AMC 10 Problems and Solutions
The American Mathematics Contest (AMC) is a challenging and prestigious national competition, administered by the Mathematical Association of America (MAA).Read More
How is HMMT different from the other math competitions you coach?
What is known about AMC, AIME and most of the other competitions, is that they are mainly individual. HMMT is different because it’s about teamwork. The whole team receives ten problems, for example, and it doesn’t matter who is solving them, it could be one person or the entire team. There are different strategies and different tactics. The team competitions provide additional flavor as you need to work as one by establishing and following strict rules. Time is very important, too. That is the main thing I am working on with my students - collaboration.
Generally speaking, you need to know all of the topics to compete in HMMT. For example: if you are working as an individual and you have a problem that you know - you either solve it or not. You have no one to ask for help, you are alone. In team competitions, even if you solved the problem, you can turn to others and ask them how they are doing.
If you have different answers, you can start from the beginning. In a team, you can always have people who are better in certain topics, like Geometry or number series, for example. And therefore, we establish rules and learn how to follow them, and who is doing what. That is why these types of competitions, like HMMT, allow you to work faster.
How do you choose your team’s members?
We have qualification rounds. In the beginning of the school year, we send invitations to our high school students who are in our competition program. Currently we have about 35 students who are interested in competing.
What do you think are the core qualities that students develop and master during HMMT?
Mainly, the students have to study how to work together as a team. For individual competitions students always have the same role but in the team contests - if they are good at one specific thing - let’s say number series - they will only do that. You can be either a solver or a checker, students can have different roles.
Basically, the qualities that students develop go from having very individualistic roles to teamwork and learning to be good at sharing their ideas with each other. All the students are more on the individual side, so it’s a full time job for me to switch them to think as team members, as a whole. We all put a lot of work in this process.
What is most beneficial for the students participating in such contests?
There are several things. For example, all good universities know about competitions like HMMT, and they also know if the student results are good. This is one practical benefit. There are many others, too. Before the year 2000, most of the mathematicians used to work individually.
There is a big change in mathematics now, because of the Internet revolution. More and more work is done in a team setting where people have different types and levels of expertise. There are very difficult problems known among the mathematicians, which just recently started to move from stage “unsolved” to at least “partially solved”. All because those problems were approached not by one or two people but the whole team. They share their results to come to very important answers. This is a beneficial thing as a whole.
For example, there is a problem known for 300 or more years which was regarded as impossible to solve. In 2014, a guy from the University of New Hampshire showed a new result on that problem, based on which a whole team was formed and they all started working together. Thanks to their work, the problem is now moved to the stage “nearly solved”. They made progress that was impossible to be done by any one individual.
And what is the most valuable benefit for you?
At HMMT, there are usually 150 teams, six students per team, so 900 participants, all together at one place. That excludes coaches, teachers and staff. During the guts round all teams sit in a very big auditorium and each team chooses one person to be the runner. In the beginning, all the runners receive the first three problems to bring to their teams. When they all come up with the answer, again - the runner goes to the jury, gives them the answers and receives the next three questions. And so on, and so on. Meanwhile, the jury has to grade the answers already given to them.
While this is happening in the main auditorium, the coaches, trainers, parents, etc. are in another hall, with a live screen showing the teams’ scoring in real time. I follow the teams, my team mostly, of course. Sometimes they solve the problems pretty fast, sometimes they are stuck. But whatever the case might be - I am there, watching, and there is absolutely nothing I can do. And the time is running so fast, especially if other teams are one or two rounds ahead of ours.
So it is thrilling, if nerve-wracking.
Can anyone become a good mathematician? Do you believe some of us are born with talent for math, and others are not, or is this not really a thing?
I think a good mentor is very important. Much more important than so-called talent. A good teacher can have a great effect. I always remember a phrase which I read in Renoir's book: talent is a question of quantity. Everyone can write one book, one story; everyone could be good one time. The real talent lies in consistency.
You have to practice what you are good at, to produce, and that will eventually make you really good. As we grow older, what we were born with is becoming less and less relevant. If we want to achieve excellence - what matters most is quantity, real results. It’s always about how much you learn and how much you work. And for that - good mentoring is essential.
Again, talent is overrated. It’s a question of work.
Is this your advice to all the students who are going to register to compete in math competitions, to work hard?
Yes, I would tell them that it’s hard, be prepared to work. You need to do it every day. The analogy is very clear: imagine how hard a swimmer needs to practice to be at his best while competing with a team on a national or international level. She or he needs to work everyday, from 5 AM to late night. Every day they swim miles and miles, back and forth. It is the same for us. The time of easy achievements passed. Now it’s time for hard work.
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