Školjka

%V0 Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC.$ Show that there exist points $D, E,$ and $F$ on sides $BC,CA,$ and $AB$ respectively such that $$OD + DH = OE +EH = OF +FH$$ and the lines $AD, BE,$ and $CF$ are concurrent.

%V0 Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$.

%V0 Let $M$ be a point in the interior of triangle $ABC$. Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$. Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define $$p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}.$$ Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?

%V0 Let $ABC$ be a triangle, and $P$ a point in the interior of this triangle. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Assume that $AP^{2}+PD^{2}=BP^{2}+PE^{2}=CP^{2}+PF^{2}$. Furthermore, let $I_{a}$, $I_{b}$, $I_{c}$ be the excenters of triangle $ABC$. Show that the point $P$ is the circumcenter of triangle $I_{a}I_{b}I_{c}$.

%V0 Let $ABC$ be an acute-angled triangle such that $\angle ABC<\angle ACB$, let $O$ be the circumcenter of triangle $ABC$, and let $D=AO\cap BC$. Denote by $E$ and $F$ the circumcenters of triangles $ABD$ and $ACD$, respectively. Let $G$ be a point on the extension of the segment $AB$ beyound $A$ such that $AG=AC$, and let $H$ be a point on the extension of the segment $AC$ beyound $A$ such that $AH=AB$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if $\angle ACB-\angle ABC=60^\circ$. comment Official version Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. Edited by orl.

%V0 Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$.