Find the greatest $n$ for which there exist $n$ lines in space, passing through a single point, such that any two of them form the same angle.

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.

Two circles $\Gamma_1$ and $\Gamma_2$are given with centres $O_1$ and $O_2$ and common exterior tangents $\ell_1$ and $\ell_2$. The line $\ell_1$ intersects $\Gamma_1$ in $A$ and $\Gamma_2$ in $B$. Let $X$ be a point on segment $O_1O_2$, not lying on $\Gamma_1$ or $\Gamma_2$. The segment $AX$ intersects $\Gamma_1$ in $Y \ne A$ and the segment $BX$ intersects $\Gamma_2$ in $Z \ne B$. Prove that the line through $Y$ tangent to $\Gamma_1$ and the line through $Z$ tangent to $\Gamma_2$ intersect each other on $\ell_2$.

Consider acute triangles $ABC$ and $BCD$, with $\angle BAC = \angle BDC$, such that $A$ and $D$ are on opposite sides of line $BC$. Denote by $E$ the foot of the perpendicular line to $AC$ through $B$ and by $F$ the foot of the perpendicular line to $BD$ through $C$. Let $H_1$ be the orthocenter of triangle $ABC$ and $H_2$ be the orthocenter of $BCD$. Show that lines $AD, EF$ and $H_1H_2$ are concurrent.

Let $ABCD$ be an inscribed quadrilateral. Let the circles with diameters $AB$ and $CD$ intersect at two points $X_1$ and $Y_1$, the circles with diameters $BC$ and $AD$ intersect at two points $X_2$ and $Y_2$, the circles with diameters $AC$ and $BD$ intersect at two points $X_3$ and $Y_3$. Prove that the lines $X_1Y_1, X_2Y_2$ and $X_3Y_3$ are concurrent. Maxim Didin

Let $ABCD$ be a convex quadrilateral. Prove that the circles inscribed in the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a common point if and only if $ABCD$ is a rhombus.

Given an isosceles triangle $ABC$ with base $BC$. Inside the side $BC$ is given a point $D$. Let $E, F$ be respectively points on the sides $AB, AC$ that $|\angle BED | = |\angle DF C| > 90^o$ . Prove that the circles circumscribed around the triangles $ABF$ and $AEC$ intersect on the line $AD$ at a point different from point $A$. (Patrik Bak, Michal Rolínek)

Let be given a circle with center $O$ and a point $P$ outside the circle. A line $l$ passes through $P$ and cuts the circle at $A$ and $B$. Let $C$ be the point symmetric to $A$ with respect to $OP$, and let $m$ be the line $BC$. Prove that all lines $m$ have a common point as $l$ varies.

In space there are given four distinct points $ A $, $ B $, $ C $, $ D $. Prove that the three segments connecting the midpoints of the segments $ AB $ and $ CD $, $ AC $ and $ BD $, $ AD $ and $ BC $ have a common midpoint.

Given is the triangle $ABC$, with $| AB | + | AC | = 3 \cdot | BC | $. Let's denote $D, E$ also points that $BCDA$ and $CBEA$ are parallelograms. On the sides $AC$ and $AB$ sides, $F$ and $G$ are selected respectively so that $| AF | = | AG | = | BC |$. Prove that the lines $DF$ and $EG$ intersect at the line segment $BC$

Let $ABC$ be a triangle with $D$ the midpoint of side $AB, E$ the midpoint of side $BC$, and $F$ the midpoint of side $AC$. Let $k_1$ be the circle passing through points $A, D$, and $F$, let $k_2$ be the circle passing through points $B, E$, and $D$, and let $k_3$ be the circle passing through $C, F$, and $E$. Prove that circles $k_1, k_2$, and $k_3$ intersect in a point.

The circle $k_a$ touches the extensions of sides $AB$ and $BC$, as well as the circumscribed circle of the triangle $ABC$ (from the outside). We denote the intersection of $k_a$ with the circumscribed circle of the triangle $ABC$ by $A'$. Analogously, we define points $B'$ and $C'$. Prove that the lines $AA',BB'$ and $CC'$ intersect in one point.

Let $O, I$ be the circumcenter and the incenter of triangle $ABC$, $P$ be an arbitrary point on segment $OI$, $P_A$, $P_B$, and $P_C$ be the second common points of lines $PA$, $PB$, and $PC$ with the circumcircle of triangle $ABC$. Prove that the bisectors of angles $BP_AC$, $CP_BA$, and $AP_CB$ concur at a point lying on $OI$.

We are given $n$ circles which have the same center. Two lines $D_1,D_2$ are concurent in $P$, a point inside all circles. The rays determined by $P$ on the line $D_i$ meet the circles in points $A_1,A_2,...,A_n$ and $A'_1, A'_2,..., A'_n$ respectively and the rays on $D_2$ meet the circles at points $B_1,B_2, ... ,B_n$ and $B'_2, B'_2 ..., B'_n$ (points with the same indices lie on the same circle). Prove that if the arcs $A_1B_1$ and $A_2B_2$ are equal then the arcs $A_iB_i$ and $A'_iB'_i$ are equal, for all $i = 1,2,... n$.

Let $ABC$ be a triangle inscribed in circle $\omega$ with center $O$, let $\omega_A$ be its $A$-mixtilinear incircle, $\omega_B$ be its $B$-mixtilinear incircle, $\omega_C$ be its $C$-mixtilinear incircle, and $X$ be the radical center of $\omega_A$, $\omega_B$, $\omega_C$. Let $A'$, $B'$, $C'$ be the points at which $\omega_A$, $\omega_B$, $\omega_C$ are tangent to $\omega$. Prove that $AA'$, $BB'$, $CC'$ and $OX$ are concurrent. [i]Proposed by Robin Park[/i]

As shown in the figure, two circles $\Gamma_1$ and $\Gamma_2$ on the plane intersect at two points $A$ and $B$. The two rays passing through $A$, $\ell_1$ and $\ell_2$ intersect $\Gamma_1$ at points $D$ and $E$ respectively, and $\Gamma_2$ at points $F$ and $C$ respectively (where $E$ and $F$ lie on line segments $AC$ and $AD$ respectively, and neither of them coincides with the endpoints). It is known that the three lines $AB$, $CF$ and $DE$ have a common point, the circumscribed circle of $\vartriangle AEF$ intersects $AB$ at point $G$, the straight line $EG$ intersects the circle $\Gamma_1$ at point $P$, the straight line $FG$ intersects the circle $\Gamma_2$ at point $Q$. Let the symmetric points of $C$ and $D$ wrt the straight line $AB$ be $C'$ and $D'$ respectively. If $PD'$ and $QC'$ intersect at point$ J$, prove that $J$ lies on the straight line $AB$. [img]https://cdn.artofproblemsolving.com/attachments/3/7/eb3acdbad52750a6879b4b6955dfdb7de19ed3.png[/img]

Let $ABC$ be a triangle in which the length of side $AB$ is $4$ units, and that of $BC$ is $2$ units. Let $D$ be the point on $AB$ at distance $3$ units from $A$. Prove that the line perpendicular to $AB$ through $D$, the angle bisector of $\angle ABC$, and the perpendicular bisector of $BC$ all meet at a single point.

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 $AB$.

In the triangle $ABC$ the altitude $BD$ and $CT$ are drawn, they intersect at the point $H$. The point $Q$ is the foot of the perpendicular drawn from the point $H$ on the bisector of the angle $A$. Prove that the bisector of the external angle $A$ of the triangle $ABC$, the bisector of the angle $BHC$ and the line $QM$, where $M$ is the midpoint of the segment $DT$, intersect at one point. (Matvsh Kursky)

Let $A_0A_1A_2$ be a triangle and let $P$ be a point in the plane, not situated on the circle $A_0A_1A_2$. The line $PA_k$ meets again the circle $A_0A_1A_2$ at point $B_k, k = 0, 1, 2$. A line $\ell$ through the point $P$ meets the line $A_{k+1}A_{k+2}$ at point $C_k, k = 0, 1, 2$. Show that the lines $B_kC_k, k = 0, 1, 2$, are concurrent and determine the locus of their concurrency point as the line $\ell$ turns about the point $P$.

Let $ABCD$ be a cyclic quadrilateral with two diagonals intersect at $E$. Let $ M$, $N$, $P$, $Q$ be the reflections of $ E $ in midpoints of $AB$, $BC$, $CD$, $DA$ respectively. Prove that the Euler lines of $ \triangle MAB$, $\triangle NBC$, $\triangle PCD,$ $\triangle QDA$ are concurrent. Trần Quang Hùng

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.

The incircle of the triangle $ABC$ touches the sides $BC, CA$ and $AB{}$ at $D,E$ and $F{}$ respectively. Let the circle $\omega$ touch the segments $CA{}$ and $AB{}$ at $Q{}$ and $R{}$ respectively, and the points $M{}$ and $N{}$ are selected on the segments $AB{}$ and $AC{}$ respectively, so that the segments $CM{}$ and $BN{}$ touch $\omega$. The bisectors of $\angle NBC$ and $\angle MCB$ intersect the segments $DE{}$ and $DF{}$ at $K{}$ and $L{}$ respectively. Prove that the lines $RK{}$ and $QL{}$ intersect on $\omega$. [i]Proposed by Tran Quang Hung[/i]

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.

Octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle $\Omega$ with center $O$. It is known that $A_1A_2\|A_5A_6$, $A_3A_4\|A_7A_8$ and $A_2A_3\|A_5A_8$. The circle $\omega_{12}$ passes through $A_1$, $A_2$ and touches $A_1A_6$; circle $\omega_{34}$ passes through $A_3$, $A_4$ and touches $A_3A_8$; the circle $\omega_{56}$ passes through $A_5$, $A_6$ and touches $A_5A_2$; the circle $\omega_{78}$ passes through $A_7$, $A_8$ and touches $A_7A_4$. The common external tangent to $\omega_{12}$ and $\omega_{34}$ cross the line passing through ${A_1A_6}\cap{A_3A_8}$ and ${A_5A_2}\cap{A_7A_4}$ at the point $X$. Prove that one of the common tangents to $\omega_{56}$ and $\omega_{78}$ passes through $X$.