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 Let $ABCD$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $AE:EB=CF:FD$. Let $P$ be the point on the segment $EF$ such that $PE:PF=AB:CD$. Prove that the ratio between the areas of triangles $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.

%V0 Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.

%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