# Somewhere on a sphere - math puzzle

What is the mean distance between two randomly choosen points within a sphere of radius 1?

# The walled garden - math puzzle

A garden is surrounded by four walls, which form a rectangular trapezoid. The parallel walls have lengths of 34 and 59. An old oak grows in the middle of the garden. The owner of the garden has figured out that the distance from the four walls to the oak are exactly the same. What is the area of the garden? |

# Risk tournament - math puzzle

The great Risk tournament is organized again. This year there are 1000 contestants. Each game in the tournament is played with 4 people. After each game, there will be one winner which goes to the next round. In the next round again groups of 4 players are formed. If one does not get an exact multiple of 4 in this way, some of the players in this round are selected randomly. These players can go to the following round automatically. In the end there will be one winner. How many games will be played?