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Quantitative
1. What is the sum of integers from 1 to 100?
2. You are presented with an antiquated scale and 12 identical looking marbles. However, 11 of these marbles identical while one weighs different from the other 11. You must identify the nonconforming marble and remove it. You may use the scale a maximum of 3 times. Note: You do not know whether the irregular marble weighs more or less than the other 11 marbles. You must identify this marble and whether or not it is heavier or lighter than the other 11.
3. What is the shortest path that you can walk from one corner of a cubic room to the extreme opposite corner of this room?
4. At some point in history, the lost city of Atlantis contained 500,000 childless, married couples. Each family wished to continue the family name through the birth of a son, but the government has a mandate to limit the risk of overpopulation. Hence, each family must have 1 child per year until the birth of a son. For example, if family A has 3 children, then there are 2 situations: (1) family A has 3 girls and will have another child; or (2) family A has 2 girls and 1 boy and will not increase the size of their family through a birth. A family is equally likely to have a male baby as it would a female baby. Let p(t) be the percentage of children in Atlantis at the end of year t. How is p(t) expected to evolve over time?
5. You have 7 red socks and 13 blue socks in your sock drawer. The light is broken in your bedroom and you must select your socks in the dark. What is the minimum number of socks you need to take out of your drawer and carry into your living room to guarantee that you have with you at least a matching pair to choose from?
6. A monkey is climbing up a 20 foot ladder. It climbs up by 6 feet every day. Each night it sleep. While sleeping, it slides down by 2 foot. When does it reach the top of the ladder?
7. A small boat is floating in a swimming pool. The boat contains a very small but very heavy rock. If the rock is tossed out of the boat into the pool, what happens to the water level in the pool?
8. You have a 52 card deck with 26 red and 26 black card. You will play a game at a casino, where you are to draw cards one by one. Drawing a red card pays you \$100, while drawing a black card costs you \$100. You may leave the game at any time. Cards are discarded from play once they are drawn from the shoe. Determine 2 items: (1) what the optimal stopping rule is in terms of maximizing your expected payoff and (2) what the expected payoff is of this strategy.
9. Carl Icahn has decided to bid for Firm A whose unknown true value is a random variable uniformly distributed between \$10M and \$100 M. Carl does not know the true value – denoted as V – of Firm A. However, as soon as people learn that Carl has made a bid, the news will cause the value of Firm A to double to 2V. Carl’s bid will only be accepted by Firm A’s Board of Directors if it is at least as large as the original value of the firm. How much should Carl bid so that he maximizes his expected payoff?
10. King Solomon requires a tax of 1,000 gold coins from each of the 10 districts of his kingdom. At the end of each year, King Solomon’s 10 tax collectors bring him the requested tax from each of the 10 districts. It is made known to King Solomon that one of his tax collectors is a cheat and is consistently delivering coins that are 10% lighter than they should be. Each gold coin weighs exactly 1 ounce. King Solomon does not know which tax collector is the cheat. How can Kind Solomon identify the rogue tax collector by using a weighing device once.
11. Can the mean of any two consecutive prime numbers ever be prime?