A convex quadrilateral $ABCD$ is inscribed into a circle $\omega$ . Suppose that there is a point $X$ on the segment $AC$ such that the $XB$ and $XD$ tangents to the circle $\omega$ . Tangent of $\omega$ at $C$, intersect $XD$ at $Q$. Let $E$ ($E\ne A$) be the intersection of the line $AQ$ with $\omega$ . Prove that $AD, BE$, and $CQ$ are concurrent.

Triangle $ABC$ has incircle $(I)$ and $P,Q$ are two points in the plane of the triangle. Let $QA,QB,QC$ meet $BA,CA,AB$ respectively at $D,E,F$. The tangent at $D$, other than $BC$, of the circle $(I)$ meets $PA$ at $X$. The points $Y$ and $Z$ are defined in the same manner. The tangent at $X$, other than $XD$, of the circle $(I)$ meets $ (I)$ at $U$. The points $V,W$ are defined in the same way. Prove that three lines $(AU,BV,CW)$ are concurrent. Tran Quang Hung, Dean of the Faculty of Science, Thanh Xuan, Hanoi.

$P$ and $Q$ are points on angle bisectors of two adjacent angles. Let $PA$, $PB$, $QC$ and $QD$ be altitudes on the sides of these adjacent angles. Prove that lines $AB$, $CD$ and $PQ$ are concurrent.

Triangle $ABC$ is inscribed into circle $k$. Points $A_1,B_1, C_1$ on its sides were marked, after this the triangle was erased. Prove that it can be restored uniquely if and only if $AA_1, BB_1$ and $CC_1$ concur.

Given that $ABC$ is a triangle, points $A_i, B_i, C_i \hspace{0.15cm} (i \in \{1,2,3\})$ and $O_A, O_B, O_C$ satisfy the following criteria: a) $ABB_1A_2, BCC_1B_2, CAA_1C_2$ are rectangles not containing any interior points of the triangle $ABC$, b) $\displaystyle \frac{AB}{BB_1} = \frac{BC}{CC_1} = \frac{CA}{AA_1}$, c) $AA_1A_3A_2, BB_1B_3B_2, CC_1C_3C_2$ are parallelograms, and d) $O_A$ is the centroid of rectangle $BCC_1B_2$, $O_B$ is the centroid of rectangle $CAA_1C_2$, and $O_C$ is the centroid of rectangle $ABB_1A_2$. Prove that $A_3O_A, B_3O_B,$ and $C_3O_C$ concur at a point. [i]Proposed by Farras Mohammad Hibban Faddila[/i]

Let $ABCDEF$ be a convex hexagon satisfying $AC = DF, CE = FB$ and $EA = BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.

Let $I_a$ be the excenter of triangle $ABC$ with respect to $A$. The line $AI_a$ intersects the circumcircle of triangle ABC at $T$. Let $X$ be a point on segment $TI_a$ such that $X I_a^2 = XA \cdot X T$ The perpendicular line from $X$ to $BC$ intersects $BC$ at $A'$. Define $B'$ and $C'$ in the same way. Prove that $AA',BB'$ and $CC'$ are concurrent.

The diagonals of the convex quadrilateral $ABCD$ intersect at the point $O$. The points $X$ and $Y$ are symmetrical to the point $O$ with respect to the midpoints of the sides $BC$ and $AD$, respectively. It is known that $AB = BC = CD$. Prove that the point of intersection of the perpendicular bisectors of the diagonals of the quadrilateral lies on the line $XY$.

Let $ABC$ be a triangle. Let $x_1$ and $x_2$ be two congruent circles, which touch each other and the segment $BC$, and which both lie within triangle $ABC$, and for which it also holds that $x_1$ touches the segment $CA$, and that $x_2$ is the segment $AB$. Let $X$ be the contact point of these two circles $x_1$ and $x_2$. Let $y_1$ and $y_2$ two congruent circles that touch each other and the segment $CA$, and both within of triangle $ABC$, and for which it also holds that $y_1$ touches the segment $AB$, and that $y_2$ the segment $BC$. Let $Y$ be the contact point of these two circles $y_1$ and $y_2$. Let $z_1$ and $z_2$ be two congruent circles that touch each other and the segment $AB$, and both within triangle $ABC$, and for which it also holds that $z_1$ touches the segment $BC$, and that $z_2$ the segment $CA$. Let $Z$ be the contact point of these two circles $z_1$ and $z_2$. Prove that the straight lines $AX, BY$ and $CZ$ intersect at a point.

Let $\omega$ be a circle touching two parallel lines $\ell_1, \ell_2$, $\omega_1$ a circle touching $\ell_1$ at $A$ and $\omega$ externally at $C$, and $\omega_2$ a circle touching $\ell_2$ at $B$, $\omega$ externally at $D$, and $\omega_1$ externally at $E$. Prove that $AD, BC$ intersect at the circumcenter of $\vartriangle CDE$.

In triangle $ABC$, the points $D$, $E$, and $F$ are the feet of the perpendiculars dropped from $A$, $B$, and $C$, respectively, onto the opposite sides. The point $X_A$ is such that a circle passing through $E$ and $F$ is tangent to the circumcircle of triangle $ABC$ at $X_A$, and $X_A$ is on a different side of $EF$ as $A$. Similarly, $X_B$ and $X_C$ are defined. Prove that the lines $AX_A$, $BX_B$, and $CX_C$ are concurrent.

Let $ABCD$ be a parallelogram such that $|AB| > |BC|$. Let $O$ be a point on the line $CD$ such that $|OB| = |OD|$. Let $\omega$ be a circle with center $O$ and radius $|OC|$. If $T$ is the second intersection of $\omega$ and $CD$, prove that $AT, BO$ and $\omega$ are concurrent. [i]Proposed by Ivan Novak[/i]

Let $ABC$ be a non isosceles triangle with circumcircle $(O)$ and incircle $(I)$. Denote $(O_1)$ as the circle that external tangent to $(O)$ at $A'$ and also tangent to the lines $AB,AC$ at $A_b,A_c$ respectively. Define the circles $(O_2), (O_3)$ and the points $B', C', B_c , B_a, C_a, C_b$ similarly. 1. Denote J as the radical center of $(O_1), (O_2), (O_3) $and suppose that $JA'$ intersects $(O_1)$ at the second point $X, JB'$ intersects $(O_2)$ at the second point Y , JC' intersects $(O_3)$ at the second point $Z$. Prove that the circle $(X Y Z)$ is tangent to $(O_1), (O_2), (O_3)$. 2. Prove that $AA', BB', CC'$ are concurrent at the point $M$ and $3$ points $I,M,O$ are collinear.

Let $ABC$ be an acute angles triangle with $O$ the center of the circumscribed circle. Point $Q$ lies on the circumscribed circle of $\vartriangle BOC$ so that $OQ$ is a diameter. Point $M$ lies on $CQ$ and point $N$ lies internally on line segment $BC$ so that $ANCM$ is a parallelogram. Prove that the circumscribed circle of $\vartriangle BOC$ and the lines $AQ$ and $NM$ pass through the same point.

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 $\vartriangle A_1B_1C_1$.

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

Let $c(O, R)$ be a circle, $AB$ a diameter and $C$ an arbitrary point on the circle different than $A$ and $B$ such that $\angle AOC > 90^o$. On the radius $OC$ we consider point $K$ and the circle $c_1(K, KC)$. The extension of the segment $KB$ meets the circle $(c)$ at point $E$. From $E$ we consider the tangents $ES$ and $ET$ to the circle $(c_1)$. Prove that the lines $BE, ST$ and $AC$ are concurrent.

In an isosceles triangle $ABC$ ($AB = BC$), the midline parallel to side $BC$ intersects the incircle at a point $F$ that does not lie on the base $AC$. Prove that the tangent to the circle at point $F$ intersects the bisector of angle $C$ on side $AB$.

Prove that in a convex quadrilateral inscribed in a circle, straight lines passing through the midpoints of the sides and perpendicular to the opposite sides intersect at one point.

Let $O$ be the circumcenter of an acute triangle $ABC$ and $A_1$ a point of the minor arc $BC$ of the circle $ABC$ . Let $A_2$ and $A_3$ be points on sides $AB$ and $AC$ respectively such that $\angle BA_1A_2=\angle OAC$ and $\angle CA_1A_3=\angle OAB$ . Points $B_2, B_3, C_2$ and $C_3$ are similarly constructed, with $B_2$ in $BC, B_3$ in $AB, C_2$ in $AC$ and $C_3$ in $BC$. Prove that lines $A_2A_3, B_2B_3$ and $C_2C_3$ are concurrent.

The incircle $(I)$ of a triangle $ABC$ touches $BC,CA,AB$ at $D,E, F$. Let $ID, IE, IF$ intersect $EF, FD,DE$ at $X,Y,Z$, respectively. The lines $\ell_a, \ell_b, \ell_c$ through $A,B,C$ respectively and are perpendicular to $YZ,ZX,XY$ . Prove that $\ell_a, \ell_b, \ell_c$ are concurrent at a point that is on the line segment joining $I$ and the centroid of triangle $ABC$ . Nguyễn Minh Hà

A triangle $ABC$ is given. Let $M$ be the midpoint of the side $AC$ of the triangle and $Z$ the image of point $B$ along the line $BM$. The circle with center $M$ and radius $MB$ intersects the lines $BA$ and $BC$ at the points $E$ and $G$ respectively. Let $H$ be the point of intersection of $EG$ with the line $AC$, and $K$ the point of intersection of $HZ$ with the line $EB$. The perpendicular from point $K$ to the line $BH$ intersects the lines $BZ$ and $BH$ at the points $L$ and $N$, respectively. If $P$ is the second point of intersection of the circumscribed circles of the triangles $KZL$ and $BLN$, prove that, the lines $BZ, KN$ and $HP$ intersect at a common point.

Given $5$ circumferences, every four of them have a common point. Prove that there exists a point that belongs to all five circumferences.

Let $ABC$ be a right triangle with $\angle ACB = 90^o$ and M the center of $AB$. Let $G$ br any point on the line $MC$ and $P$ a point on the line $AG$, such that $\angle CPA = \angle BAC$ . Further let $Q$ be a point on the straight line $BG$, such that $\angle BQC = \angle CBA$ . Show that the circles of the triangles $AQG$ and $BPG$ intersect on the segment $AB$.

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.