a point $X$ on $k$ with the following property: If lines $XA$ and $XB$ intersect $CD$ at points $P$ and $Q$ respectively, then $PQ = a$. Show how to construct all such points $X$ and prove that the obtained points indeed have the desired property.

Let $ABC$ be a right triangle with $\angle ACB = 90^o$ and M the center of $AB$. Let $G$ br any point on the line $MC$ and $P$ a point on the line $AG$, such that $\angle CPA = \angle BAC$ . Further let $Q$ be a point on the straight line $BG$, such that $\angle BQC = \angle CBA$ . Show that the circles of the triangles $AQG$ and $BPG$ intersect on the segment $AB$.

Given is a triangle $ABC$ with circumcentre $O$. The circumcircle of triangle $AOC$ intersects side $BC$ at $D$ and side $AB$ at $E$. Prove that the triangles $BDE$ and $AOC$ have circumradiuses of equal length.

A circle through the vertices $A$ and $B$ of $\triangle ABC$ intersects segments $AC$ and $BC$ at points $P$ and $Q$ respectively. If $AQ=AC$, $\angle BAQ=\angle CBP$ and $AB=(\sqrt{3}+1)PQ$, find the measures of the angles of $\triangle ABC$.

Let $ABC$ be a triangle with $AB > AC$ with incenter $I{}$. The internal bisector of the angle $BAC$ intersects the $BC$ at the point $D{}$. Let $M{}$ the midpoint of the segment $AD{}$, and let $F{}$ be the second intersection point of $MB$ with the circumcircle of the triangle $BIC$. Prove that $AF$ is perpendicular to $FC$.

In a rectangle $ABCD$ be a rectangle and $BC = 2AB$, let $E$ be the midpoint of $BC$ and $P$ an arbitrary inner point of $AD$. Let $F$ and $G$ be the feet of perpendiculars drawn correspondingly from $A$ to $BP$ and from $D$ to $CP$. Prove that the points $E,F,P,G$ are concyclic.

The lines \(AB\) and \(CD\), containing the lateral sides of the trapezoid \(ABCD\), intersect at point \(Q\). Inside the trapezoid \(ABCD\), a point \(P\) is chosen such that \(\angle APB = \angle CPD\). Prove that the circumcircles of triangles \(BPD\) and \(APC\) intersect again on the line \(PQ\). [i]Proposed by Mykhailo Shtandenko[/i]

What is the largest possible area of a quadrilateral with sides $1,4,7,8$ ? $ \textbf{(A)}\ 7\sqrt 2 \qquad\textbf{(B)}\ 10\sqrt 3 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 12\sqrt 3 \qquad\textbf{(E)}\ 9\sqrt 5 $

Let $ ABC$ be an equilateral triangle and $ P$ in its interior. The distances from $ P$ to the triangle's sides are denoted by $ a^2, b^2,c^2$ respectively, where $ a,b,c>0$. Find the locus of the points $ P$ for which $ a,b,c$ can be the sides of a non-degenerate triangle.

A circle $\omega$ through the incentre$ I$ of a triangle $ABC$ and tangent to $AB$ at $A$, intersects the segment $BC$ at $D$ and the extension of$ BC$ at $E$. Prove that the line $IC$ intersects $\omega$ at a point $M$ such that $MD=ME$.

A right circular cone pointing downward forms an angle of $60^\circ$ at its vertex. Sphere $S$ with radius $1$ is set into the cone so that it is tangent to the side of the cone. Three congruent spheres are placed in the cone on top of S so that they are all tangent to each other, to sphere $S$, and to the side of the cone. The radius of these congruent spheres can be written as $\tfrac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime. Find $a + b + c$. [asy] size(150); real t=0.12; void ball(pair x, real r, real h, bool ww=true) { pair xx=yscale(t)*x+(0,h); path P=circle(xx,r); unfill(P); draw(P); if(ww) draw(ellipse(xx-(0,r/2),0.85*r,t*r)); } pair X=(0,0); real H=17, h=5, R=h/2; draw(H*dir(120)--(0,0)--H*dir(60)); draw(ellipse((0,0.87*H),H/2,t*H/2)); pair Y=(R,h+2*R),C=(0,h); real r; for(int k=0;k<20;++k) { r=-(dir(30)*Y).x; Y-=(sqrt(3)/2*Y.x-r,abs(Y-C)-R-r)/3; } ball(Y.x*dir(90),r,Y.y,false); ball(X,R,h); ball(Y.x*dir(-30),r,Y.y); ball(Y.x*dir(210),r,Y.y);[/asy]

A triangle whose all sides have lengths greater than $1$ is contained in a unit square. Show that the center of the square lies inside the triangle.

The pentagons $P_1P_2P_3P_4P_5$ and$I_1I_2I_3I_4I_5$ are cyclic, where $I_i$ is the incentre of the triangle $P_{i-1}P_iP_{i+1}$ (reckoned cyclically, that is $P_0=P_5$ and $P_6=P_1$). Show that the lines $P_1I_1, P_2I_2, P_3I_3, P_4I_4$ and $P_5I_5$ meet in a single point.

Let $ABCD$ be a square, and let $P$ be a point on the minor arc $CD$ of its circumcircle. The lines $PA, PB$ meet the diagonals $BD, AC$ at points $K, L$ respectively. The points $M, N$ are the projections of $K, L$ respectively to $CD$, and $Q$ is the common point of lines $KN$ and $ML$. Prove that $PQ$ bisects the segment $AB$.

$\triangle{AOB}$ is a right triangle with $\angle{AOB}=90^{o}$. $C$ and $D$ are moving on $AO$ and $BO$ respectively such that $AC=BD$. Show that there is a fixed point $P$ through which the perpendicular bisector of $CD$ always passes.

Given $2N$ real numbers. It is known that if they are arbitrarily divided into two groups of $N$ numbers each then the products of the numbers of each group differ by $2$ at most. Is it necessarily true that if we arbitrarily place these numbers along a circle then there are two neighboring numbers that differ by $2$ at most, for a) $N=50$; (3 marks) b) $N=25$? (5 marks)

A triangle $ABC$ is inscribed in a circle $\omega$. A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$). Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$. Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$. [i](Russia) Vasily Mokin and Fedor Ivlev[/i]

Given a convex pentagon $ABCDE$ such that $AE$ is parallel to $CD$ and $AB=BC$. Angle bisectors of angles $A$ and $C$ intersect at $K$. Prove that $BK$ and $AE$ are parallel.