Disjoint circles $ o_1, o_2$, with centers $ I_1, I_2$ respectively, are tangent to the line $ k$ at $ A_1, A_2$ respectively and they lie on the same side of this line. Point $ C$ lies on segment $ I_1I_2$ and $ \angle A_1CA_2 \equal{} 90^{\circ}$. Let $ B_1$ be the second intersection of $ A_1C$ with $ o_1$, and let $ B_2$ be the second intersection of $ A_2C$ with $ o_2$. Prove that $ B_1B_2$ is tangent to the circles $ o_1, o_2$.

The circle $ \Gamma $ is inscribed to the scalene triangle $ABC$. $ \Gamma $ is tangent to the sides $BC, CA$ and $AB$ at $D, E$ and $F$ respectively. The line $EF$ intersects the line $BC$ at $G$. The circle of diameter $GD$ intersects $ \Gamma $ in $R$ ($ R\neq D $). Let $P$, $Q$ ($ P\neq R , Q\neq R $) be the intersections of $ \Gamma $ with $BR$ and $CR$, respectively. The lines $BQ$ and $CP$ intersects at $X$. The circumcircle of $CDE$ meets $QR$ at $M$, and the circumcircle of $BDF$ meet $PR$ at $N$. Prove that $PM$, $QN$ and $RX$ are concurrent. [i]Author: Arnoldo Aguilar, El Salvador[/i]

Find the self-intersection index of the surface $x^4+y^4=1$ in the projective plane $\text{CP}^2$.

Let $ABCD$ be a square and let $\Gamma$ be the circumcircle of $ABCD$. $M$ is a point of $\Gamma$ belonging to the arc $CD$ which doesn't contain $A$. $P$ and $R$ are respectively the intersection points of $(AM)$ with $[BD]$ and $[CD]$, $Q$ and $S$ are respectively the intersection points of $(BM)$ with $[AC]$ and $[DC]$. Prove that $(PS)$ and $(QR)$ are perpendicular.

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

A half-circle of diameter $EF$ is placed on the side $BC$ of a triangle $ABC$ and it is tangent to the sides $AB$ and $AC$ in the points $Q$ and $P$ respectively. Prove that the intersection point $K$ between the lines $EP$ and $FQ$ lies on the altitude from $A$ of the triangle $ABC$. [i]Albania[/i]

Let $ABC$ be a triangle with circumcircle $w$. Let $D$ be the midpoint of arc $BC$ that contains $A$. Define $E$ and $F$ similarly. Let the incircle of $ABC$ touches $BC,CA,AB$ at $K,L,M$ respectively. Prove that $DK,EL,FM$ are concurrent.

Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$. Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.

A point $B$ lies on a chord $AC$ of circle $\omega.$ Segments $AB$ and $BC$ are diameters of circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$ respectively. These circles intersect $\omega$ for the second time in points $D$ and $E$ respectively. The rays $O_1D$ and $O_2E$ meet in a point $F,$ and the rays $AD$ and $CE$ do in a point $G.$ Prove that the line $FG$ passes through the midpoint of the segment $AC.$

Let $AB$ be a diameter of a circle $(O)$ and let $P$ be any point on the tangent drawn at $B$ to $(O).$ Define $AP\cap (O)=C\neq A,$ and let $D$ be the point diametrically opposite to $C.$ If $DP$ meets $(O)$ second time in $E,$ then, [b](i)[/b] Prove that $AE, BC, PO$ concur at $M.$ [b](ii)[/b] If $R$ is the radius of $(O),$ find $P$ such that the area of $\triangle AMB$ is maximum, and calculate the area in terms of $R.$

Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.

Let $ABCDEF$ be a non-degenerate cyclic hexagon with no two opposite sides parallel, and define $X=AB\cap DE$, $Y=BC\cap EF$, and $Z=CD\cap FA$. Prove that \[\frac{XY}{XZ}=\frac{BE}{AD}\frac{\sin |\angle{B}-\angle{E}|}{\sin |\angle{A}-\angle{D}|}.\][i]Proposed by Victor Wang[/i]

Let $A$ be point in the exterior of the circle $\mathcal C$. Two lines passing through $A$ intersect the circle $\mathcal C$ in points $B$ and $C$ (with $B$ between $A$ and $C$) respectively in $D$ and $E$ (with $D$ between $A$ and $E$). The parallel from $D$ to $BC$ intersects the second time the circle $\mathcal C$ in $F$. Let $G$ be the second point of intersection between the circle $\mathcal C$ and the line $AF$ and $M$ the point in which the lines $AB$ and $EG$ intersect. Prove that \[ \frac 1{AM} = \frac 1{AB} + \frac 1{AC}. \]

In a scalene triangle $ABC , AD, BE , CF$ are the angle bisectors $(D \in BC , E \in AC , F \in AB)$. Points $K_{a}, K_{b}, K_{c}$ on the incircle of triangle $ABC$ are such that $DK_{a}, EK_{b}, FK_{c}$ are tangent to the incircle and $K_{a}\not\in BC , K_{b}\not\in AC , K_{c}\not\in AB$. Let $A_{1}, B_{1}, C_{1}$ be the midpoints of sides $BC , CA, AB$ , respectively. Prove that the lines $A_{1}K_{a}, B_{1}K_{b}, C_{1}K_{c}$ intersect on the incircle of triangle $ABC$.

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]

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

Let be given two parallel lines $k$ and $l$, and a circle not intersecting $k$. Consider a variable point $A$ on the line $k$. The two tangents from this point $A$ to the circle intersect the line $l$ at $B$ and $C$. Let $m$ be the line through the point $A$ and the midpoint of the segment $BC$. Prove that all the lines $m$ (as $A$ varies) have a common point.

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Four segments divide a convex quadrilateral into nine quadrilaterals. The points of intersections of these segments lie on the diagonals of the quadrilateral (see figure). It is known that the quadrilaterals 1, 2, 3, 4 admit inscribed circles. Prove that the quadrilateral 5 also has an inscribed circle. [asy] pair A,B,C,D,E,F,G,H,I,J,K,L; A=(-4.0,4.0);B=(-1.06,4.34);C=(-0.02,4.46);D=(4.14,4.93);E=(3.81,0.85);F=(3.7,-0.42); G=(3.49,-3.05);H=(1.37,-2.88);I=(-1.46,-2.65);J=(-2.91,-2.52);K=(-3.14,-1.03);L=(-3.61,1.64); draw(A--D);draw(D--G);draw(G--J);draw(J--A); draw(A--G);draw(D--J); draw(B--I);draw(C--H);draw(E--L);draw(F--K); pair R,S,T,U,V; R=(-2.52,2.56);S=(1.91,2.58);T=(-0.63,-0.11);U=(-2.37,-1.94);V=(2.38,-2.06); label("1",R,N);label("2",S,N);label("3",T,N);label("4",U,N);label("5",V,N); [/asy] [i]Proposed by Nairi M. Sedrakyan, Armenia[/i]

Find the volume of the region of points $(x,y,z)$ such that \[\left(x^{2}+y^{2}+z^{2}+8\right)^{2}\le 36\left(x^{2}+y^{2}\right). \]

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.

Let $S_1$ and $S_2$ be two circumferences, with centers $O_1$ and $O_2$ respectively, and secants on $M$ and $N$. The line $t$ is the common tangent to $S_1$ and $S_2$ closer to $M$. The points $A$ and $B$ are the intersection points of $t$ with $S_1$ and $S_2$, $C$ is the point such that $BC$ is a diameter of $S_2$, and $D$ the intersection point of the line $O_1O_2$ with the perpendicular line to $AM$ through $B$. Show that $M$, $D$ and $C$ are collinear.

Given a triangle $ ABC$, let $ r$ be the external bisector of $ \angle ABC$. $ P$ and $ Q$ are the feet of the perpendiculars from $ A$ and $ C$ to $ r$. If $ CP \cap BA \equal{} M$ and $ AQ \cap BC\equal{}N$, show that $ MN$, $ r$ and $ AC$ concur.

The incircle of triangle $ABC$ touches $BC$, $CA$, $AB$ at points $A_1$, $B_1$, $C_1$, respectively. The perpendicular from the incenter $I$ to the median from vertex $C$ meets the line $A_1B_1$ in point $K$. Prove that $CK$ is parallel to $AB$.

Given an acute and scalene triangle $ ABC$, let $ H$ be its orthocenter, $ O$ its circumcenter, $ E$ and $ F$ the feet of the altitudes drawn from $ B$ and $ C$, respectively. Line $ AO$ intersects the circumcircle of the triangle again at point $ G$ and segments $ FE$ and $ BC$ at points $ X$ and $ Y$ respectively. Let $ Z$ be the point of intersection of line $ AH$ and the tangent line to the circumcircle at $ G$. Prove that $ HX$ is parallel to $ YZ$.