JBMO-SL 2003

Is there is a convex quadrilateral which the diagonals divide into four triangles with areas of distinct primes?

Is there a triangle with $12 cm^2$ area and $12$ cm perimeter?

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 $GA \perp GC$ .

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 $ABC$ .

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 $[MN]$ .

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 $\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\ge \frac{3}{2}$