To the Rules of the Contest.
There has been a misunderstanding that I accept only the first correct solution to each problem. This is wrong: all correct solutions are valued equally!
To Problem 3.
It is assumed that the loot can be divided into any number of parts, but it is hard to evaluate and compare the parts (since the items are not just identical coins but more sophisticated pieces of jewelry). The goal is to find a division procedure which would make each pirate sure that he got at least one third of the loot . Here is an example of some procedure which does not solve the problem because it does not fulfil this requirement.
Example. Pirate A divides the loot into 3 parts which he thinks are equal, Pirate B leaves one part (smallest in his opinion) to Pirate A, then divides the rest into 2 parts which he considers equal, and Pirate C finally chooses the part that he thinks is bigger than the other one.
The procedure seems to make sense. However if you try to prove that each pirate should be satisfied, you'll find a flaw. Indeed, Pirate A is fine: he evaluates his share as exactly 1/3 of the whole loot because he gets one of the 3 parts which he considers equal. Pirate B is also fine: he evaluates his share as at least 1/3 since he gets exactly 1/2 (in his opinion) of what he thought was at least 2/3 of the whole loot. However, when Pirate B picks the smallest (in his opinion) third to leave for Pirate A, Pirate C does not participate in making this decision. He may disagree with Pirate B and think that the part left by Pirate B to Pirate A exceeds 1/3 of the whole loot. This gives Pirate C a good reason to think that 1/2 of the rest is smaller than 1/3 of the whole loot.
To Problem 4.
As it is clear from the gossips on Problem 3, a correct solution should not only include a plausible answer but also provide an explanation why this answer is correct. Without such an explanation it is not possible to tell right and wrong apart.
The same applies to Problem 4. In order to evaluate your solution, I need not only to receive your answer for the number of triangles but also to see your reasoning which would convince me that your answer is correct.
To Problem 5.
There has been a number of misconceptions about this problem (partly caused by occurence of Tehiyah in the formulation) which led some people to think that Alisa's statement is neither true nor false. Of course, mentioning Tehiyah as well as referring to students in a class is unnecessary. One can therefore rephrase Alisa's claim this way:
In each group of people there are two people who have the same number of friends within the group.
It is silently assumed that "a group of people" has at least two members, and that if A is a friend of B then B has to be a friend of A. With this explanations, Alisa's statement is precise and thus got to be either true or false (but not both!)
The claim starts with "In each ... ". Thus in order to disprove the claim one has to come up with a counter-example : an (imaginary) list of names A,B, ... and a scheme of who is friends with who such that all members of the list have different numbers of friends.
On the other hand, if it impossible to come up with such scheme, then one should be able to find out why this is impossible. Such an explanation why will therefore prove that Alisa is right.
To Problem 6.
The main rule in approaching game problems is that you should play the game first. This is fun, helps to develop a winning strategy and also to get rid of some illusions - such as the impression that the outcome of the game does not depend on player's strategies.
To Problem 8.
The plane is divided by several straight lines into regions called states. The problem is to paint the map thus formed into two colors so that nearby states don't look the same (that's how maps are usually colored). In other words, if two states have a segment of common border, they have to be of different colors (if however they only have a common point - corner - it is OK to have the same color).