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IMO Shortlist 2007 problem G5

Let $ABC$ be a fixed triangle, and let $A_1$ , $B_1$ , $C_1$ be the midpoints of sides $BC$ , $CA$ , $AB$ , respectively. Let $P$ be a variable point on the circumcircle. Let lines $PA_1$ , $PB_1$ , $PC_1$ meet the circumcircle again at $A'$ , $B'$ , $C'$ , respectively. Assume that the points $A$ , $B$ , $C$ , $A'$ , $B'$ , $C'$ are distinct, and lines $AA'$ , $BB'$ , $CC'$ form a triangle. Prove that the area of this triangle does not depend on $P$ .

Author: Christopher Bradley, United Kingdom

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IMO Shortlist 2005 problem G5

Let $\triangle ABC$ be an acute-angled triangle with $AB \not= AC$ . Let $H$ be the orthocenter of triangle $ABC$ , and let $M$ be the midpoint of the side $BC$ . Let $D$ be a point on the side $AB$ and $E$ a point on the side $AC$ such that $AE=AD$ and the points $D$ , $H$ , $E$ are on the same line. Prove that the line $HM$ is perpendicular to the common chord of the circumscribed circles of triangle $\triangle ABC$ and triangle $\triangle ADE$ .

IMO Shortlist 2005 problem G6

Let $ABC$ be a triangle, and $M$ the midpoint of its side $BC$ . Let $\gamma$ be the incircle of triangle $ABC$ . The median $AM$ of triangle $ABC$ intersects the incircle $\gamma$ at two points $K$ and $L$ . Let the lines passing through $K$ and $L$ , parallel to $BC$ , intersect the incircle $\gamma$ again in two points $X$ and $Y$ . Let the lines $AX$ and $AY$ intersect $BC$ again at the points $P$ and $Q$ . Prove that $BP = CQ$ .

IMO Shortlist 2006 problem G5

In triangle $ABC$ , let $J$ be the center of the excircle tangent to side $BC$ at $A_{1}$ and to the extensions of the sides $AC$ and $AB$ at $B_{1}$ and $C_{1}$ respectively. Suppose that the lines $A_{1}B_{1}$ and $AB$ are perpendicular and intersect at $D$ . Let $E$ be the foot of the perpendicular from $C_{1}$ to line $DJ$ . Determine the angles $\angle{BEA_{1}}$ and $\angle{AEB_{1}}$ .

IMO Shortlist 2007 problem G6

Determine the smallest positive real number $k$ with the following property. Let $ABCD$ be a convex quadrilateral, and let points $A_1$ , $B_1$ , $C_1$ , and $D_1$ lie on sides $AB$ , $BC$ , $CD$ , and $DA$ , respectively. Consider the areas of triangles $AA_1D_1$ , $BB_1A_1$ , $CC_1B_1$ and $DD_1C_1$ ; let $S$ be the sum of the two smallest ones, and let $S_1$ be the area of quadrilateral $A_1B_1C_1D_1$ . Then we always have $kS_1\ge S$ .

Author: unknown author, USA

IMO Shortlist 2007 problem G7

Given an acute triangle $ABC$ with $\angle B > \angle C$ . Point $I$ is the incenter, and $R$ the circumradius. Point $D$ is the foot of the altitude from vertex $A$ . Point $K$ lies on line $AD$ such that $AK = 2R$ , and $D$ separates $A$ and $K$ . Lines $DI$ and $KI$ meet sides $AC$ and $BC$ at $E,F$ respectively. Let $IE = IF$ .

Prove that $\angle B\leq 3\angle C$ .

Author: Davoud Vakili, Iran

IMO Shortlist 2007 problem G8

Point $P$ lies on side $AB$ of a convex quadrilateral $ABCD$ . Let $\omega$ be the incircle of triangle $CPD$ , and let $I$ be its incenter. Suppose that $\omega$ is tangent to the incircles of triangles $APD$ and $BPC$ at points $K$ and $L$ , respectively. Let lines $AC$ and $BD$ meet at $E$ , and let lines $AK$ and $BL$ meet at $F$ . Prove that points $E$ , $I$ , and $F$ are collinear.

Author: Waldemar Pompe, Poland