#1.

   Show that for any five points chosen within an equilateral triangle with sides of length  2, there are two points whose distance apart is at most  1.

#2.

   In a group of 5 people, show that there are two who have the same number of acquaintances in the group. Generalize your result to show that in a group of  n people, there are two who have the same number of acquaintances.

#3.

   A chess master spends 11 weeks preparing for a tournament. He plays at least one game each day but no more than 12 games per week. Show that there is a string of days during which he will have played exactly 21 games. For this problem, consider a week to be the 7­day period from Monday through Sunday.

#4.

   Show that for any gathering of 6 people, there must be three in the group who are mutual friends or three who are mutual enemies.

#5.

   A 6-by-6 checkerboard is perfectly covered with 18 dominoes. Provide justification that for any such cover, it is possible to cut the cover either horizontally or vertically into two rectangles without cutting through a domino.