Sequences subsequences and remainders

Question

Assignment:

Q1. How do I prove that in a group of n people there are two people with the same number of acquaintances within the group?  (Am thinking this is a pigeon hole problem)

Q2. Prove that given a sequence of twelve integers, a₁, a₂, …,a₁₂, there is a subsequence a_j, a_j+1, …, a_k where 12 divides ∑^k_n= a^a n.

Q3. A scrape of paper is found in an old desk that read:
   
72 turkeys $X67.9Y
The first and last digits of the price were smudged.  What are the two smudged digits?

Q4. Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by 3.

Provide complete and step by step solution for the question and show calculations and use formulas.