1. Consider a scenario in which 10 pigeons roost in 11 holes. Is it necessarily true that there

exists a hole containing two or more pigeons? Justify your answer.

2. Prove the following: in a set of 3 integers, either at least two of them are odd or at least two

of them are even.

3. There are 365 days in a year. Consider a set S of 2000 people. For each day d = {1,..., 365},

let na denote the number of people in S whose birthday lies on day d. Let d, denote the day

such that n, is maximized, over all possible days (i.e., it's the day that is the birthday of

the largest number of people in S).

What is the minimum possible value for d.? Justify your answer.