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.
A carpet has a square shape with side 275 cm. A moth has eaten 4 holes through it. Will it always be possible to cut a square section of side 1 m out of the carpet, so that the section does not contain any holes? Treat the holes as points.
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 \(\frac{1}{n}\).
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.