In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

A point $E$ lies on the altitude $BD$ of triangle $ABC$, and $\angle AEC=90^\circ.$ Points $O_1$ and $O_2$ are the circumcenters of triangles $AEB$ and $CEB$; points $F, L$ are the midpoints of the segments $AC$ and $O_1O_2.$ Prove that the points $L,E,F$ are collinear.

In convex quadrilateral $AEBC$, $\angle BEA = \angle CAE = 90^{\circ}$ and $AB = 15$, $BC = 14$ and $CA = 13$. Let $D$ be the foot of the altitude from $C$ to $\overline{AB}$. If ray $CD$ meets $\overline{AE}$ at $F$, compute $AE \cdot AF$. [i]Proposed by David Stoner[/i]

Let $P$ be a point inside a triangle $ABC$ such that $$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$ Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.

Given an equilateral triangle $ABC$ and a point $P$ so that the distances $P$ to $A$ and to $C$ are not farther than the distances $P$ to $B$. Prove that $PB = PA + PC$ if and only if $P$ lies on the circumcircle of $\vartriangle ABC$.

A point $D$ lies inside triangle $ABC$. Let $A_1, B_1, C_1$ be the second intersection points of the lines $AD$, $BD$, and $CD$ with the circumcircles of $BDC$, $CDA$, and $ADB$, respectively. Prove that $$\frac{AD}{AA_1} + \frac{BD}{BA_1} + \frac{CD}{CC_1} = 1.$$

Let $ABC$ be an acute triangle and $\omega$ be its circumcircle. The bisector of $\angle BAC$ intersects $\omega$ at [another point] $M$. Let $P$ be a point on $AM$ and inside $\triangle ABC$. Lines passing $P$ that are parallel to $AB$ and $AC$ intersects $BC$ on $E, F$ respectively. Lines $ME, MF$ intersects $\omega$ at points $K, L$ respectively. Prove that $AM, BL, CK$ are concurrent.

Let $\mathcal{C}$ be the circumcircle of the square with vertices $(0, 0), (0, 1978), (1978, 0), (1978, 1978)$ in the Cartesian plane. Prove that $\mathcal{C}$ contains no other point for which both coordinates are integers.

Let $I$ be the center of the inscribed circle of $ABC$ and let $I_A$ be the center of the exscribed circle touching the side $BC$. Let $M$ be the midpoint of the side $BC$, and $N$ be the midpoint of the arc $BAC$ of the circumscribed circle of $ABC$ . The point $T$ is symmetric to the point $N$ wrt point $A$. Prove that the points $I_A,M,I,T$ lie on the same circle. (Danilo Hilko)

Let $ABC$ be a triangle and let $X$ be on $BC$ such that $AX=AB$. let $AX$ meet circumcircle $\omega$ of triangle $ABC$ again at $D$. prove that circumcentre of triangle $BDX$ lies on $\omega$.

In a triangle $ABC$ with $AB<AC<BC$, the perpendicular bisectors of $AC$ and $BC$ intersect $BC$ and $AC$ at $K$ and $L$, respectively. Let $O$, $O_1$, and $O_2$ be the circumcentres of triangles $ABC$, $CKL$, and $OAB$, respectively. Prove that $OCO_1O_2$ is a parallelogram.

A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called [i]unstuck[/i] if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form $\sqrt{N}$, for a positive integer $N$. Find $N$.

Let $O$ be the circumcircle of $\triangle ABC$, with $AC=BC$ end let $D=AO\cap BC$. If $BD$ and $CD$ are integer numbers and $AO-CD$ is prime, determine such three numbers.

Let $ABC$ be a triangle with $AB \neq BC$ and let $BD$ the interior bisectrix of $ \angle ABC$ with $D \in AC$ . Let $M$ be the midpoint of the arc $AC$ that contains the point $B$ in the circumcircle of the triangle $ABC$ .The circumcircle of the triangle $BDM$ intersects the segment $AB$ in $K \neq B$ . Denote by $J$ the symmetric of $A$ with respect to $K$ .If $DJ$ intersects $AM$ in $O$ then prove that $J,B,M,O$ are concyclic.

Let $ABC$ be a triangle with circumcircle $\omega$, and let $X$ be the reflection of $A$ in $B$. Line $CX$ meets $\omega$ again at $D$. Lines $BD$ and $AC$ meet at $E$, and lines $AD$ and $BC$ meet at $F$. Let $M$ and $N$ denote the midpoints of $AB$ and $ AC$. Can line $EF$ share a point with the circumcircle of triangle $AMN?$ [i]Proposed by Sayandeep Shee[/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 $H$ be the orthocentre and $O$ be the circumcentre of an acute triangle $ABC$. Let $AD$ and $BE$ be the altitudes of the triangle with $D$ on $BC$ and $E$ on $CA$. Let $K =OD \cap BE, L = OE \cap AD$. Let $X$ be the second point of intersection of the circumcircles of triangles $HKD$ and $HLE$, and let $M$ be the midpoint of side $AB$. Prove that points $K, L, M$ are collinear if and only if $X$ is the circumcentre of triangle $EOD$.

Given acute angle triangle $ ABC$. Let $ CD$be the altitude , $ H$ be the orthocenter and $ O$ be the circumcenter of $ \triangle ABC$ The line through point $ D$ and perpendicular with $ OD$ , is intersect $ BC$ at $ E$. Prove that $ \angle DHE \equal{} \angle ABC$.

(B.Frenkin) Construct the triangle, given its centroid and the feet of an altitude and a bisector from the same vertex.

If the incircle of a right triangle with area $a$ is the circumcircle of a right triangle with area $b$, what is the minimum value of $\frac{a}{b}$? $ \textbf{(A)}\ 3 + 2\sqrt2 \qquad\textbf{(B)}\ 1+\sqrt2 \qquad\textbf{(C)}\ 2\sqrt2 \qquad\textbf{(D)}\ 2+\sqrt3 \qquad\textbf{(E)}\ 2\sqrt3$

The bisectors of the angles $\angle{A},\angle{B},\angle{C}$ of a triangle $\triangle{ABC}$ intersect with the circumcircle $c_1(O,R)$ of $\triangle{ABC}$ at $A_2,B_2,C_2$ respectively.The tangents of $c_1$ at $A_2,B_2,C_2$ intersect each other at $A_3,B_3,C_3$ (the points $A_3,A$ lie on the same side of $BC$,the points $B_3,B$ on the same side of $CA$,and $C_3,C$ on the same side of $AB$).The incircle $c_2(I,r)$ of $\triangle{ABC}$ is tangent to $BC,CA,AB$ at $A_1,B_1,C_1$ respectively.Prove that $A_1A_2,B_1B_2,C_1C_2,AA_3,BB_3,CC_3$ are concurrent. [hide=Diagram][asy]import graph; size(11cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -9.26871978147865, xmax = 19.467150423463277, ymin = -6.150626456647122, ymax = 10.10782642246474; /* image dimensions */ pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); draw((1.0409487561836381,4.30054785243355)--(0.,0.)--(6.,0.)--cycle, aqaqaq); /* draw figures */ draw((1.0409487561836381,4.30054785243355)--(0.,0.), uququq); draw((0.,0.)--(6.,0.), uququq); draw((6.,0.)--(1.0409487561836381,4.30054785243355), uququq); draw(circle((3.,1.550104087253063), 3.376806580383107)); draw(circle((1.9303371951242874,1.5188413314630436), 1.5188413314630436)); draw((1.0226422135625703,7.734611112525813)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((-1.2916762981259242,-1.8267024931300444)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((-0.2820306621765219,2.344520485530311)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((1.0559139088339535,1.4932847901569466)--(5.212367857300808,4.101231513568902), linetype("2 2")); draw((1.0559139088339535,1.4932847901569466)--(3.,-1.8267024931300442), linetype("2 2")); draw((12.047991949367804,-1.8267024931300444)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((1.0226422135625703,7.734611112525813)--(-1.2916762981259242,-1.8267024931300444)); draw((-1.2916762981259242,-1.8267024931300444)--(12.047991949367804,-1.8267024931300444)); draw((12.047991949367804,-1.8267024931300444)--(1.0226422135625703,7.734611112525813)); /* dots and labels */ dot((1.0409487561836381,4.30054785243355),linewidth(3.pt) + dotstyle); label("$A$", (0.5889800538632699,4.463280489351154), NE * labelscalefactor); dot((0.,0.),linewidth(3.pt) + dotstyle); label("$B$", (-0.5723380089304358,-0.10096957139619551), NE * labelscalefactor); dot((6.,0.),linewidth(3.pt) + dotstyle); label("$C$", (6.233525986976863,0.06107480945873997), NE * labelscalefactor); label("$c_1$", (1.9663572911302232,5.111458012770896), NE * labelscalefactor); dot((3.,-1.8267024931300442),linewidth(3.pt) + dotstyle); label("$A_2$", (2.9386235762598374,-2.3155761097469805), NE * labelscalefactor); dot((5.212367857300808,4.101231513568902),linewidth(3.pt) + dotstyle); label("$B_2$", (5.315274495465561,4.274228711687063), NE * labelscalefactor); dot((-0.2820306621765219,2.344520485530311),linewidth(3.pt) + dotstyle); label("$C_2$", (-0.9234341674494632,2.6807922999468636), NE * labelscalefactor); dot((1.0226422135625703,7.734611112525813),linewidth(3.pt) + dotstyle); label("$A_3$", (1.1291279900463889,7.893219884113956), NE * labelscalefactor); dot((-1.2916762981259242,-1.8267024931300444),linewidth(3.pt) + dotstyle); label("$B_3$", (-1.8146782621516093,-1.4783468086631473), NE * labelscalefactor); dot((12.047991949367804,-1.8267024931300444),linewidth(3.pt) + dotstyle); label("$C_3$", (12.148145888182015,-1.6673985863272387), NE * labelscalefactor); dot((1.9303371951242874,1.5188413314630436),linewidth(3.pt) + dotstyle); label("$I$", (2.047379481557691,1.681518618008095), NE * labelscalefactor); dot((1.9303371951242878,0.),linewidth(3.pt) + dotstyle); label("$A_1$", (1.4532167517562602,-0.5600953171518461), NE * labelscalefactor); label("$c_2$", (1.5072315453745722,3.247947632939138), NE * labelscalefactor); dot((2.9254299438737803,2.666303492733126),linewidth(3.pt) + dotstyle); label("$B_1$", (2.8576013858323694,3.1129106488933584), NE * labelscalefactor); dot((0.45412477306806903,1.8761589424582812),linewidth(3.pt) + dotstyle); label("$C_1$", (0,2.3296961414278368), NE * labelscalefactor); dot((1.0559139088339535,1.4932847901569466),linewidth(3.pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/hide]

Given an acute triangle $ABC$ with $AC>BC$ and the circumcenter of triangle $ABC$ is $O$. The altitude of triangle $ABC$ from $C$ intersects $AB$ and the circumcircle at $D$ and $E$, respectively. A line which passed through $O$ which is parallel to $AB$ intersects $AC$ at $F$. Show that the line $CO$, the line which passed through $F$ and perpendicular to $AC$, and the line which passed through $E$ and parallel with $DO$ are concurrent. [i]Fajar Yuliawan, Bandung[/i]

A triangle $ABC$ is inscribed in the circle $(O,R)$. A circle $(O',R')$ is internally tangent to $(O)$ at $I$ such that $R < R'$. $P$ is a point on the circle $(O)$. Rays $PA, PB, PC$ meet $(O')$ at $A_1,B_1,C_1$. Let $A_2B_2C_2$ be the triangle formed by the intersections of the line symmetric to $B_1C_1$ about $BC$, the line symmetric to $C_1A_1$ about $CA$ and the line symmetric to $A_1B_1$ about $AB$. Prove that the circumcircle of $A_2B_2C_2$ is tangent to $(O)$. Nguyễn Văn Linh

$ABCDEF$ is a convex hexagon, such that $|\angle FAB| = |\angle BCD| =|\angle DEF|$ and $|AB| =|BC|,$ $|CD| = |DE|$, $|EF| = |FA|$. Prove that the lines $AD$, $BE$ and $CF$ are concurrent.

In triangle $ABC$, $P$ is a point on altitude $AD$. $Q,R$ are the feet of the perpendiculars from $P$ to $AB,AC$, and $QP,RP$ meet $BC$ at $S$ and $T$ respectively. the circumcircles of $BQS$ and $CRT$ meet $QR$ at $X,Y$. a) Prove $SX,TY, AD$ are concurrent at a point $Z$. b) Prove $Z$ is on $QR$ iff $Z=H$, where $H$ is the orthocenter of $ABC$. [i]Ray Li.[/i]