Each digit, 1, 2, 3, 4, 5, 6, 7, 8 and 9 is represented by a different letter A, B, C, D, E, F, G, H and I but not necessarily in this order. Further, each of A + B + C, C + D + E, E + F + G and G + H + I is equal to 13.
Determine the value of each of the alphabets such that the above conditions are satisfied.

A survey is conducted in a school among 150 students and it was found that 100 students play cricket, 90 students play football and 80 students play hockey. As physical education was given emphasis in the school, every student had to play at least one sport.
What can be the maximum number of students who play exactly two sports?
What can be the maximum number of students who play exactly one sport?

Prove or Disprove:

There exists some power of 3 which ends in 001.

How many 5 digit numbers are there such that the sum of its digits is odd?
Example: 12345 is a five digit number such that the sum of its digits = 1 + 2 + 3 + 4 + 5 = 15 = an odd number.

Prove that:
"In any group of 6 people, there are either 3 mutual friends or 3 mutual strangers".

Mr. A, Mr. B, Mr. C and Mr. D are four men who are Physicist, Mathematician, Doctor and Policeman, though not necessarily in that order. Mr. A and Mr. B are neighbours and they go together to their work by a bike. Mr. B makes more money than Mr. C. Mr. A meets Mr. D regularly in chess playing. The Physicist always goes by car. The only time the Doctor met the Policeman ever was when the Policeman arrested the Doctor for speeding. The policeman makes more money than the Mathematician. The Doctor makes more money than Mr. D

Who is the Physicist?
(a) Mr. A
(b) Mr. C
(c) Mr. D
(d) Mr. B
(e) Cannot be determined

Who is the second highest earner?
(a) Mr. C
(b) Mr. D
(c) Mr. A
(d) Either (a) or (b)
(e) Cannot be determined

Who is the Mathematician?
(a) Mr. A
(b) Mr. D
(c) Mr. B
(d) Mr. C
(e) Cannot be determined

What is the profession of Mr. B?
(a) Physicist
(b) Policeman
(c) Mathematician
(d) Doctor
(e) Cannot be determined

Let us suppose that there are 100 students in my class. Prove that among them there are at least two of them who have an equal number of friends.
Note: If A is friends with B then it also implies that B is friends with A or else A and B both are strangers to each other.