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Math Problems: Seemingly Easy, Yet intense!

Try 3 levels of Math problems that seem easy, yet it is intense. Challenge yourself and your friends with these problems.

Level 1 - Easy - 10 points

The first problem is something that is somewhat elementary. 

From a biased coin(a coin where the probability of heads is not 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 the 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-known 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 that appear only once. (Here [] means the greatest integer function.

Seems easy right? Intuition says induction might kill the problem. Well, try-hard.

Some more beautiful problems for you:-

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