Junior Balkan Mathematical Olympiad 2005 [Shortlists & Solutions]

1. Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$. Prove that the lines $AP$ and $CS$ are parallel.
   
2. Let $ABCD$ be an isosceles trapezoid with $AB=AD=BC$, $AB||CD$, $AB>CD$. Let $E=AC\cap BD$ and $N$ symmetric to $B$ wrt $AC$. Prove that the quadrilateral $ANDE$ is cyclic.
3. Let $ABCDEF$ be a regular hexagon and $M\in (DE)$, $N\in (CD)$ such that $m(\widehat{AMN})={90}^{\circ }$ and $AN=CM\sqrt{2}$. Find the value of ${\displaystyle \frac{DM}{ME}}$.
4. Let $ABC$ be an isosceles triangle $(AB=AC)$ so that $\mathrm{\angle }A<2\mathrm{\angle }B$ . Let $D$, $Z$ points on the extension of height $AM$ so that $\mathrm{\angle }CBD=\mathrm{\angle }A$ and $\mathrm{\angle }ZBA={90}^{\circ }$. Let $E$ the orthogonal projection of $M$ on height $BF$, and let $K$ the orthogonal projection of $Z$ on $AE$. Prove that $$\mathrm{\angle }KDZ=\mathrm{\angle }KDB=\mathrm{\angle }KZB.$$
5. Let $O$ be the center of the concentric circles $C_1$, $C_2$ of radii $3$ and $5$ respectively. Let $A\in C_1$, $B\in C_2$ and $C$ point so that triangle $ABC$ is equilateral. Find the maximum length of $[OC]$.
6. Let $C_1$, $C_2$ be two circles intersecting at points $A$, $P$ with centers $O$, $K$ respectively. Let $B$, $C$ be the symmetric of $A$ wrt $O$, $K$ in circles $C_1$, $C_2$ respectively. A random line passing through $A$ intersects circles $C_1$, $C_2$ at $D$, $E$ respectively. Prove that the center of circumcircle of triangle $DEP$ lies on the circumcircle of triangle $OKP$.
7. Let $ABCD$ be a parallelogram, $P\in (CD)$, $Q\in (AB)$, $M=AP\cap DQ$, $N=BP\cap CQ$, $K=MN\cap AD$, $L=MN\cap BC$. Prove that $BL=DK$.

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