Most people who know about the University of Waterloo also know that it is recognized internationally as a centre that excels in teaching math, engineering, and computer science. Its programs are hard to get into and extremely competitive.
As a broad celebration of knowledge, and equally as a low-key recruitment tool, the University also runs annual math and computer science competitions for high school students. The Centre for Education in Mathematics and Computing (CEMC), an educational outreach organization housed at UWaterloo’s Faculty of Mathematics, has become very well-regarded because of these competitions.
Recently, our wrote the Senior and Intermediate Mathematics Contests. The contest asks nine questions; three of these need full answers that show the student’s work, and the other six only require the answer. In preparation, our students took a short break from regular math lessons to discuss some contest-style problems. As you will see below, contest problems have their own style to them. They read almost like a joke, having a lengthy set up to a short punchline.
A strong result earns the student a certificate of accomplishment and a very strong result could earn the student a scholarship at UWaterloo. If a student is less successful, they will at least have had an opportunity to stretch their mathematical muscles without affecting their grades. The CEMC’s stated purpose is to “increase interest, enjoyment, confidence, and ability in mathematics and computer science” which is, we think, is exactly what they accomplish.
We won’t have our own students’ results for a little while yet, but if you think you’ve got math chops of your own and want to get a sense of the challenge that our students undertook, try one of last year’s questions.(And don’t forget to show your work!)
For each positive integer n, the Murray number of n is the smallest positive integer M, with M > n, for which there exist one or more distinct integers greater than n and less than or equal to M whose product times n is a perfect square. For example, the Murray number of 3 is 8 since 3 × 6 × 8 = 144 and it can be shown that it is not possible to multiply 3 by one or more distinct integers that are greater than 3 and less than 8 to obtain a perfect square.
- The Murray number of 6 is 12. Show why this is true.
- Determine the Murray number of 8. (No justification is required.)
- Prove that there are infinitely many positive integers n for which n is not a perfect square and the Murray number of n is less than 2n.
- Prove that, for all positive integers n, the Murray number of n exists and is greater than or equal to n + 3.