## Medium puzzles

After the revolution, each of the 66 citizens of a certain city, including the king, has a salary of 1. King cannot vote, but has the power to suggest changes - namely, redistribution of salaries. Each person's salary must be a whole number of dollars, and the salaries must sum to 66. He suggests a new salary plan for every person including himelf in front of the city. Citizens are greedy, and vote yes if their salary is raised, no if decreased, and don't vote otherwise. The suggested plan will be implemented if the number of "yes" votes are more than "no" votes. The king is both, selfish and clever. He proposes a series of such plans. What is the maximum salary he can obtain for himself?

Hint
Solution
Source: P. Winkler
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At a party of N people, some have a symmetric friendship. Symmetric means that if A is friends with B, then B is in turn friends with A. Prove that there are at-least two people with same number of friends.

Solution
Source: Top Quant Interview
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p and q are two points chosen at random between 0 & 1. What is the probability that the ratio p/q lies between 1 & 2?

Hint
Solution
Source: Written Test
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Roll a die, and you get paid what the dice shows. But if you want, you can request a second chance & roll the die again; get paid what the second roll shows instead of the first. What is the expected value?

A very innocent monkey throws a fair die. The monkey will eat as many bananas as are shown on the die, from 1 to 5. But if the die shows '6', the monkey will eat 5 bananas and throw the die again. This may continue indefinitely. What is the expected number of bananas the monkey will eat?

Hint
Solution
Source: Self, to show an example of recursive probability
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