Olympiad problems

1949 Moscow Mathematical Olympiad, 159

Consider a closed broken line of perimeter $1$ on a plane. Prove that a disc of radius $\frac14$ can cover this line.

1941 Moscow Mathematical Olympiad, 087

Tags: points , discs , combinatorial geometry , geometry

On a plane, several points are chosen so that a disc of radius $1$ can cover every $3$ of them. Prove that a disc of radius $1$ can cover all the points.

1998 Tournament Of Towns, 2

Tags: combinatorial geometry , combinatorics , discs , perimeter , concurrent

On the plane are $n$ paper disks of radius $1$ whose boundaries all pass through a certain point, which lies inside the region covered by the disks. Find the perimeter of this region. (P Kozhevnikov)

1984 All Soviet Union Mathematical Olympiad, 384

Tags: areas , polygon , perimeter , circle , discs , geometry

The centre of the coin with radius $r$ is moved along some polygon with the perimeter $P$, that is circumscribed around the circle with radius $R$ ($R>r$). Find the coin trace area (a sort of polygon ring).

1987 Tournament Of Towns, (145) 2

Tags: Cover , discs , disc , combinatorial geometry , distance

Α disk of radius $1$ is covered by seven identical disks. Prove that their radii are not less than $\frac12$ .

1983 Swedish Mathematical Competition, 5

Tags: geometry , combinatorial geometry , combinatorics , disks , discs , Geometric Inequalities , min

Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$. What is the smallest possible radius?

1983 Austrian-Polish Competition, 3

Tags: area , geometry , discs , covering

A bounded planar region of area $S$ is covered by a finite family $F$ of closed discs. Prove that $F$ contains a subfamily consisting of pairwise disjoint discs, of joint area not less than $S/9$.