Problem #1

Today a company has only one asset – an empty building - which was bought at $100 one year ago by spending $100 worth of cash which the only shareholder of the company, Vaughn, put into the company by taking a loan from a bank. John is a friend of Vaughn and he is interested in buy 50% of this Company. He thinks that the market value of this asset after a year will actually become $115. How much should he pay to Vaughn as a fair price for 50% of the company?

(a) $15
(b) $7.50
(c) $57.50
(d) $50

Problem #2

There is a cake shop very near to your house and in that shop there are only two types of pastries – black forest and vanilla. Both sell for $2 a piece though you believe that black forest is more delicious and popular. You have $50 to spend and you need to buy pastries for a group of friends who are coming for dinner tonight. You are unsure who likes black forest and who likes vanilla. How will you spend your money?

(a) You will buy 25 pieces of black forest pastries so no one will have any reason
      to feel unhappy;
(b) You will buy 10 pieces of vanilla pastries and 15 pieces of black forest pastries
      and bet that more people like black forest than vanilla;
(c) Your will buy 12 pieces of vanilla and 12 pieces of black forest pastries
      and keep the $2 to buy one additional pastry if need be;
(d) You will buy 5 pieces of each and keep $30 with you to buy additional pastries
      after the guests have arrived but this could be a hassle for you;

Problem #3

What is common between stocks (shares of companies) and elevators (lifts, that take people up and down a building):

(a) each has three states of existence;
(b) each has two states of existence;
(c) each moves according to the same mechanical laws;
(d) there is nothing common, and this is a ridiculous question;

Problem #4

Which of the following statements is true about the following identity

(a) the above identity will generate a prime number for x equal to zero;
(b) on a positive number line (from zero to plus infinity) the above identity contains at
     least one prime number ;
(c) the above identity will generate a prime number for all values of x up to 28;
(d) all the above statements are true;

Problem #5

The arc sine law is an important principle in statistical (random walk) theory and an arc sine density function is given by:

This function has the property of blowing up at extremes such as at one and zero (i.e. f(x) becomes infinitely large and indeterminate numerically when x=0 and x=1. In between the function is fairly restraint. Statistically speaking, there is a very high probability mass at the tails of this probability distribution function and the mass is very thinly, but consistently, spread in between the tails.

A derivatives trader trades options, which are financial derivatives on underlying financial assets, such as stocks, bonds, commodities, etc. His profit and loss is calculated in a simple way by valuing his positions – the value of his purchases – each day and comparing it with his original purchase price

Now you are told that this derivatives trader's profit and loss distribution is governed by the same arc sine law mentioned above and the probability density function of the trader's profit and loss is given by the above function. You will:

(a) think that this is a lot of baloney and would try your luck in genetic engineering
     instead;
(b) think that this is plausible but a few constraints need to be placed on this function;
(c) agree with this statement;

If your answer is (b) or (c) then please explain your answer. (If your answer is (a) then let us know and we can suggest a few good schools in the U.S. which offer graduate studies in genetic engineering.)

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