Olympiad problems

1949 Putnam, B3

Tags: Putnam , curves

Let $K$ be a closed plane curve such that the distance between any two points of $K$ is always less than $1.$ Show that $K$ lies in a circle of radius $\frac{1}{\sqrt{3}}.$

2004 Miklós Schweitzer, 9

Tags: college contests , Miklos Schweitzer , analytic geometry , curves

Let $F$ be a smooth (i.e. $C^{\infty}$) closed surface. Call a continuous map $f\colon F\rightarrow \mathbb{R}^2$ an [i]almost-immersion[/i] if there exists a smooth closed embedded curve $\gamma$ (possibly disconnected) in $F$ such that $f$ is smooth and of maximal rank (i.e., rank 2) on $F\backslash \gamma$ and each point $p\in\gamma$ admits local coordinate charts $(x,y)$ and $(u,v)$ about $p$ and $f(p)$, respectively, such taht the coordinates of $p$ and $f(p)$ are zero and the map $f$ is given by $(x,y)\rightarrow (u,v), u=|x|, v=y$. Determine the genera of those smooth, closed, connected, orientable surfaces $F$ that admit an almost-immersion in the plane with the curve $\gamma$ having a given positive number $n$ of connected components.

1969 Putnam, B4

Tags: Putnam , curves

Show that any curve of unit length can be covered by a closed rectangle of area $1 \slash 4$.