Pigeonhole Principle Questions; please answer ASAP (dont need to answer all at once) 1. Given twelve integers, show that two of them can be chosen whose difference is divisible by 11

2. Given 11 different natural numbers, none greater than 20. Prove that two of these can be chosen, one of which divides the other.

3. Show that in any group of five people, there are two who have an identical number of friends within the group.

4. For a group of n people, show that at least two of them have same number of friends within the group.

5. The digits 1, 2, . . . , 9 are divided into three groups. Prove that the product of the numbers in one of the groups must exceed 71.

6. Random pick 6 points from a square with sides with a length of 1. Show that you can always find 2 points from these 6 that their distance is less or equal to

7. Randomly pick 5 points from a sphere. Show that you can always find a closed semi-sphere ( half a sphere and boundary) that contains 4 points. Please show the work.