In a little kingdom, the king and the other 65 citizens each have a salary of one coin. The king cannot vote, but he has the power to suggest changes — in particular, redistribution of salaries. Each person's salary must be a whole number of coins, and the salaries must sum to 66. Each suggestion is voted on and carried if there are more votes for than against. Each voter will definitely vote "yes" if his/her salary is to be increased, "no" if decreased, and otherwise not to bother voting.

The king is both selfish and clever. What is the maximum salary he can obtain for himself?

Hint 1: The king may want, temporarily, to give up his own salary to get things started.
Hint 2: Try to reduce the number of salaried citizens with each voting.

Step-by-step explanation:

Step 1: The king begins by proposing that 33 citizens have their salaries doubled to 2 coins, at the expense of the remaining 33 citizens (himself included). 33 citizens will vote "yes" (because they get more salary) and 32 will vote no (because they get less), and the voting passes. After the voting, 33 citizens will have 2 coins each, the other 33 people (including the king himself) has 0 coins.

Step 2: Next, he increases the salaries of 17 of the 33 salaried voters (some to 3 and some to 4 coins) while reducing the remaining 16 to no salary at all. 17 will vote yes and 16 will vote no, and the remaining citizens will not vote (their salary goes from 0 to 0 so they don't care). The voting passes. After the voting, 17 citizens will have 3 or 4 coins, the rest (including the king himself) has 0 coins.

Step 3: the king does the same thing again: he increases the salaries of 9 of the 17 salaried voters while reducing the remaining 8 to no salary at all. 9 will vote yes, 8 will vote no, the rest doesn't vote, and the voting passes. After the voting, only 9 citizens will share the 66 coins, the other 57 people (including the king himself) has 0 coins.

Step 4, 5, and 6: In successive turns, the number of salaried voters falls to 5, 3, and 2. At this stage, only 2 citizens have all the total of 66 coins and no one else (including the king himself) has anything.

Step 7: Finally, the king bribes three citizens with 1 coin each to over throw the two with big salary, and he will keep the rest to himself. The vote will be 3 yes, 2 no, and rest doesn't vote. Thus the king finishes with a royal salary of 63 coins.

More generally, if the original number of citizens (king included) is N, the king can achieve a maximum salary of N-3 coins with this method.

(a) the volume of the cone is 20/3 in3

hence, option a is correct.