Školjka

%V0 Let $ABC$ be a triangle, and $I$ its incenter. Consider a circle which lies inside the circumcircle of triangle $ABC$ and touches it, and which also touches the sides $CA$ and $BC$ of triangle $ABC$ at the points $D$ and $E$, respectively. Show that the point $I$ is the midpoint of the segment $DE$.

%V0 For a triangle $T = ABC$ we take the point $X$ on the side $(AB)$ such that $AX/AB=4/5$, the point $Y$ on the segment $(CX)$ such that $CY = 2YX$ and, if possible, the point $Z$ on the ray ($CA$ such that $\widehat{CXZ} = 180 - \widehat{ABC}$. We denote by $\Sigma$ the set of all triangles $T$ for which $\widehat{XYZ} = 45$. Prove that all triangles from $\Sigma$ are similar and find the measure of their smallest angle.

%V0 Let $ABC$ be a triangle for which there exists an interior point $F$ such that $\angle AFB=\angle BFC=\angle CFA$. Let the lines $BF$ and $CF$ meet the sides $AC$ and $AB$ at $D$ and $E$ respectively. Prove that $$AB+AC\geq4DE.$$

%V0 Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.

%V0 Denote by $M$ midpoint of side $BC$ in an isosceles triangle $\triangle ABC$ with $AC = AB$. Take a point $X$ on a smaller arc $\widehat{MA}$ of circumcircle of triangle $\triangle ABM$. Denote by $T$ point inside of angle $BMA$ such that $\angle TMX = 90$ and $TX = BX$. Prove that $\angle MTB - \angle CTM$ does not depend on choice of $X$. Author: unknown author, Canada

%V0 Find the largest possible integer $k$, such that the following statement is true: Let $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain $$b_1 \leqslant b_2 \leqslant \cdots \leqslant b_{2009} \qquad \text{the lengths of the blue sides,}$$ $$r_1 \leqslant r_2 \leqslant \cdots \leqslant r_{2009} \qquad \text{the lengths of the red sides,}$$ $$w_1 \leqslant w_2 \leqslant \cdots \leqslant w_{2009} \qquad \text{the lengths of the white sides.}$$ Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_j$, $r_j$, $w_j$. Proposed by Michal Rolinek, Czech Republic