## Junior Balkan Mathematical Olympiad Shortlist 2014

### Geometry

1. Let ${ABC}$ be a triangle with $m\left( \angle B \right)=m\left( \angle C \right)={{40}^{{}^\circ }}$ Line bisector of ${\angle{B}}$ intersects ${AC}$ at point ${D}$. Prove that $BD+DA=BC$.
2. Acute-angled triangle ${ABC}$ with ${AB<AC<BC}$ and let be ${c(O,R)}$ it’s circumicircle. Diametes ${BD}$ and ${CE}$ are drawn. Circle ${c_1(A,AE)}$ interescts ${AC}$ at ${K}$ Circle ${{c}_{2}(A,AD)}$ intersects ${BA}$ at ${L}$ (${A}$ lies between ${B}$ and ${L}$). Prove that lines $EK$ and $DL$ intersect at circle.
3. Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.
4. Let $ABC$ be an acute triangle such that $AB\not=AC$. Let $M$ be the midpoint $BC$, $H$ the orthocenter of $\triangle ABC$, $O_1$ the midpoint of $AH$ and $O_2$ the circumcenter of $\triangle BCH$. Prove that $O_1AMO_2$ is a parallelogram.
5. Let $ABC$ be a triangle with ${AB\ne BC}$ and let ${BD}$ be the internal bisector of $\angle ABC$ $\left( D\in AC \right)$. Denote by ${M}$ the midpoint of the arc ${AC}$ which contains point ${B}$. The circumscribed circle of the triangle $\triangle BDM$ intersects the segment ${AB}$ at point ${K\neq B}$. Let ${J}$ be the reflection of ${A}$ with respect to ${K}$. Prove that if ${DJ\cap AM=\left\{O\right\}}$ then the points $J$, $B$, $M$, $O$ belong to the same circle.
6. Let $ABCD$ be a quadrilateral whose diagonals are not perpendicular and whose sides $AB$ and $CD$ are not parallel.Let $O$ be the intersection of its diagonals. Denote with $H_1$ and $H_2$ the orthocenters of triangles $AOB$ and $COD$, respectively. If $M$ and $N$ are the midpoints of the segment lines $AB$ and $CD$, respectively. Prove that the lines $H_1H_2$ and $MN$ are parallel if and only if $AC=BD$.