Let $M$be an interior point of triangle $ABC$. Let the line $AM$ intersect the circumcircle of the triangle $MBC$ for the second time at point $D$, the line $BM$ intersect the circumcircle of the triangle $MCA$ for the second time at point $E$, and the line $CM$ intersect the circumcircle of the triangle $MAB$ for the second time at point $F$. Prove that $\frac{AD}{MD} + \frac{BE}{ME} + \frac{CF}{MF} \geq \frac{9}{2}$. [i]Proposed by Nairy Sedrakyan[/i]

In a non-isosceles triangle $ABC$,$O$ and $I$ are circumcenter and incenter,respectively.$B^\prime$ is reflection of $B$ with respect to $OI$ and lies inside the angle $ABI$.Prove that the tangents to circumcirle of $\triangle BB^\prime I$ at $B^\prime$,$I$ intersect on $AC$. (A. Kuznetsov)

Let $O$ be the circumcenter of triangle $ABC$ with $|AB|=5$, $|BC|=6$, $|AC|=7$. Let $A_1$, $B_1$, $C_1$ be the reflections of $O$ over the lines $BC$, $AC$, $AB$, respectively. What is the distance between $A$ and the circumcenter of triangle $A_1B_1C_1$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ \sqrt {29} \qquad\textbf{(C)}\ \dfrac {19}{2\sqrt 6} \qquad\textbf{(D)}\ \dfrac {35}{4\sqrt 6} \qquad\textbf{(E)}\ \sqrt {\dfrac {35}3} $

Let $ABC$ be a triangle, let $I$ be its incenter, let $\Omega$ be its circumcircle, and let $\omega$ be the $A$- mixtilinear incircle. Let $D,E$ and $T$ be the intersections of $\omega$ and $AB,AC$ and $\Omega$, respectively, let the line $IT$ cross $\omega$ again at $P$, and let lines $PD$ and $PE$ cross the line $BC$ at $M$ and $N$ respectively. Prove that points $D,E,M,N$ are concyclic. What is the center of this circle?

Consider an acute triangle $ABC$ with $AB>AC$ inscribed in a circle of center $O$. From the midpoint $D$ of side $BC$ we draw line $(\ell)$ perpendicular to side $AB$ that intersects it at point $E$. If line $AO$ intersects line $(\ell)$ at point $Z$, prove that points $A,Z,D,C$ are concyclic.

Bisectors of triangle ABC of an angles A and C intersect with BC and AB at points A1 and C1 respectively. Lines AA1 and CC1 intersect circumcircle of triangle ABC at points A2 and C2 respectively. K is intersection point of C1A2 and A1C2. I is incenter of ABC. Prove that the line KI divides AC into two equal parts.

In an acute-angled triangle $ABC$, we consider the feet $H_a$ and $H_b$ of the altitudes from $A$ and $B$, and the intersections $W_a$ and $W_b$ of the angle bisectors from $A$ and $B$ with the opposite sides $BC$ and $CA$ respectively. Show that the centre of the incircle $I$ of triangle $ABC$ lies on the segment $H_aH_b$ if and only if the centre of the circumcircle $O$ of triangle $ABC$ lies on the segment $W_aW_b$.

Let $ABC$ be a triangle with $AB=26$, $AC=28$, $BC=30$. Let $X$, $Y$, $Z$ be the midpoints of arcs $BC$, $CA$, $AB$ (not containing the opposite vertices) respectively on the circumcircle of $ABC$. Let $P$ be the midpoint of arc $BC$ containing point $A$. Suppose lines $BP$ and $XZ$ meet at $M$ , while lines $CP$ and $XY$ meet at $N$. Find the square of the distance from $X$ to $MN$. [i]Proposed by Michael Kural[/i]

A circle is circumscribed about a triangle with sides $ 20$, $ 21$, and $ 29$, thus dividing the interior of the circle into four regions. Let $ A$, $ B$, and $ C$ be the areas of the non-triangular regions, with $ C$ being the largest. Then $ \textbf{(A)}\ A \plus{} B \equal{} C\qquad \textbf{(B)}\ A \plus{} B \plus{} 210 \equal{} C\qquad \textbf{(C)}\ A^2 \plus{} B^2 \equal{} C^2\qquad \\ \textbf{(D)}\ 20A \plus{} 21B \equal{} 29C\qquad \textbf{(E)}\ \frac{1}{A^2} \plus{} \frac{1}{B^2} \equal{} \frac{1}{C^2}$

Let $ABC$ be a scalene triangle with incenter $I$ and incircle $\omega$. Let the tangency points of $\omega$ to $BC,AC\text{ and } AB$ be $D,E,F$ respectively. Let the line $EF$ intersect the circumcircle of $ABC$ at the points $G, H$. Assume that $E$ lies between the points $F$ and $G$. Let $\Gamma$ be a circle that passes through $G$ and $H$ and that is tangent to $\omega$ at the point $M$ which lies on different semi-planes with $D$ with respect to the line $EF$. Let $\Gamma$ intersect $BC$ at points $K$ and $L$ and let the second intersection point of the circumcircle of $ABC$ and the circumcircle of $AKL$ be $N$. Prove that the intersection point of $NM$ and $AI$ lies on the circumcircle of $ABC$ if and only if the intersection point of $HB$ and $GC$ lies on $\Gamma$.

Let $ABCD$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $AE:EB=CF:FD$. Let $P$ be the point on the segment $EF$ such that $PE:PF=AB:CD$. Prove that the ratio between the areas of triangles $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.

Let $D$, $E$, $F$ be the midpoints of the arcs $BC$, $CA$, $AB$ on the circumcircle of a triangle $ABC$ not containing the points $A$, $B$, $C$, respectively. Let the line $DE$ meets $BC$ and $CA$ at $G$ and $H$, and let $M$ be the midpoint of the segment $GH$. Let the line $FD$ meet $BC$ and $AB$ at $K$ and $J$, and let $N$ be the midpoint of the segment $KJ$. a) Find the angles of triangle $DMN$; b) Prove that if $P$ is the point of intersection of the lines $AD$ and $EF$, then the circumcenter of triangle $DMN$ lies on the circumcircle of triangle $PMN$.

Consider an isosceles triangle. let $R$ be the radius of its circumscribed circle and $r$ be the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circle is \[ d=\sqrt{R(R-2r)} \]

In acute-angled triangle $ABC$, the height $AH$ and median $BM$ were drawn. Point $D$ lies on the circumcircle of triangle $BHM$ such that $AD \parallel BM$ and $B, D$ are on opposite sides of line $AC$. Prove that $BC=BD$.

The circle inscribed in the triangle $ABC$ touches the sides $AB$ and $AC$ at the points $K$ and $L$ , respectively. The angle bisectors from $B$ and $C$ intersect the altitude of the triangle from the vertex $A$ at the points $Q$ and $R$ , respectively. Prove that one of the points of intersection of the circles circumscribed around the triangles $BKQ$ and $CPL$ lies on $BC$.

Let $ABCD$ be a cyclic quadrilateral that inscribed in the circle $\omega$.Let $I_{1},I_{2}$ and $r_{1},r_{2}$ be incenters and radii of incircles of triangles $ACD$ and $ABC$,respectively.assume that $r_{1}=r_{2}$. let $\omega'$ be a circle that touches $AB,AD$ and touches $\omega$ at $T$. tangents from $A,T$ to $\omega$ meet at the point $K$.prove that $I_{1},I_{2},K$ lie on a line.

Let $L$ be an arbitrary point on the minor arc $CD$ of the circumcircle of square $ABCD$. Let $K,M,N$ be the intersection points of $AL,CD$; $CL,AD$; and $MK,BC$ respectively. Prove that $B,M,L,N$ are concyclic.

In $\triangle ABC$, $AC>BC$, $CM$ is the median, and $CH$ is the altitude emanating from $C$, as shown in the figure on the right. Determine the measure of $\angle MCH$ if $\angle ACM$ and $\angle BCH$ each have measure $17^\circ$. [asy] size(200); defaultpen(linewidth(0.8)); pair A=origin,B=(10,0),C=(7,5),M=(5,0),H=(7,0); draw(A--C--B--cycle^^H--C--M); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,NE); label("$M$",M,NW); label("$H$",H,NE); [/asy]

Let $ \gamma$ be circumcircle of $ \triangle ABC$.Let $ R_a$ be radius of circle touching $ AB,AC$&$ \gamma$ internally.Define $ R_b,R_c$ similarly. Prove That $ \frac {1}{aR_a} \plus{} \frac {1}{bR_b} \plus{} \frac {1}{cR_c} \equal{} \frac {s^2}{rabc}$.

It is known that $\odot O$ is the circumcircle of $\vartriangle ABC$ wwith diameter $AB$. The tangents of $\odot O$ at points $B$ and $C$ intersect at $P$ . The line perpendicular to $PA$ at point $A$ intersects the extension of $BC$ at point $D$. Extend $DP$ at length $PE = PB$. If $\angle ADP = 40^o$ , find the measure of $\angle E$.

Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.

With the medians of an acute-angled triangle another triangle is constructed. If $R$ and $R_m$ are the radii of the circles circumscribed about the first and the second triangle, respectively, prove that \[R_m>\frac{5}{6}R\]

In cyclic quadrilateral $ABXC$, $\measuredangle XAB = \measuredangle XAC$. Denote by $I$ the incenter of $\triangle ABC$ and by $D$ the projection of $I$ on $\overline{BC}$. If $AI = 25$, $ID = 7$, and $BC = 14$, then $XI$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$. [i]Proposed by Aaron Lin[/i]

Given an acute triangle $ABC$. $\Gamma _{B}$ is a circle that passes through $AB$, tangent to $AC$ at $A$ and centered at $O_{B}$. Define $\Gamma_C$ and $O_C$ the same way. Let the altitudes of $\triangle ABC$ from $B$ and $C$ meets the circumcircle of $\triangle ABC$ at $X$ and $Y$, respectively. Prove that $A$, the midpoint of $XY$ and the midpoint of $O_{B}O_{C}$ is collinear.

Let $ ABCD$ be a cyclic convex quadrilateral. Let $ E $ be the intersection of lines $ AB, CD $. $ P $ is the intersection of line passing $ B $ and perpendicular to $ AC $, and line passing $ C $ and perpendicular to $ BD$. $ Q $ is the intersection of line passing $ D $ and perpendicular to $ AC $, and line passing $ A $ and perpendicular to $ BD $. Prove that three points $ E, P, Q $ are collinear.