by Brandon Ewonus, Grade 12
The Canadian Mathematical Olympiad (CMO) is Canada’s premier national advanced mathematics competition and is staged by the Canadian Mathematical Society (CMS). The CMO is a closed competition, and only the top Canadian students demonstrating excellence in mathematics are invited to participate. This year, I was invited to write the CMO. I qualified, along with only 97 other Canadian students, because my score on the Sun Life Financial Canadian Open Mathematics Challenge (COMC) contest was high and put me in the top 1% of contest writers.
I am no stranger to math contests. In the past two years, I’ve received first place on the BC Colleges High School Mathematics contest, gold standard on the Hypatia contest, a gold medal on the Fermat contest, two gold medals on the Euclid contest, and an invitation to write the American Invitational Mathematics Examination (AIME), in addition to a gold medal on the Canadian Open.
St. Michaels University School math teacher, Mrs. Margaret Skinner was instrumental in getting me excited about attempting a much higher-level math. I have been interested in math related problem solving for as long as I can remember, but she encouraged me to pursue my desire to explore math’s challenging questions. I asked to take AP Calculus BC when I was in Grade 10 and SMUS made it happen. Ms. Linda Rajotte was my AP Calculus BC teacher and she encouraged me to join math club and participate in math contests as an enrichment activity. Mr. Smith, also a SMUS math instructor, organized math contest takers and sent reminders about current contests, which was a great help for busy SMUS students.
Some students across Canada attend math-training camps in order to secure an invitation to this prestigious Olympiad competition. However, other students like myself, pursue math on their own and participate in math competitions and contests through school organized math clubs. The math department at SMUS encourages students interested in mathematics to join the math club and participate in local, regional, and national math competitions.
In my case, since I completed all of the SMUS math courses by Grade 10, I have been taking university level math courses for the past two years, while enrolled at SMUS. I enjoy the challenges math has to offer and am thankful to have been associated with the many SMUS teachers who have supported my interests.
The Math Olympiad is a three-hour contest with only 5 questions. The questions were quite difficult and I had to work quickly and creatively, making sure to answer as fully as possible. I thought it was fun!
Below is a problem I recently came across. Though not as difficult as some of the problems I encountered on the Olympiad, it has a surprising result.
Problem: “Bob has two children. One is a boy born on Thursday. What is the probability that Bob has two boys?”
Solution: First, let’s consider the different possibilities of children.
1st Child | 2nd Child | Possibilities | |
Case 1 | Boy born on Thursday | Boy born on any day | 7 |
Case 2 | Boy born on Thursday | Girl born on any day | 7 |
Case 3 | Girl born on any day | Boy born on Thursday | 7 |
Case 4 | Boy born on any day | Boy born on Thursday | 7 |
Now, the total number of combinations of children with specified gender and birth day is 7+7+7+7-1=27. Note that we must subtract 1 because we double counted the possibility where both children were boys born on Thursday (in cases 1 and 4). Of these 27 combinations, there are 7+7-1=13 possibilities of two boys (once again, 1 must be subtracted for double counting). Thus, the probability that Bob has two boys is 13/27.
Note that in this problem, the trait specified had a 1/7 chance of occurring (being born on Thursday). As a generalization, if the trait specified has a 1/n chance of occurring, then the probability that Bob has two sons is (2n-1)/(4n-1). If no trait is specified (i.e. one child is a boy, what are the odds that the other one is a boy?) then n=1 and the probability that Bob has two boys is 1/3. On the other hand, as n approaches infinity, the probability that Bob has two boys approaches 1/2. Thus if an exact birth date had been specified for Bob’s son, say January 1st, then the probability that he has two sons would have been 0.4996…~1/2.
I was honoured to represent SMUS as an invited participant in the 2010 Canadian Mathematics Olympiad. I look forward to pursuing a degree in a mathematics-related field at Stanford University in California this fall.
Well done, Brandon. Sure glad you knew how to figure that Olympiad problem! Good luck at Stanford.