Mathematical Olympiads for Undergrad Students

Mathematical Olympiads for Undergrad Students

Mathematical Olympiads are popular among high school students. However, there is nothing similar for college students, except maybe IMC. Even IMC is not popular. It focuses mostly on the same kind of problems as high school Olympiads, and you can not participate if you are over 23 years old. In addition, it is organized by country, as opposed to globally, thus favoring countries with a large population. Topics such as probability are never considered.

This is an opportunity to create Mathematical Olympiads for college students, with no age or country restrictions. It could be organized online, offering interesting, varied, and challenging problems, allowing participants to read literature about the problems, and have a few weeks to submit a solution. In short, something like Kaggle competitions, except that Kaggle focuses exclusively on machine learning and data processing. Not sure where the funding could come from, but if I decided to organize this kind of competition, I would be able to fund it myself. 

Below are examples of problems that I would propose. They do not require knowledge beyond advanced undergrad level in math, statistics, or probabilities. They are are more difficult, and more original, than typical exam questions. Participants are encouraged to use tools such as WolframAlpha to automatically compute integrals or solve systems of equations involved in these problems.

Is anyone interested in this potential new venture?

Problem 1

Using complex analysis or other means, prove that 

See solution here

Problem 2

Points are randomly distributed on the plane, with an average of m points per unit area. A circle of radius R is drawn around each point. What is the proportion of the plane covered by these (possibly overlapping) circles? How can you use this problem to compute an approximation of exp(-Pi)?

See solution here

Problem 3

What is the minimal correlation between two random variables if the marginal distributions are exponential? Prove that it can not be lower than 1 – (Pi^2 / 6) = -0.644. Provide an example where the lower bound is attained.

See solution here (in section 9.)

Problem 4

A special version of the logistic map is produced by an iterative algorithm, as follows: the seed x = x(1) is anywhere in [0, 1] and x(n+1) = g(x(n)) with g(y) = SQRT(4*y*(1-y)). The equilibrium distribution satisfies the stochastic integral equation P(X < y) = P(g(X) < y). You can look at this as if X is a random variable with the observed values being the successive values of x(n). The equilibrium distribution does not depend on the seed x except for very rare, bad seeds, that do not behave well. Solve the stochastic integral equation to derive the equilibrium distribution. Also, what is the theoretical correlation between x(n) and x(n+1), at equilibrium? (answer: -1/2.)

See solution here (in last section.)

DSC Resources

Link: Mathematical Olympiads for Undergrad Students