Leo’s grandma placed five empty plates on a square 1 metre\({}\times{}\)1 metre table for dinner. Show that some two of these plates were less than 75 cm apart.
Prove that, in a circle of radius 10, you cannot place 400 points so that the distance between each two points is greater than 1.
A square area of size \(100\times 100\) is covered in tiles of size \(1\times 1\) in 4 different colours – white, red, black, and grey. No two tiles of the same colour touch one another, that is share a side or a corner. How many red tiles can there be?
One corner square was cut from a chessboard. What is the smallest number of equal triangles that can be cut into this shape?
10 magazines lie on a coffee table, completely covering it. Prove that you can remove five of them so that the remaining magazines will cover at least half of the table.
In a square which has sides of length 1 there are 100 figures, the total area of which sums to more than 99. Prove that in the square there is a point which belongs to all of these figures.
On the planet Tau Ceti, the landmass takes up more than half the surface area. Prove that the Tau Cetians can drill a hole through the centre of their planet that connects land to land.
120 unit squares are placed inside a \(20 \times 25\) rectangle. Prove that it will always be possible to place a circle with diameter 1 inside the rectangle, without it overlapping with any of the unit squares.