Something recently caused me to reflect on my math journey in high school. Perhaps, the memories of old days came back when I glanced through the Mathematics magazine “Toán Học và Tuổi Trẻ” (Mathematics and the Youth). Each issue of this monthly magazine challenges readers with a set of problems, and also gives the solutions to problems given in the previous month. Each question is very tricky. I remember that in my 10th grade my math teacher required us to individually submit solutions to the problem set by the end of each month. Of course, that was not the only assignments we had. We were also supposed to write solution to every problem in a trigonometry book and turn in many weekly problems. To me and my classmates, that was an immense stress.
The purpose of such training is to prepare us for a number of upcoming math contests: the Ho Chi Minh City Math Olympiad, the Vietnam Math Olympiad, the 30/4 Math Olympiad of South Vietnam, and even the International Math Olympiad (IMO). The kind of problems in these contests is similar to the kind of problems found in the Putnam Competition. Each problem is neatly stated and doesn’t require much background to understand. However, it requires a great deal of creativity and brain power to solve. I will refer to this kind of problem as the “olympiad” problems, although they might not be found in an IMO exam. Prof. Arthur Engel wrote an interesting article on how olympiad problems are created.
For my personal training in high school, I often used the websites mathlinks.ro (now called artofproblemsolving.com) and diendantoanhoc.org. On those websites, people post challenging problems for which they might or might not have a solution. When it comes to Number Theory, it is actually not hard to make up an extremely difficult problem without knowing how to solve it. In fact, I made up a few problems in 2005 which were standing until 2013. I did not know how to solve them, nor did I expect anybody to be able to solve them. But I am happy to see that there are ingenious people who are able to solve them. Those problems are:
Problem 1: Show that for each positive integer n, there exists a multiple of
that has exactly n digit 1’s in its decimal form. [Solution here]
Problem 2: Denote by a(n) the number of digit 1’s in the decimal form of n. Show that there exists a positive integer n such that
[Solution here]
Problem 3: Show that there exist infinitely many square numbers n that has an odd number of digits (in the decimal form) in which the digit 1 occurs only once and it occurs at the middle of n. [Solution here]
There are similarity and difference between the olympiad problems which I did back then and the research problems which I do now. I don’t attempt to give a comprehensive comparison here. Rather, I would like to mention just a few points.
An olympiad problem found on an actual exam always has a solution. If following the “right” track, a person can solve the problem in a reasonable amount of time (an hour or so). A research problem does not have a solution available somewhere. It may take weeks, months, or years to solve (or completely abandoned). An olympiad problem is usually cooked up, i.e. artificially created or complexified from a much easier problem. Its purpose is to test and strengthen the intellectual ability of students. It is usually a stand-alone problem and is not viewed as part of a bigger problem. Unless one can point out how such a problem fits in the math literature, it will be hard to be published on a peer-reviewed journal. On the other hand, a research problem typically stems from an existing line of research, tackling a well-known problem. Knowledge on a research problem is usually incremental. The solution might not be found all at once, but is known little by little over time. Any discovery enriches the literature on that line of research, and is suitable to be published on a peer-reviewed journal. Unlike olympiad problems which are typically intended to be solved individually, research problems often require human collaboration to cope with.
I am grateful for the training that I underwent in high school despite the excessive stress it gave me then. Now as an educator and researcher, I have numerous opportunities to use the skills acquired then to assist my students and research groups.