Školjka

%V0 The incircle $\Omega$ of the acute-angled triangle $ABC$ is tangent to its side $BC$ at a point $K$. Let $AD$ be an altitude of triangle $ABC$, and let $M$ be the midpoint of the segment $AD$. If $N$ is the common point of the circle $\Omega$ and the line $KM$ (distinct from $K$), then prove that the incircle $\Omega$ and the circumcircle of triangle $BCN$ are tangent to each other at the point $N$.

%V0 For a given triangle $ABC$, let $X$ be a variable point on the line $BC$ such that $C$ lies between $B$ and $X$ and the incircles of the triangles $ABX$ and $ACX$ intersect at two distinct points $P$ and $Q.$ Prove that the line $PQ$ passes through a point independent of $X$. comment An extension by Darij Grinberg can be found here.

%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 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 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 Given an acute triangle $ABC$ with $\angle B > \angle C$. Point $I$ is the incenter, and $R$ the circumradius. Point $D$ is the foot of the altitude from vertex $A$. Point $K$ lies on line $AD$ such that $AK = 2R$, and $D$ separates $A$ and $K$. Lines $DI$ and $KI$ meet sides $AC$ and $BC$ at $E,F$ respectively. Let $IE = IF$. Prove that $\angle B\leq 3\angle C$. Author: Davoud Vakili, Iran