UofA MCM/ICM Qualification Competition

Winter semester 2012

Q1: Within a typical university, there are generally certain courses that have a reputation for being "easy" and others that have a reputation for being "difficult". Therefore, grade point average (GPA) may not be an accurate measure of academic achievement; a student taking easier courses will typically have a higher GPA than an equally strong student taking more difficult courses.

Develop a new measure of overall academic achievement to take this into account. You should be able to implement your model objectively, without making any assumptions about which classes are "easy". That is, if you know the grades of each student in each course, you should be able to determine from that data which courses are more difficult.

Keep in mind that courses with higher average grades may not necessarily be easier; it could be that the students who take these courses are stronger.

Q2: Consider city traffic. The usual city speed limit is 50km/h, but the average speed that cars actually travel due to stops and congestion is much lower (for instance in Vancouver it has been measured to 24.9km/h average speed). A driver might be tempted to speed by thinking that driving 60km/h he gets to his destination 20% faster. However this is not typically true.

Consider a city environment with roads, stop signs, and traffic lights. (and if you like add congestion).
Let top speed be the maximum speed it can travel (ie the highest speed a driver chooses to drive)
Let average speed be the total distance driven divided by total duration of a trip.
Make a reasonable model and mathematical simulation or derivation that shows how the average distance traveled per hour (average speed) varies as a function of the top speed. Plot a curve for different top speeds and discuss relevant cases (e.g how much faster is a car than a bike (assume a bike can go top 20km/h)? How much does speeding gain?
Validate your model on reasonable data (Real data, or data you generate that you can convincinly argue give realistic results.)