MathDB

12

Part of 2007 Stanford Mathematics Tournament

Problems(2)

SMT 2007 Team #12

Source:

6/30/2012
Brownian motion (for example, pollen grains in water randomly pushed by collisions from water molecules) simplified to one dimension and beginning at the origin has several interesting properties. If B(t)B(t) denotes the position of the particle at time tt, the average of B(t)B(t) is x=0x=0, but the averate of B(t)2B(t)^{2} is tt, and these properties of course still hold if we move the space and time origins (x=0x=0 and t=0t=0) to a later position and time of the particle (past and future are independent). What is the average of the product B(t)B(s)B(t)B(s)?
Judgment Issue

Source:

3/15/2010
Pete has some trouble slicing a 20-inch (diameter) pizza. His first two cuts (from center to circumference of the pizza) make a 30º slice. He continues making cuts until he has gone around the whole pizza, each time trying to copy the angle of the previous slice but in fact adding 2º each time. That is, he makes adjacent slices of 30º, 32º, 34º, and so on. What is the area of the smallest slice?
geometry