At a certain convention participate 100
persons. It is known that each participant personally knows (and is known by) at least 67 others.
A certain lady would like to dine with other 3 at the same table, provided they know her and they know each other.
Prove that this is possible.
Advise how actually she can arrange it.
Using Ady's Hint:
Let us call the lady A. From the list of 100 names ERASE A and all those that don't know her. We are erasing at most 33 names, leaving at least 67.
Invite any participant on the list (call her B) and then erase B and all those that don't know B. We are erasing at most 33 names, leaving at least 34.
Invite any participant on the list (call her C) and then erase C and all those that don't know C. We are erasing at most 33 names, leaving at least 1.
Invite anyone remaining on the list. Thus, it is always possible, and this method is one way to accomplish it.
Solution (spoiler)
