Heres my solution for the first one
if 3 or more of the ai's are equal, we can easily form an equilateral triangle. so assume not more than 2 ai's to be equal.
divide the interval (1,13) as follows
(1,5] , (5,9) , [9,13)
using pigeon hole principle, atleast three of the ai's lie in the same interval which can be used to form a triangle.
third one seems an exercise on induction. please post your solutions people