There are eight points inside a circle of radius 1. Show that there are at least two points with distance between them less then 1.
Leo’s dad was making a pizza for lunch. He decided to place 7 pieces of pineapple on it. Assuming the pizza is a circle of a \(20\) cm radius, show that some two pieces of pineapple were placed closer than \(20\) cm apart.
Each point on a circle was painted red or green. Show that there is an isosceles triangle whose vertices are on the circumference of the circle, such that all three vertices are red or all three are green.
Anna has a garden shaped like an equilateral triangle of side \(8\) metres. She wants to plant \(17\) plants, but they need space – they need to be at least \(2\) metres apart in order for their roots to have access to all the microelements in the ground. Show that Anna’s garden is unfortunately too small.
12 straight lines passing through the origin are drawn on a plane. Prove that it is possible to choose two of these lines such that the angle between them is less than 17 degrees.
There are 7 points placed inside a regular hexagon of side length 1 unit. Prove that among the points there are two which are less than 1 unit apart.
A straight corridor of length 100 m is covered with 20 rugs that have a total length of 1 km. The width of each rug is equal to the width of the corridor. What is the longest possible total length of corridor that is not covered by a rug?
A plane contains \(n\) straight lines, of which no two are parallel. Prove that some of the angles will be smaller than \(180^\circ/n\).