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IMO Shortlist 2000 problem G2

Two circles $G_1$ and $G_2$ intersect at two points $M$ and $N$ . Let $AB$ be the line tangent to these circles at $A$ and $B$ , respectively, so that $M$ lies closer to $AB$ than $N$ . Let $CD$ be the line parallel to $AB$ and passing through the point $M$ , with $C$ on $G_1$ and $D$ on $G_2$ . Lines $AC$ and $BD$ meet at $E$ ; lines $AN$ and $CD$ meet at $P$ ; lines $BN$ and $CD$ meet at $Q$ . Show that $EP = EQ$ .

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IMO Shortlist 2008 problem G1

Let $H$ be the orthocenter of an acute-angled triangle $ABC$ . The circle $\Gamma_{A}$ centered at the midpoint of $BC$ and passing through $H$ intersects the sideline $BC$ at points $A_{1}$ and $A_{2}$ . Similarly, define the points $B_{1}$ , $B_{2}$ , $C_{1}$ and $C_{2}$ .

Prove that six points $A_{1}$ , $A_{2}$ , $B_{1}$ , $B_{2}$ , $C_{1}$ and $C_{2}$ are concyclic.

Author: Andrey Gavrilyuk, Russia

IMO Shortlist 2006 problem G1

Let $ABC$ be triangle with incenter $I$ . A point $P$ in the interior of the triangle satisfies $\angle PBA+\angle PCA = \angle PBC+\angle PCB.$ Show that $AP \geq AI$ , and that equality holds if and only if $P=I$ .

IMO Shortlist 2004 problem G1

1. Let $ABC$ be an acute-angled triangle with $AB\neq AC$ . The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ respectively. Denote by $O$ the midpoint of the side $BC$ . The bisectors of the angles $\angle BAC$ and $\angle MON$ intersect at $R$ . Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the side $BC$ .

IMO Shortlist 2001 problem G2

Consider an acute-angled triangle $ABC$ . Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$ , and let $O$ be the circumcenter of triangle $ABC$ . Assume that $\angle C \geq \angle B+30^{\circ}$ . Prove that $\angle A+\angle COP < 90^{\circ}$ .

IMO Shortlist 2000 problem G1

In the plane we are given two circles intersecting at $X$ and $Y$ . Prove that there exist four points with the following property:

(P) For every circle touching the two given circles at $A$ and $B$ , and meeting the line $XY$ at $C$ and $D$ , each of the lines $AC$ , $AD$ , $BC$ , $BD$ passes through one of these points.

IMO Shortlist 1995 problem G1

Let $A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$ . The line $XY$ meets $BC$ at $Z$ . Let $P$ be a point on the line $XY$ other than $Z$ . The line $CP$ intersects the circle with diameter $AC$ at $C$ and $M$ , and the line $BP$ intersects the circle with diameter $BD$ at $B$ and $N$ . Prove that the lines $AM,DN,XY$ are concurrent.