In a triangle $ABC$, let $H, I$ and $O$ be the orthocentre, incentre and circumcentre, respectively. If the points $B, H, I, C$ lie on a circle, what is the magnitude of $\angle BOC$ in degrees?

An arbitrary point $M$ is selected in the interior of the segment $AB$. The square $AMCD$ and $MBEF$ are constructed on the same side of $AB$, with segments $AM$ and $MB$ as their respective bases. The circles circumscribed about these squares, with centers $P$ and $Q$, intersect at $M$ and also at another point $N$. Let $N'$ denote the point of intersection of the straight lines $AF$ and $BC$. a) Prove that $N$ and $N'$ coincide; b) Prove that the straight lines $MN$ pass through a fixed point $S$ independent of the choice of $M$; c) Find the locus of the midpoints of the segments $PQ$ as $M$ varies between $A$ and $B$.

In a triangle $ABC$, $\angle BAC =90^{\circ}$. Point $D$ lies on the side $BC$ and satisfies $\angle BDA=2\angle BAD$. Prove that \[\frac{2}{AD}=\frac{1}{BD}+\frac{1}{CD} \]

$\omega$ is circumcircle of triangle $ABC$ and $BB_1, CC_1$ are bisectors of ABC. $I$ is center incirle . $B_1 C1$ and $\omega$ intersects at $M$ and $N$ . Find the ratio of circumradius of $ABC$ to circumradius $MIN$.

Let $ABCD$ be a trapezium such that $AB||CD$ and $AB+CD=AD$. Let $P$ be the point on $AD$ such that $AP=AB$ and $PD=CD$. $a)$ Prove that $\angle BPC=90^{\circ}$. $b)$ $Q$ is the midpoint of $BC$ and $R$ is the point of intersection between the line $AD$ and the circle passing through the points $B,A$ and $Q$. Show that the points $B,P,R$ and $C$ are concyclic.

$ABC$ is an acute angle triangle such that $AB>AC$ and $\hat{BAC}=60^{\circ}$. Let's denote by $O$ the center of the circumscribed circle of the triangle and $H$ the intersection of altitudes of this triangle. Line $OH$ intersects $AB$ in point $P$ and $AC$ in point $Q$. Find the value of the ration $\frac{PO}{HQ}$.

Given a convex quadrilateral with $\angle B+\angle D<180$.Let $P$ be an arbitrary point on the plane,define $f(P)=PA*BC+PD*CA+PC*AB$. (1)Prove that $P,A,B,C$ are concyclic when $f(P)$ attains its minimum. (2)Suppose that $E$ is a point on the minor arc $AB$ of the circumcircle $O$ of $ABC$,such that$AE=\frac{\sqrt 3}{2}AB,BC=(\sqrt 3-1)EC,\angle ECA=2\angle ECB$.Knowing that $DA,DC$ are tangent to circle $O$,$AC=\sqrt 2$,find the minimum of $f(P)$.

The excircle of a triangle $ABC$ corresponding to $A$ touches the lines $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. The excircle corresponding to $B$ touches $BC,CA,AB$ at $A_2,B_2,C_2$, and the excircle corresponding to $C$ touches $BC,CA,AB$ at $A_3,B_3,C_3$, respectively. Find the maximum possible value of the ratio of the sum of the perimeters of $\triangle A_1B_1C_1$, $\triangle A_2B_2C_2$ and $\triangle A_3B_3C_3$ to the circumradius of $\triangle ABC$.

Let $ ABC$ be an isoceles triangle with $ AC\equal{}BC$. A point $ P$ lies inside $ ABC$ such that \[ \angle PAB \equal{} \angle PBC, \angle PAC \equal{} \angle PCB.\] Let $ M$ be the midpoint of $ AB$ and $ K$ be the intersection of $ BP$ and $ AC$. Prove that $ AP$ and $ PK$ trisect $ \angle MPC$.

Let $\Omega$ be the circle circumscribed to the triangle $ABC$. Tangents taken to the circle $\Omega$ at points $A$ and $B$ intersects at the point $P$ , and the perpendicular bisector of $ (BC)$ cuts line $AC$ at point $Q$. Prove that lines $BC$ and $PQ$ are parallel.

Given a right triangle $ABC$, the point $M$ is the midpoint of the hypotenuse $AB$. A circle is circumscribed around the triangle $BCM$, which intersects the segment $AC$ at a point $Q$ other than $C$. It turned out that the segment $QA$ is twice as large as the side $BC$. Find the acute angles of triangle $ABC$. (Mykola Moroz)

Given an acute triangle $ABC$ satisfying $\angle ACB<\angle ABC<\angle ACB+\dfrac{\angle BAC}{2}$. Let $D$ be a point on $BC$ such that $\angle ADC=\angle ACB+\dfrac{\angle BAC}{2}$. Tangent of circumcircle of $ABC$ at $A$ hits $BC$ at $E$. Bisector of $\angle AEB$ intersects $AD$ and $(ADE)$ at $G$ and $F$ respectively, $DF$ hits $AE$ at $H.$ a) Prove that circle with diameter $AE,DF,GH$ go through one common point. b) On the exterior bisector of $\angle BAC $ and ray $AC$ given point $K$ and $M$ respectively satisfying $KB=KD=KM$, On the exterior bisector of $\angle BAC$ and ray $AB$ given point $L$ and $N$ respectively satisfying $LC=LD=LN.$ Circle throughs $M,N$ and midpoint $I$ of $BC$ hits $BC$ at $P$ ($P\neq I$). Prove that $BM,CN,AP$ concurrent.

In an acute triangle $ ABC$, let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[AB\right]$ such that $ \angle ADB\equal{}\angle AEC\equal{}90{}^\circ$. If perimeter of triangle $ AED$ is 9, circumradius of $ AED$ is $ \frac{9}{5}$ and perimeter of triangle $ ABC$ is 15, then $ \left|BC\right|$ is $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ \frac{24}{5} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \frac{48}{5}$

The incircle of triangle $ABC$ touches its sides $AB$ and $BC$ at points $P$ and $Q.$ The line $PQ$ meets the circumcircle of triangle $ABC$ at points $X$ and $Y.$ Find $\angle XBY$ if $\angle ABC = 90^\circ.$ [i]Proposed by A. Smirnov[/i]

The bisectors of the angles $\sphericalangle CAB$ and $\sphericalangle BCA$ intersect the circumcircle of $ABC$ in $P$ and $Q$ respectively. These bisectors intersect each other in point $I$. Prove that $PQ \perp BI$.

Let $K,L,M$ be points on sides $AB,BC,CA$, respectively, of a triangle $ABC$ such that $AK/AB = BL/BC = CM/CA = 1/3$. Show that if the circumcircles of the triangles $AKM, BLK, CML$ are equal, then so are the incircles of these triangles.

Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$. [i]Proposed by Sammy Luo[/i]

Three circles of equal radius have a common point $O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle are collinear with the point $O$.

Let $ P$, $ Q$, and $ R$ be the points on sides $ BC$, $ CA$, and $ AB$ of an acute triangle $ ABC$ such that triangle $ PQR$ is equilateral and has minimal area among all such equilateral triangles. Prove that the perpendiculars from $ A$ to line $ QR$, from $ B$ to line $ RP$, and from $ C$ to line $ PQ$ are concurrent.

Given a trapezoid, divide it by a line into two quadrilaterals in such a way that both of them are cyclic with the same circumradius. Discuss conditions of solvability.

Triangles $MA_2B_2$ and $MA_1B_1$ are similar to each other and have the same orientation. Prove that the circles circumcribed around these triangles and the straight lines $A_1A_2$ , $B_1B_2$ have a common point.

$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.

Two circles intersect at two points $A$ and $B$. A line $\ell$ which passes through the point $A$ meets the two circles again at the points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ (which do not contain the point $A$) on the respective circles. Let $K$ be the midpoint of the segment $CD$. Prove that $\measuredangle MKN = 90^{\circ}$.

Points $X$ and $Y$ are the midpoints of arcs $AB$ and $BC$ of the circumscribed circle of triangle $ABC$. Point $T$ lies on side $AC$. It turned out that the bisectors of the angles $ATB$ and $BTC$ pass through points $X$ and $Y$ respectively. What angle $B$ can be in triangle $ABC$?

Let $ ABC $ be an acute triangle having $ AB<AC, I $ be its incenter, $ D,E,F $ be intersection of the incircle with $ BC, CA, $ respectively, $ AB, X $ be the middle of the arc $ BAC, $ which is an arc of the circumcicle of it, $ P $ be the projection of $ D $ on $ EF $ and $ Q $ be the projection of $ A $ on $ ID. $ [b]a)[/b] Show that $ IX $ and $ PQ $ are parallel. [b]b)[/b] If the circle of diameter $ AI $ intersects the circumcircle of $ ABC $ at $ Y\neq A, $ prove that $ XQ $ intersects $ PI $ at $ Y. $