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.
One corner square was cut from a chessboard. What is the smallest number of equal triangles that can be cut into this shape?
Every day, James bakes a square cake size \(3\times3\). Jack immediately cuts out for himself four square pieces of size \(1\times1\) with sides parallel to the sides of the cake (not necessarily along the \(3\times3\) grid lines). After that, Sarah cuts out from the rest of the cake a square piece with sides, also parallel to the sides of the cake. What is the largest piece of cake that Sarah can count on, regardless of Jack’s actions?
A unit square is divided into \(n\) triangles. Prove that one of the triangles can be used to completely cover a square with side length \(\frac{1}{n}\).
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.
A spherical sun is observed to have a finite number of circular sunspots, each of which covers less than half of the sun’s surface. These sunspots are said to be enclosed, that is no two sunspots can touch, and they do not overlap with one another. Prove that the sun will have two diametrically opposite points that are not covered by sunspots.