That third problem is just a variation on the famous 'four corner bug problem'. They even used this on the show 'NUMB3Rs'.
Google it and you will find lots. It is based on a logarithmic spiral.
In how many ways can the number 2008 be written as the sum of consecutive positive integers?
Arrange 5 red queens and 3 blue queens on a 5×5 board so that no queen attacks a queen of the other colour.
Consider 4 (dimensionless) flies, 2 males and 2 females. They are situated at the corners of 1 square meter. Every fly tries to reach the male/female fly in front of her/him.
Since the flies are flying towards another, they will meet each other at a certain time in the center of the square. How far does each fly fly?
Suppose 2008 can be written as a sum of n positive integers.In how many ways can the number 2008 be written as the sum of consecutive positive integers?
If n is odd then the number in the middle of the consecutive integers is 2008/n. Since this is an integer n|2008 => n=1 or 251. n=1 works but n=251 includes negative integers.
If n is even, the two middle numbers are 2008/n-0.5 and 2008/n+0.5, so 2(2008)/n is odd =>16|n. Let n = 16m. m|2008 since 16m|4016 and m is odd. m=1 or 251 but 16(251) is too big and will get negative integers so the only possible values for n are 16 and 1