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 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

%V0 Regard a plane with a Cartesian coordinate system; for each point with integer coordinates, draw a circular disk centered at this point and having the radius $\frac{1}{1000}$. a) Prove the existence of an equilateral triangle whose vertices lie in the interior of different disks; b) Show that every equilateral triangle whose vertices lie in the interior of different disks has a sidelength 96. Radu Gologan, Romania Remark [The " 96" in (b) can be strengthened to " 124". By the way, part (a) of this problem is the place where I used the well-known "Dedekind" theorem.]

%V0 For any set $S$ of five points in the plane, no three of which are collinear, let $M(S)$ and $m(S)$ denote the greatest and smallest areas, respectively, of triangles determined by three points from $S$. What is the minimum possible value of $M(S)/m(S)$ ?

%V0 Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.

%V0 Let $ABC$ be an acute triangle. Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$, with $DA=DC, EA=EB$, and $FB=FC$, such that $$\angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad \angle CFB = 2 \angle ACB.$$ Let $D'$ be the intersection of lines $DB$ and $EF$, let $E'$ be the intersection of $EC$ and $DF$, and let $F'$ be the intersection of $FA$ and $DE$. Find, with proof, the value of the sum $$\frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}.$$