Let $ABC$ an acute triangle and $\Gamma$ its circumcircle. The bisector of $BAC$ intersects $\Gamma$ at $M\neq A$. A line $r$ parallel to $BC$ intersects $AC$ at $X$ and $AB$ at $Y$. Also, $MX$ and $MY$ intersect $\Gamma$ again at $S$ and $T$, respectively. If $XY$ and $ST$ intersect at $P$, prove that $PA$ is tangent to $\Gamma$.

Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.

Let $\triangle ABC$ be a triangle and let $D$ be a point on side $\overline{BC}$. Let $U$ and $V$ be the circumcenters of triangles $\triangle ABD$ and $\triangle ADC$, respectively. Show, that $\triangle ABC$ and $\triangle AUV$ are similar. [i](41th Austrian Mathematical Olympiad, regional competition, problem 3)[/i]

Let $D$ and $E$ be points on the sides $BC$ and $AC$ of the triangle $\triangle ABC$, respectively. The circumcircle of $\triangle ADC$ meets the circumcircle of $\triangle BCE$ for the second time at $F$. The line $FE$ meets the line $AD$ at $G$, while the line $FD$ meets the line $BE$ at $H$. Prove that the lines $CF$, $AH$ and $BG$ pass through the same point. [i]Authored by Petar Filipovski[/i]

Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB $ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD=PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.

Let $O$ be the center of the circle circumscribed around $\vartriangle ABC$, let $A_1, B_1, C_1$ be symmetric to $O$ through respective sides of $\vartriangle ABC$. Prove that all altitudes of $\vartriangle A_1B_1C_1$ pass through $O$, and all altitudes of $\vartriangle ABC$ pass through the center of the circle circumscribed around $\vartriangle A_1B_1C_1$.

A circle passing through $B$ and $C$ meets the side $[AB]$ of $\triangle ABC$ at $D$, and $[AC]$ at $E$. The circumcircle of $\triangle ACD$ intersects with $BE$ at a point $F$ outside $[BE]$. If $|AD| = 4, |BD|= 8$, then what is $|AF|$? $\textbf{(A)}\ \sqrt3 \qquad\textbf{(B)}\ 2\sqrt6 \qquad\textbf{(C)}\ 4\sqrt6 \qquad\textbf{(D)}\ \sqrt6 \qquad\textbf{(E)}\ \text{None}$

Let $AL$ be the angle bisector of the acute-angled triangle $ABC$. and $\omega$ be the circle circumscribed about it. Denote by $P$ the intersection point of the extension of the altitude $BH$ of the triangle $ABC$ with the circle $\omega$ . Prove that if $\angle BLA= \angle BAC$, then $BP = CP$.

In triangle $ABC$ angle bisectors $AA_{1}$ and $CC_{1}$ intersect at $I$. Line through $B$ parallel to $AC$ intersects rays $AA_{1}$ and $CC_{1}$ at points $A_{2}$ and $C_{2}$ respectively. Let $O_{a}$ and $O_{c}$ be the circumcenters of triangles $AC_{1}C_{2}$ and $CA_{1}A_{2}$ respectively. Prove that $\angle{O_{a}BO_{c}} = \angle{AIC} $

Let $C_1$ be a circle inside the circle $C_2$ and let $P$ in the interior of $C_1$, $Q$ in the exterior of $C_2$. One draws variable lines $l_i$ through $P$, not passing through $Q$. Let $l_i$ intersect $C_1$ in $A_i,B_i$, and let the circumcircle of $QA_iB_i$ intersect $C_2$ in $M_i,N_i$. Show that all lines $M_i,N_i$ are concurrent.

In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}\equal{}\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.

Let $M$ be an arbitrary point of a circle circumscribed around an acute-angled triangle $ABC$. Prove that the product of the distances from the point $M$ to the sides $AC$ and $BC$ is equal to the product of the distances from $M$ to the side $AB$ and to the tangent to the circumscribed circle at point $C$.

Let $ABCD$ be a cyclic quadrilateral. Suppose that $E$ is the intersection of $AB$ and $CD, F$ is the intersection of $AD$ and $CB, I$ is the intersection of $AC$ and $BD$. The circumcircles $(FAB), (FCD)$ meet $FI$ at $K, L$. Prove that $EK = EL$ Nguyễn Minh Hà

In triangle $ABC$, $AB = AC$. Point $D$ is the midpoint of side $BC$. Point $E$ lies outside the triangle $ABC$ such that $CE \perp AB$ and $BE = BD$. Let $M$ be the midpoint of segment $BE$. Point $F$ lies on the minor arc $\widehat{AD}$ of the circumcircle of triangle $ABD$ such that $MF \perp BE$. Prove that $ED \perp FD.$ [asy] defaultpen(fontsize(10)); size(6cm); pair A = (3,10), B = (0,0), C = (6,0), D = (3,0), E = intersectionpoints( Circle(B, 3), C--(C+100*dir(B--A)*dir(90)) )[1], M = midpoint(B--E), F = intersectionpoints(M--(M+50*dir(E--B)*dir(90)), circumcircle(A,B,D))[0]; dot(A^^B^^C^^D^^E^^M^^F); draw(B--C--A--B--E--D--F--M^^circumcircle(A,B,D)); pair point = extension(M,F,A,D); pair[] p={A,B,C,D,E,F,M}; string s = "A,B,C,D,E,F,M"; int size = p.length; real[] d; real[] mult; for(int i = 0; i<size; ++i) { d[i] = 0; mult[i] = 1;} d[4] = -50; string[] k= split(s,","); for(int i = 0;i<p.length;++i) { label("$"+k[i]+"$",p[i],mult[i]*dir(point--p[i])*dir(d[i])); }[/asy]

$ABCD$ is a cyclic quadrilateral in which $AB=3$, $BC=5$, $CD=6$, and $AD=10$. $M$, $I$, and $T$ are the feet of the perpendiculars from $D$ to lines $AB$, $AC$, and $BC$ respectively. Determine the value of $MI/IT$.

Let $\triangle ABC$ be a triangle with $BC = 4, CA= 5, AB= 6$, and let $O$ be the circumcenter of $\triangle ABC$. Let $O_b$ and $O_c$ be the reflections of $O$ about lines $CA$ and $AB$ respectively. Suppose $BO_b$ and $CO_c$ intersect at $T$, and let $M$ be the midpoint of $BC$. Given that $MT^2 = \frac{p}{q}$ for some coprime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Sreejato Bhattacharya[/i]

Let $ABC$ be a triangle with $(O), (I)$ being the circumcircle, and incircle respectively. Let $(I)$ touch $BC,CA$, and $AB$ at $A_0, B_0, C_0$ let $BC,CA$, and $AB$ intersect $B_0C_0, C_0A_0, A_0Bv$ at $A_1, B_1$, and $C_1$ respectively. Prove that $OI$ passes through the orthocenter of four triangles $A_0B_0C_0, A_0B_1C_1, B_0C_1A_1,C_0A_1B_1$. Nguyễn Minh Hà

Given a circle $\omega$ with radius $1$. Let $T$ be a set of triangles good, if the following conditions apply: (a) the circumcircle of each triangle in the set $T$ is $\omega$; (b) The interior of any two triangles in the set $T$ has no common point. Find all positive real numbers $t$, for which for each positive integer $n$ there is a good set of $n$ triangles, where the perimeter of each triangle is greater than $t$.

Let triangle ABC with $ AB=c$ $AC=b$ $BC=a$ $R$ circumradius, $p$ half peremetr of $ABC$. I f $\frac{acosA+bcosB+ccosC}{asinA+bsinB+csinC}=\frac{p}{9R}$ then find all value of $cosA$.

Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.

Let $ABC$ be a triangle such that $AC>AB.$ A circle tangent to the sides $AB$ and $AC$ at $D$ and $E$ respectively, intersects the circumcircle of $ABC$ at $K$ and $L$. Let $X$ and $Y$ be points on the sides $AB$ and $AC$ respectively, satisfying \[ \frac{AX}{AB}=\frac{CE}{BD+CE} \quad \text{and} \quad \frac{AY}{AC}=\frac{BD}{BD+CE} \] Show that the lines $XY, BC$ and $KL$ are concurrent.

Bisectors $AA_1$ and $BB_1$ of a right triangle $ABC \ (\angle C=90^\circ )$ meet at a point $I.$ Let $O$ be the circumcenter of triangle $CA_1B_1.$ Prove that $OI \perp AB.$

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles $AEF, BFD, CDE$ intersect lines $AI, BI, CI$, respectively, at points $K, L, M$ (different from $A, B, C$), respectively. Prove that $K, L, M, I$ are concyclic.

Vertex $B$ of the $\angle ABC$ lies out the circle, and the $[BA)$ and $[BC)$ beams intersect it. Point $K$ belongs to the intersection of the $[BA)$ beam and the circumference. Chord $KP$ is orthogonal to the angle bisector of $\angle ABC$ . Line $(KP)$ intersects the beam $BC$ in the point $M$. Prove that the segment $[PM]$ is twice as long as the distance from the circle centre to the angle bisector of $\angle ABC$ .

$(BEL 2) (a)$ Find the equations of regular hyperbolas passing through the points $A(\alpha, 0), B(\beta, 0),$ and $C(0, \gamma).$ $(b)$ Prove that all such hyperbolas pass through the orthocenter $H$ of the triangle $ABC.$ $(c)$ Find the locus of the centers of these hyperbolas. $(d)$ Check whether this locus coincides with the nine-point circle of the triangle $ABC.$