Two circles with centers $O_1$ and $O_2$ meet at points $A$ and $B$. The bisector of angle $O_1AO_2$ meets the circles for the second time at points $C $and $D$. Prove that the distances from the circumcenter of triangle $CBD$ to $O_1$ and to $O_2$ are equal.

Let $P$ and $Q$ be two distinct points in the plane. Let us denote by $m(PQ)$ the segment bisector of $PQ$. Let $S$ be a finite subset of the plane, with more than one element, that satisfies the following properties: (i) If $P$ and $Q$ are in $S$, then $m(PQ)$ intersects $S$. (ii) If $P_1Q_1, P_2Q_2, P_3Q_3$ are three diferent segments such that its endpoints are points of $S$, then, there is non point in $S$ such that it intersects the three lines $m(P_1Q_1)$, $m(P_2Q_2)$, and $m(P_3Q_3)$. Find the number of points that $S$ may contain.

Acute-angled triangle $ABC$ is inscribed into circle $\Omega$. Lines tangent to $\Omega$ at $B$ and $C$ intersect at $P$. Points $D$ and $E$ are on $AB$ and $AC$ such that $PD$ and $PE$ are perpendicular to $AB$ and $AC$ respectively. Prove that the orthocentre of triangle $ADE$ is the midpoint of $BC$.

In a cyclic quadrilateral $ABCD$ with $AB=AD$ points $M$,$N$ lie on the sides $BC$ and $CD$ respectively so that $MN=BM+DN$ . Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$ .

Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i \equal{} 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i\plus{}1}$ and $ A_iA_{i\plus{}2}$ (the addition of indices being mod 3). Let $ B_i, i \equal{} 1, 2, 3,$ be the second point of intersection of $ C_{i\plus{}1}$ and $ C_{i\plus{}2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.

In the segment $AC$ of an isosceles triangle $\triangle ABC$ with base $BC$ is chosen a point $D$. On the smaller arc $CD$ of the circumcircle of $\triangle BCD$ is chosen a point $K$. Line $CK$ intersects the line through $A$ parallel to $BC$ at $T$. $M$ is the midpoint of segment $DT$. Prove that $\angle AKT=\angle CAM$. [i](A.Kuznetsov)[/i]

Let $E$ be a point on the median $CD$ of a triangle $ABC$. The circle $\mathcal S_1$ passing through $E$ and touching $AB$ at $A$ meets the side $AC$ again at $M$. The circle $S_2$ passing through $E$ and touching $AB$ at $B$ meets the side $BC$ at $N$. Prove that the circumcircle of $\triangle CMN$ is tangent to both $\mathcal S_1$ and $\mathcal S_2$.

Let $M$ be the intersection of the diagonals $AC$ and $BD$ of cyclic quadrilateral $ABCD$. If $|AB|=5$, $|CD|=3$, and $m(\widehat{AMB}) = 60^\circ$, what is the circumradius of the quadrilateral? $ \textbf{(A)}\ 5\sqrt 3 \qquad\textbf{(B)}\ \dfrac {7\sqrt 3}{3} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{34} $

Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.

The circumradius of a tetrahedron $ ABCD$ is $ R$, and the lenghts of the segments connecting the vertices $ A,B,C,D$ with the centroids of the opposite faces are equal to $ m_a,m_b,m_c$ and $ m_d$, respectively. Prove that: $ m_a\plus{}m_b\plus{}m_c\plus{}m_d\leq \dfrac{16}{3}R$

Let $ABC$ be an acute triangle and $(O)$ be its circumcircle. Denote by $H$ its orthocenter and $I$ the midpoint of $BC$. The lines $BH, CH$ intersect $AC,AB$ at $E, F$ respectively. The circles $(IBF$) and $(ICE)$ meet again at $D$. a) Prove that $D, I,A$ are collinear and $HD, EF, BC$ are concurrent. b) Let $L$ be the foot of the angle bisector of $\angle BAC$ on the side $BC$. The circle $(ADL)$ intersects $(O)$ again at $K$ and intersects the line $BC$ at $S$ out of the side $BC$. Suppose that $AK,AS$ intersects the circles $(AEF)$ again at $G, T$ respectively. Prove that $TG = TD$.

A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object? $ \textbf{(A)}\ \dfrac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \dfrac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \dfrac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} $

Let $O$ be the circumcenter and $I$ be the incenter of $\vartriangle$, $P$ is the reflection from $I$ through $O$, the foot of perpendicular from $P$ to $BC,CA,AB$ is $X,Y,Z$, respectively. Prove that $AP^2+PX^2=BP^2+PY^2=CP^2+PZ^2$.

In the following, the point of intersection of two lines $ g$ and $ h$ will be abbreviated as $ g\cap h$. Suppose $ ABC$ is a triangle in which $ \angle A \equal{} 90^{\circ}$ and $ \angle B > \angle C$. Let $ O$ be the circumcircle of the triangle $ ABC$. Let $ l_{A}$ and $ l_{B}$ be the tangents to the circle $ O$ at $ A$ and $ B$, respectively. Let $ BC \cap l_{A} \equal{} S$ and $ AC \cap l_{B} \equal{} D$. Furthermore, let $ AB \cap DS \equal{} E$, and let $ CE \cap l_{A} \equal{} T$. Denote by $ P$ the foot of the perpendicular from $ E$ on $ l_{A}$. Denote by $ Q$ the point of intersection of the line $ CP$ with the circle $ O$ (different from $ C$). Denote by $ R$ be the point of intersection of the line $ QT$ with the circle $ O$ (different from $ Q$). Finally, define $ U \equal{} BR \cap l_{A}$. Prove that \[ \frac {SU \cdot SP}{TU \cdot TP} \equal{} \frac {SA^{2}}{TA^{2}}. \]

Given are two circles $\omega_1,\omega_2$ which intersect at points $X,Y$. Let $P$ be an arbitrary point on $\omega_1$. Suppose that the lines $PX,PY$ meet $\omega_2$ again at points $A,B$ respectively. Prove that the circumcircles of all triangles $PAB$ have the same radius.

Let $ABC$ be an acute angled triangle, $O$ be the circumcenter of $ABC$, and $R$ be the cicumradius. $AO$ meets the circumcircle of $BOC$ at $A'$, $BO$ meets the circumcircle of $COA$, and $CO$ meets the circumcircle of $AOB$ at $C'$. Prove that \[OA' \cdot OB' \cdot OC' \geq 8R^3.\] When does inequality occur?

Given triangle $ABC$ . From the $P$ point three lines $(PA),(PB),(PC)$ are drawn. They cross the circumscribed circle at $A_1, B_1,C_1$ points respectively. It comes out that the $A_1B_1C_1$ triangle equals to the initial one. Prove that there are not more than eight such a points $P$ in a plane.

Let $ABC$ be a triangle with $\angle C=90$. The tangent points of the inscribed circle with the sides $BC, CA$ and $AB$ are $M, N$ and $P.$ Points $M_1, N_1, P_1$ are symmetric to points $M, N, P$ with respect to midpoints of sides $BC, CA$ and $AB.$ Find the smallest value of $\frac{AO_1+BO_1}{AB},$ where $O_1$ is the circumcenter of triangle $M_1N_1P_1.$

Prove that for every convex polygon can be found such three sequential vertices for which a circle that they lie on covers the polygon. [i]Jordan Tabov[/i]

Let $N$ be the midpoint of arc $ABC$ of the circumcircle of triangle $ABC$, let $M$ be the midpoint of $AC$ and let $I_1, I_2$ be the incentres of triangles $ABM$ and $CBM$. Prove that points $I_1, I_2, B, N$ lie on a circle. [i]M. Kungojin[/i]

The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle? $ \textbf{(A)}\ \frac{3\sqrt2}{\pi} \qquad \textbf{(B)}\ \frac{3\sqrt3}{\pi} \qquad \textbf{(C)}\ \sqrt3 \qquad \textbf{(D)}\ \frac{6}{\pi} \qquad \textbf{(E)}\ \sqrt3\pi$

Let $ABC$ be a triangle, $G$ it`s centroid, $H$ it`s orthocenter, and $M$ the midpoint of the arc $\widehat{AC}$ (not containing $B$). It is known that $MG=R$, where $R$ is the radius of the circumcircle. Prove that $BG\geq BH$. [i]Proposed by F. Bakharev[/i]

A quadrilateral $KLMN$ is given. A circle with center $O$ meets its side $KL$ at points $A$ and $A_1$, side $LM$ at points $B$ and $B_1$, etc. Prove that if the circumcircles of triangles $KDA, LAB, MBC$ and $NCD$ concur at point $P$, then a) the circumcircles of triangles $KD_1A_1, LA_1B_1, MB_1C_1$ and $NC1D1$ also concur at some point $Q$; b) point $O$ lies on the perpendicular bisector to $PQ$.

[b]i.)[/b] Consider a circle $K$ with diameter $AB;$ with circle $L$ tangent to $AB$ and to $K$ and with a circle $M$ tangent to circle $K,$ circle $L$ and $AB.$ Calculate the ration of the area of circle $K$ to the area of circle $M.$ [b]ii.)[/b] In triangle $ABC, AB = AC$ and $\angle CAB = 80^{\circ}.$ If points $D,E$ and $F$ lie on sides $BC, AC$ and $AB,$ respectively and $CE = CD$ and $BF = BD,$ then find the size of $\angle EDF.$

Let $ABC$ be an acute triangle inscribed in circle $(O)$, with orthocenter $H$. Median $AM$ of triangle $ABC$ intersects circle $(O)$ at $A$ and $N$. $AH$ intersects $(O)$ at $A$ and $K$. Three lines $KN, BC$ and line through $H$ and perpendicular to $AN$ intersect each other and form triangle $X Y Z$. Prove that the circumcircle of triangle $X Y Z$ is tangent to $(O)$.