Olympiad problems

1966 Vietnam National Olympiad, 2

$a, b$ are two fixed lines through $O$. Variable lines $x, y$ are parallel. $x$ intersects a at $A$ and $b$ at $C$, $y$ intersects $a$ at $B$ and $b$at $D$. The lines $AD$ and $BC$ meet at $M$. The line through $M$ parallel to $x$ meets $a$ at $L$ and $b$ at $N$. What can you say about $L, M, N$? Find the locus $M$.

Cono Sur Shortlist - geometry, 2005.G1

Tags: geometry , trigonometry , analytic geometry , geometry open

Construct triangle given all lenght of it altitudes. Please, do it elementary with Euclidian geometry (no trigonometry or coordinate geometry).

1991 Balkan MO, 2

Tags: geometry , number theory , relatively prime , geometry open

Show that there are infinitely many noncongruent triangles which satisfy the following conditions: i) the side lengths are relatively prime integers; ii)the area is an integer number; iii)the altitudes' lengths are not integer numbers.

1987 Balkan MO, 3

Tags: trigonometry , geometry open , geometry

In the triangle $ABC$ the following equality holds: \[\sin^{23}{\frac{A}{2}}\cos^{48}{\frac{B}{2}}=\sin^{23}{\frac{B}{2}}\cos^{48}{\frac{A}{2}}\] Determine the value of $\frac{AC}{BC}$.