Problems

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.