1978 Dutch Mathematical Olympiad

Part of Dutch Mathematical Olympiad

Subcontests

(4)

1

1978 points on plane

1978

points in the flat plane. Each point has a circular disk with that point as its center and the radius is the distance to a fixed point. Prove that there are five of these circular disks, which together cover all

1978

points (circular disk means: the circle and its inner area).

1

rectangles from lattice points

On the plane with a rectangular coordinate system, a set of infinitely many rectangles is given. Every rectangle has the origin as one of its vertices. The sides of all rectangles are parallel to the coordinate axes, and all sides have integer lengths. Prove that there are at least two rectangles in the set, one of which completely covers the other.

1

3x^2 = 9 + y^3

Prove that no integer

x

y

3x^2 = 9 + y^3.

1

square tiles on a x b floor

One tiles a floor of

a \times b

^2

with square tiles,

a,b \in N

. Tiles do not overlap, and sides of floor and tiles are parallel. Using tiles of

2\times 2

^2

leaves the same amount of floor uncovered as using tiles of

4\times 4

^2

3\times 3

^2

29

^2

floor uncovered. Determine

a

b