Junior Balkan Mathematical Olympiad 2004 [Shortlists & Solutions]

1. Two circles $C_1$ and $C_2$ intersect in points $A$ and $B$. A circle $C$ with center in $A$ intersect $C_1$ in $M$ ​​and $P$ and $C_2$ in $N$ and $Q$ so that $N$ and $Q$ are located on different sides wrt $MP$, and $AB> AM$. Prove that $\angle MBQ = \angle NBP$.
   
2. Let $E$, $F$ be two distinct points inside a parallelogram $ABCD$. Determine the maximum possible number of triangles having the same area with three vertices from points $A$, $B$, $C$, $D$, $E$, $F$.
3. Let $ABC$ be a triangle inscribed in circle $C$. Circles $C_1$, $C_2$, $C_3$ are tangent internally with circle $C$ in $A_1$, $B_1$, $C_1$ and tangent to sides $[BC]$, $[CA]$, $[AB]$ in points $A_2$, $B_2$, $C_2$ respectively, so that $A$, $A_1$ are on one side of $BC$ and so on. Lines $A_1A_2$, $B_1B_2$ and $C_1C_2$ intersect the circle $C$ for second time at points $A’$, $B’$ and $C’$, respectively. Prove that if $ M = BB’ \cap CC’$ then $m (\angle MAA’) = 90^\circ$.
4. Let $ABC$ be a triangle with $m (\angle C) = 90^\circ$ and the points $D \in [AC]$, $E\in [BC]$. Inside the triangle we construct the semicircles $C_1$, $C_2$, $C_3$, $C_4$ of diameters $[AC]$, $[BC]$, $[CD]$, $[CE]$ and let $\{C, K\} = C_1 \cap C_2$, $\{C, M\} =C_3 \cap C_4$, $\{C, L\} = C_2 \cap C_3$, $\{C, N\} =C_1 \cap C_4$. Show that points $K$, $L$, $M$, $N$ are concyclic.
5. Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$.

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