IMC09 problem 2
For a couple of years I have thought about starting a mathematical blog, but I decided not to, because I thought I didn’t have enough time and ideas to keep a high enough post rate. But now and then I have something I want to say, and no place to say it, so I have finally decided to start this blog anyway, but also that I won’t have to post something every month.
This post and the next are partly inspired by these two posts by Gowers, where he tries to explore how people think when they do math, by asking his readers to solve a very simple equation, and explain how they solved it. In this and my next post I will consider a more difficult problem: It is problem 2 from IMC09 (IMC, International Mathematics Competition, is a math completion for university students). The reason I’m using an IMC-problem is that University of Copenhagen is going to participate in IMC for the first time this year, and I hope that talking about how we solve problems will be a good preparation for the contest both for me and for the rest of the team. Still, I hope that this discussion will be of interest to both IMO contestants and to professional mathematicians.
I solved this problem about two months ago, but I thought a lot about what problem solving strategies I used, so I still remember most of my thoughts. I have chosen to write about this particular problem because I noticed that I used many general problem solving strategies, that are good to know. You are welcome to post your thoughts on the problem in a comment, even if you haven’t solved the problem. If you have solved the problem, please try to describe the whole thought process, not just the thoughts that led you to the solution. I will post a description of how I solved it in a couple of days.
The problem is:
Let , , and be real square matrices of the same size, and suppose that is invertible. Prove that if then .