Let $ABC$ be an acute triangle with $AB \neq AC$. Let $D$ be the foot of the altitude from $A$ and $\omega$ the circumcircle of the triangle. Let $\omega_1$ be the circle tangent to $AD$, $BD$ and $\omega$. Let $\omega_2$ be the circle tangent to $AD$, $CD$ and $\omega$. Let $\ell$ be the interior common tangent to both $\omega_1$ and $\omega_2$, different from $AD$. Prove that $\ell$ passes through the midpoint of $BC$ if and only if $2BC = AB + AC$.

Let $ P$ be a point inside a triangle $ ABC$ such that \[ \angle APB \minus{} \angle ACB \equal{} \angle APC \minus{} \angle ABC. \] Let $ D$, $ E$ be the incenters of triangles $ APB$, $ APC$, respectively. Show that the lines $ AP$, $ BD$, $ CE$ meet at a point.

Let $P$ be an arbitrary point on the side $BC$ of triangle $ABC$ and let $D$ be the tangency point between the incircle of the triangle $ABC$ and the side $BC$. If $Q$ and $R$ are respectively the incenters in the triangles $ABP$ and $ACP$, prove that $\angle QDR$ is a right angle. Prove that the triangle $QDR$ is isosceles if and only if $P$ is the foot of the altitude from $A$ in the triangle $ABC$.

Let $I$ be the incenter of $\triangle ABC$ with $AB>AC$. Let $\Gamma$ be the circle with diameter $AI$. The circumcircle of $\triangle ABC$ intersects $\Gamma$ at points $A,D$, with point $D$ lying on $\overarc{AC}$ (not containing $B$). Let the line passing through $A$ and parallel to $BC$ intersect $\Gamma$ at points $A,E$. If $DI$ is the angle bisector of $\angle CDE$, and $\angle ABC = 33^{\circ}$, find the value of $\angle BAC$.

Let $M$ be the midpoint of side $BC$ of triangle $ABC$, $Q$ the point of intersection of its angle bisectors. It is known that $MQ=QA$. Find the smallest possible value of angle $\angle MQA$.

As shown on the figure, square $PQRS$ is inscribed in right triangle $ABC$, whose right angle is at $C$, so that $S$ and $P$ are on sides $BC$ and $CA$, respectively, while $Q$ and $R$ are on side $AB$. Prove that $A B\geq3QR$ and determine when equality occurs. [asy] defaultpen(linewidth(0.7)+fontsize(10)); size(150); real a=8, b=6; real y=a/((a^2+b^2)/(a*b)+1), r=degrees((a,b))+180; pair A=b*dir(-r)*dir(90), B=a*dir(180)*dir(-r), C=origin, S=y*dir(-r)*dir(180), P=(y*b/a)*dir(90-r), Q=foot(P, A, B), R=foot(S, A, B); draw(A--B--C--cycle^^R--S--P--Q); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$S$", S, dir(point--S)); label("$R$", R, dir(270)); label("$Q$", Q, dir(270)); label("$P$", P, dir(point--P));[/asy]

Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid. Prove that these three Euler lines pass through one common point. [i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$. [i]Floor van Lamoen[/i]

In acute triangle $ABC, AB<AC$, $I$ is its incenter, $D$ is the foot of perpendicular from $I$ to $BC$, altitude $AH$ meets $BI,CI$ at $P,Q$ respectively. Let $O$ be the circumcenter of $\triangle IPQ$, extend $AO$ to meet $BC$ at $L$. Circumcircle of $\triangle AIL$ meets $BC$ again at $N$. Prove that $\frac{BD}{CD}=\frac{BN}{CN}$.

Let $ABC$ be a triangle, $I$ its incenter and $\omega$ its incircle. Let $D$,$E$ and $F$ be the points of tangency of $\omega$ with $BC$,$AC$ and $AB$, respectively and $M$,$N$ and $P$ be the midpoints of $BC$, $AC$ and $AB$. Let $D'$ be the second intersection of $DI$ with $\omega$, $Q$ the intersection of $DI$ with $EF$ and $U \ne Q$ be the intersection of $(AD'Q)$ with $(DMQ)$. Suppose that $U$ lies on the circumcircle of $BDF$. Prove that $PN, AM, UF$ concur.

$M$ is midpoint of side $BC$ of triangle $ABC$, and $I$ is incenter of triangle $ABC$, and $T$ is midpoint of arc $BC$, that does not contain $A$. Prove that \[\cos B+\cos C=1\Longleftrightarrow MI=MT\]

A circle $ C$ has two parallel tangents $ L'$ and$ L"$. A circle $ C'$ touches $ L'$ at $ A$ and $ C$ at $ X$. A circle $ C"$ touches $ L"$ at $ B$, $ C$ at $ Y$ and $ C'$ at $ Z$. The lines $ AY$ and $ BX$ meet at $ Q$. Show that $ Q$ is the circumcenter of $ XYZ$

Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.

Given a triangle $ABC$, let $D$ and $E$ be points on the side $BC$ such that $\angle BAD = \angle CAE$. If $M$ and $N$ are, respectively, the points of tangency of the incircles of the triangles $ABD$ and $ACE$ with the line $BC$, then show that \[\frac{1}{MB}+\frac{1}{MD}= \frac{1}{NC}+\frac{1}{NE}. \]

Let $XYZ$ be a triangle with $ \angle X = 60^\circ $ and $ \angle Y = 45^\circ $. A circle with center $P$ passes through points $A$ and $B$ on side $XY$, $C$ and $D$ on side $YZ$, and $E$ and $F$ on side $ZX$. Suppose $AB=CD=EF$. Find $ \angle XPY $ in degrees.

Let $ABC$ be an acute triangle and let $X$ and $Y$ be points on the segments $AB$ and $AC$ such that $BX = CY$. If $I_{B}$ and $I_{C}$ are centers of inscribed circles in triangles $ABY$ and $ACX$, and $T$ is the second intersection point of the circumcircles of $ABY$ and $ACX$, show that: $$\frac{TI_{B}}{TI_{C}} = \frac{BY}{CX}.$$ [i]Proposed by Nikola Velov[/i]

Let $A_1$ and $C_1$ be the tangency points of the incircle of triangle $ABC$ with $BC$ and $AB$ respectively, $A'$ and $C'$ be the tangency points of the excircle inscribed into the angle $B$ with the extensions of $BC$ and $AB$ respectively. Prove that the orthocenter $H$ of triangle $ABC$ lies on $A_1C_1$ if and only if the lines $A'C_1$ and $BA$ are orthogonal.

A sphere is inscribed in the tetrahedron whose vertices are $A=(6,0,0), B=(0,4,0), C=(0,0,2),$ and $D=(0,0,0).$ The radius of the sphere is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Consider an acute triangle $ABC$ with circumcircle $\mathcal{C}$. Let $H$ be the orthocenter of $ABC$ and $M$ the midpoint of $BC$. Lines $AH$, $BH$ and $CH$ cut $\mathcal{C}$ again at points $D$, $E$, and $F$ respectively; line $MH$ cuts $\mathcal{C}$ at $J$ such that $H$ lies between $J$ and $M$. Let $K$ and $L$ be the incenters of triangles $DEJ$ and $DFJ$ respectively. Prove $KL$ is parallel to $BC$.

In given triangle $\triangle ABC$, difference between sizes of each pair of sides is at least $d>0$. Let $G$ and $I$ be the centroid and incenter of $\triangle ABC$ and $r$ be its inradius. Show that $$[AIG]+[BIG]+[CIG]\ge\frac{2}{3}dr,$$ where $[XYZ]$ is (nonnegative) area of triangle $\triangle XYZ$.

In a triangle $ABC$ with the incentre $I,$ the angle bisector $AD$ meets the circumcircle of triangle $BIC$ at point $N\neq I$. a) Express the angles of $\triangle BCN$ in terms of the angles of triangle $ABC$. b) Show that the circumcentre of triangle $BIC$ is at the intersection of $AI$ and the circumcentre of $ABC$.

The incircle of triangle $ABC$ touches sides $BC, CA$ and $AB$ at points $D, E$ and $F$, respectively. Let $P$ be the intersection of lines $AD$ and $BE$. The reflections of $P$ with respect to $EF, FD$ and $DE$ are $X,Y$ and $Z$, respectively. Prove that lines $AX, BY$ and $CZ$ are concurrent at a point on line $IO$, where $I$ and $O$ are the incenter and circumcenter of triangle $ABC$.

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 $MD=ME$.

Consider the three disjoint arcs of a circle determined by three points of the circle. We construct a circle around each of the midpoint of every arc which goes the end points of the arc. Prove that the three circles pass through a common point.

Let $ABC$ be a non-isosceles triangle with incenter $I$. Let $D$ be a point on the segment $BC$ such that the circumcircle of $BID$ intersects the segment $AB$ at $E\neq B$, and the circumcircle of $CID$ intersects the segment $AC$ at $F\neq C$. The circumcircle of $DEF$ intersects $AB$ and $AC$ at the second points $M$ and $N$ respectively. Let $P$ be the point of intersection of $IB$ and $DE$, and let $Q$ be the point of intersection of $IC$ and $DF$. Prove that the three lines $EN, FM$ and $PQ$ are parallel. [i]Proposed by Saudi Arabia[/i]

Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other