Junior Balkan Mathematical Olympiad 2003 [Shortlists & Solutions]

1. Is there is a convex quadrilateral which the diagonals divide into four triangles with areas of distinct primes?
2. Is there a triangle with $12 \, cm^2$ area and $12$ cm perimeter?
3. Let $G$ be the centroid of triangle $ABC$, and $A'$ the symmetric of $A$ wrt $C$. Show that $G$, $B$, $C$, $A'$ are concyclic if and only if $GA \perp GC$.
   
4. Three equal circles have a common point $M$ and intersect in pairs at points $A$, $B$, $C$. Prove that that $M$ is the orthocenter of triangle $ABC$.
5. Let $ABC$ be an isosceles triangle with $AB = AC$. A semi-circle of diameter $[EF] $ with $E, F \in [BC]$, is tangent to the sides $AB$, $AC$ in $M$, $N$ respectively and $AE$ intersects the semicircle at $P$. Prove that $PF$ passes through the midpoint of $[MN]$.
6. Parallels to the sides of a triangle passing through an interior point divide the inside of a triangle into $6$ parts with the marked areas as in the figure. Show that $$\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\ge \frac{3}{2}$$
7. Let $D$, $E$, $F$ be the midpoints of the arcs $BC$, $CA$, $AB$ on the circumcircle of a triangle $ABC$ not containing the points $A$, $B$, $C$, respectively. Let the line $DE$ meets $BC$ and $CA$ at $G$ and $H$, and let $M$ be the midpoint of the segment $GH$. Let the line $FD$ meet $BC$ and $AB$ at $K$ and $J$, and let $N$ be the midpoint of the segment $KJ$.
   a) Find the angles of triangle $DMN$;
   b) Prove that if $P$ is the point of intersection of the lines $AD$ and $EF$, then the circumcenter of triangle $DMN$ lies on the circumcircle of triangle $PMN$.

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