Olympiad problems

1986 Tournament Of Towns, (128) 3

Does there exist a set of $100$ triangles in which not one of the triangles can be covered by the other $99$?

1988 Tournament Of Towns, (172) 5

Is it possible to cover a plane with circles in such a way that exactly $1988$ circles pass through each point? ( N . Vasiliev)

circles combinatorics combinatorial geometry cover

1987 Tournament Of Towns, (145) 2

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

distance combinatorial geometry disc cover

2013 Argentina National Olympiad Level 2, 6

Is there a square with side lenght $\ell < 1$ that can completely cover any rectangle of diagonal $1$?

rectangle geometry square cover

1985 Tournament Of Towns, (095) 4

The convex set $F$ does not cover a semi-circle of radius $R$. Is it possible that two sets, congruent to $F$, cover the circle of radius $R$ ? What if $F$ is not convex? ( N . B . Vasiliev , A. G . Samosvat)

convex geometry circles combinatorial geometry cover number theory