Level 1 – Easy – 10 points

The first problem is something which is somewhat elementary.

From a biased coin(a coin where probability of heads is not be 1/2) how can you generate two events which are equally likely.(same probability).

To restate it suppose there is a chocolate cake.There are two persons Nilasha and Diganta,both of them want to eat the cake.They decide both of them should have the same probability of eating the cake.But they have a biased coin.How should they design the toss so that both of them are equally likely to eat the cake.(What it means is that you need to find two events with same probability with a biased coin)

Level 2 – Medium – 20 points

The next problem was asked to me in a mock interview by some seniors which was actually a very nice application of a well know theorem.

Prove that there exists a series of 100 consecutive natural numbers with exactly 13 primes between them.

Hint-I could have given any number less than 26 in place of 13 in the above problem

 

Level 3 – Hard – 20 points

This problem was given in MOP 2005 and none of the students could answer this one.Then two former gold medalists along with a hint from the proposer of the problem, solved this seemingly easy yet intense problem.
So here goes the problem,

There are n co-linear points on a straight line.No two distances between any two points can appear more than twice.(That is the same length cannot be repeated more than twice).Prove that there are at least [n/2] distances which appear only once.(Here [] means the greatest integer function.

Seems easy right? Intuition say induction might kill the problem.Well try hard. The solutions will be out by next week.
P.S-And with this I will commence a new section called problems of the month.

The full solutions are to be given at officialbhattacharya@gmail.com and the top scorer of MARCH and APRIL will receive a book from us.You may discuss your observations in the forum section

Best of Luck