Olympiad problems

1983 Swedish Mathematical Competition, 5

Tags: geometric inequalities , min , geometry , disc , combinatorics , disks , combinatorial geometry

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: disc , covering , geometry , area

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 $S/9$.

2003 IMAR Test, 1

Tags: disc , regular , covering , pentagon , geometry

Prove that the interior of a convex pentagon whose sides are all equal, is not covered by the open disks having the sides of the pentagon as diameter.

1998 Tournament Of Towns, 2

Tags: combinatorial geometry , disc , perimeter , combinatorics , concurrency

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)

2018 Stars of Mathematics, 4

Tags: convex polygon , diameter , disc , convex , geometry , geometric inequality

Given an integer $n \ge 3$, prove that the diameter of a convex $n$-gon (interior and boundary) containing a disc of radius $r$ is (strictly) greater than $r(1 + 1/ \cos( \pi /n))$. The Editors

1984 All Soviet Union Mathematical Olympiad, 384

Tags: geometry , perimeter , polygon , disc , circles , area

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).

1941 Moscow Mathematical Olympiad, 087

Tags: combinatorial geometry , geometry , point , disc

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.

1949 Moscow Mathematical Olympiad, 159

Tags: geometry , disc , perimeter , combinatorial geometry

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

1999 Tournament Of Towns, 4

Tags: combinatorial geometry , combinatorics , diameter , disc , coloring

$n$ diameters divide a disk into $2n$ equal sectors. $n$ of the sectors are coloured blue , and the other $n$ are coloured red (in arbitrary order) . Blue sectors are numbered from $1$ to $n$ in the anticlockwise direction, starting from an arbitrary blue sector, and red sectors are numbered from $1$ to $n$ in the clockwise direction, starting from an arbitrary red sector. Prove that there is a semi-disk containing sectors with all numbers from $1$ to $n$. (V Proizvolov)

1987 Tournament Of Towns, (145) 2

Tags: combinatorial geometry , cover , disc , distance

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