Školjka

%V0 Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$) intersect at point $P$. Points $O_1$ and $O_2$ are circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$, respectively. Prove that the perpendicular from $E_1$ on $CD$, the perpendicular from $E_2$ on $AB$ and the lines $H_1H_2$ are concurrent. Proposed by Ukraine

%V0 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

%V0 In a triangle $ABC$, let $M_{a}$, $M_{b}$, $M_{c}$ be the midpoints of the sides $BC$, $CA$, $AB$, respectively, and $T_{a}$, $T_{b}$, $T_{c}$ be the midpoints of the arcs $BC$, $CA$, $AB$ of the circumcircle of $ABC$, not containing the vertices $A$, $B$, $C$, respectively. For $i \in \left\{a, b, c\right\}$, let $w_{i}$ be the circle with $M_{i}T_{i}$ as diameter. Let $p_{i}$ be the common external common tangent to the circles $w_{j}$ and $w_{k}$ (for all $\left\{i, j, k\right\}= \left\{a, b, c\right\}$) such that $w_{i}$ lies on the opposite side of $p_{i}$ than $w_{j}$ and $w_{k}$ do. Prove that the lines $p_{a}$, $p_{b}$, $p_{c}$ form a triangle similar to $ABC$ and find the ratio of similitude.

%V0 Circles $w_{1}$ and $w_{2}$ with centres $O_{1}$ and $O_{2}$ are externally tangent at point $D$ and internally tangent to a circle $w$ at points $E$ and $F$ respectively. Line $t$ is the common tangent of $w_{1}$ and $w_{2}$ at $D$. Let $AB$ be the diameter of $w$ perpendicular to $t$, so that $A, E, O_{1}$ are on the same side of $t$. Prove that lines $AO_{1}$, $BO_{2}$, $EF$ and $t$ are concurrent.

%V0 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 $\triangle ADE$.

%V0 Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$. comment (According to my team leader, last year some of the countries wanted a geometry question that was even easier than this...that explains IMO 2003/4...) [Note by Darij: This was also Problem 6 of the German pre-TST 2004, written in December 03.] Edited by Orl.