Školjka

%V0 Let triangle $ABC$ be such that its circumradius is $R = 1.$ Let $r$ be the inradius of $ABC$ and let $p$ be the inradius of the orthic triangle $A'B'C'$ of triangle $ABC.$ Prove that $$p \leq 1 - \frac{1}{3 \cdot (1+r)^2}.$$

%V0 Given a triangle $ABC$, let $D$ and $E$ be points on the side $BC$ such that $\angle BAD = \angle CAE$. If $M$ and $N$ are, respectively, the points of tangency of the incircles of the triangles $ABD$ and $ACE$ with the line $BC$, then show that $$\frac{1}{MB}+\frac{1}{MD}= \frac{1}{NC}+\frac{1}{NE}.$$

%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 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.

%V0 In the coordinate plane consider the set $S$ of all points with integer coordinates. For a positive integer $k$, two distinct points $a$, $B\in S$ will be called $k$-friends if there is a point $C\in S$ such that the area of the triangle $ABC$ is equal to $k$. A set $T\subset S$ will be called $k$-clique if every two points in $T$ are $k$-friends. Find the least positive integer $k$ for which there exits a $k$-clique with more than 200 elements. Proposed by Jorge Tipe, Peru

%V0 In an acute triangle $ABC$ segments $BE$ and $CF$ are altitudes. Two circles passing through the point $A$ anf $F$ and tangent to the line $BC$ at the points $P$ and $Q$ so that $B$ lies between $C$ and $Q$. Prove that lines $PE$ and $QF$ intersect on the circumcircle of triangle $AEF$. Proposed by Davood Vakili, Iran