1. (BOfA, ML) There are a cup of milk and a cup of water. Take one teaspoon of milk,

put into the water cup; mix well. Take one teaspoon of the mixture in the water cup and put into the milk cup then mix well.

Which is higher: the percentage of water in the milk cup or the percentage of milk in the water cup ?

2. (Barclays, ML) \(W_t\) is brownian motion. N is cdf of normal \(N(0,1)\). Calculate \(E(N(W_t))\).

3. (GS) There are 5 thieves numbered 1,..,5 trying to divide 100 gold coins using this algorithm: the number 1 will come up with a way to divide money, if there is more than 50% agreement among them about his method (including the dividing thief) then it's done. If not, then they will kill the first thief and the second thief will divide money coming up with his own method. If you are the first one, what method you will use to divide money ?

4. (SG) There is a cubic cheese 3x3x3. There is a rat eating this cheese in the following manner: it east a corner (1x1x1) of the cubic the first day. The next day, it will eat another 1x1x1 cell which has the same outer face as the one it eats the day before. Find an algorithm so that the rate can eat the center cell the last day.

5. (ML, LB, SG, Bear, DB) The today price of a certain stock is 20$. Here is an option: if the stock reaches 40$ then the payoff is 1$. Price this option.

6. (SG) \(t < T\). W is brownian motion. Calculate \(E(W_T|W_t), E(W_t|W_T), E(W_t| |W_T| )\) (\(W_t\) conditioning to the absolute value of \(W_T\))

7. (JPM) There are parallel lines with distance d lying on the same 2-D plane. There is a line segment with length l>d. Find probability that this line segment not crossing any other lines.

8. (ML) \(T_1 < T_2\). Pricing forward-start option \(E(\frac{S_{T_2}}{S_{T_1}}-K)^{+}\)