Olympiad problems

1951 Miklós Schweitzer, 15

Let the line $ z\equal{}x, \, y\equal{}0$ rotate at a constant speed about the $ z$-axis; let at the same time the point of intersection of this line with the $ z$-axis be displaced along the $ z$-axis at constant speed. (a) Determine that surface of rotation upon which the resulting helical surface can be developed (i.e. isometrically mapped). (b) Find those lines of the surface of rotation into which the axis and the generators of the helical surface will be mapped by this development.

1980 Miklós Schweitzer, 9

Tags: geometry , perimeter , advanced fields

Let us divide by straight lines a quadrangle of unit area into $ n$ subpolygons and draw a circle into each subpolygon. Show that the sum of the perimeters of the circles is at most $ \pi \sqrt{n}$ (the lines are not allowed to cut the interior of a subpolygon). [i]G. and L. Fejes-Toth[/i]

1972 Miklós Schweitzer, 2

Tags: advanced fields

Let $ \leq$ be a reflexive, antisymmetric relation on a finite set $ A$. Show that this relation can be extended to an appropriate finite superset $ B$ of $ A$ such that $ \leq$ on $ B$ remains reflexive, antisymmetric, and any two elements of $ B$ have a least upper bound as well as a greatest lower bound. (The relation $ \leq$ is extended to $ B$ if for $ x,y \in A , x \leq y$ holds in $ A$ if and only if it holds in $ B$.) [i]E. Freid[/i]

1979 Miklós Schweitzer, 6

Tags: analytic geometry , conic , advanced fields

Let us defined a pseudo-Riemannian metric on the set of points of the Euclidean space $ \mathbb{E}^3$ not lying on the $ z$-axis by the metric tensor \[ \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \minus{}\sqrt{x^2\plus{}y^2} \\ \end{array} \right),\] where $ (x,y,z)$ is a Cartesian coordinate system $ \mathbb{E}^3$. Show that the orthogonal projections of the geodesic curves of this Riemannian space onto the $ (x,y)$-plane are straight lines or conic sections with focus at the origin [i]P. Nagy[/i]