Olympiad problems

1957 Putnam, A1

The normals to a surface all intersect a fixed straight line. Show that the surface is a portion of a surface of revolution.

Answer either (i) or (ii): (i) Let $a>0.$ Three straight lines pass through the three points $(0,-a,a), (a,0,-a)$ and $(-a,a,0),$ parallel to the $x-,y-$ and $z-$axis, respectively. A variable straight line moves so that it has one point in common with each of the three given lines. Find the equation of the surface described by the variable line. (II) Which planes cut the surface $xy+yz+xz=0$ in (1) circles, (2) parabolas?

2001 SNSB Admission, 4

Tags: differential geometry , surfaces

Let $ p,q $ be the two most distant points (in the Euclidean sense) of a closed surface $ M $ embedded in the Euclidean space. [b]a)[/b] Show that the tangent planes of $ M $ at $ p $ and $ q $ are parallel. [b]b)[/b] What happened if $ M $ would be a closed curve of $ \mathcal{C}^{\infty } \left(\mathbb{R}^3\right) $ class, instead?

1957 Putnam, A1

1949 Putnam, A1

2001 SNSB Admission, 4