Olympiad problems

1971 All Soviet Union Mathematical Olympiad, 155

$N$ unit squares on the infinite sheet of cross-lined paper are painted with black colour. Prove that you can cut out the finite number of square pieces and satisfy two conditions all the black squares are contained in those pieces the area of black squares is not less than $1/5$ and not greater than $4/5$ of every piece area.

1985 All Soviet Union Mathematical Olympiad, 416

Tags: combinatorial geometry , rectangle , unit square , area , geometry

Given big enough sheet of cross-lined paper with the side of the squares equal to $1$. We are allowed to cut it along the lines only. Prove that for every $m>12$ we can cut out a rectangle of the greater than $m$ area such, that it is impossible to cut out a rectangle of $m$ area from it.

1961 All Russian Mathematical Olympiad, 012

Tags: rectangle , unit square , combinatorics

Given $120$ unit squares arbitrarily situated in the $20\times 25$ rectangle. Prove that you can place a circle with the unit diameter without intersecting any of the squares.

1971 All Soviet Union Mathematical Olympiad, 147

Tags: circles , distance , combinatorial geometry , unit square , geometry

Given an unit square and some circles inside. Radius of each circle is less than $0.001$, and there is no couple of points belonging to the different circles with the distance between them $0.001$ exactly. Prove that the area, covered by the circles is not greater than $0.34$.

1987 All Soviet Union Mathematical Olympiad, 446

Tags: combinatorics , combinatorial geometry , unit square

An $L$ is an arrangement of $3$ adjacent unit squares formed by deleting one unit square from a $2 \times 2$ square. a) How many $L$s can be placed on an $8 \times 8$ board (with no interior points overlapping)? b) Show that if any one square is deleted from a $1987 \times 1987$ board, then the remaining squares can be covered with $L$s (with no interior points overlapping).

1965 All Russian Mathematical Olympiad, 064

Tags: combinatorial geometry , combinatorics , unit square

Is it possible to place $1965$ points in a square with side $1$ so that any rectangle of area $1/200$ with sides parallel to the sides of the square contains at least one of these points inside?