National high school mathematics competition, first held in 1976 as the Atlantic Regions Mathematics League. There are two somewhat disjointed parts: the National Competition, taking place the weekend after Memorial Day, and the Power Contest, a smaller affair designed for the classroom.

The National Competition is an all-day affair held simultaneously at three sites: Penn State University, the University of Iowa, and San Jose State University. Each team of 15 people completes 4 rounds:

  • The team round: Each team collaboratively answers a set of 10 numerical-answer questions.
  • The power round: Each team collaboratively answers questions and proves conjectures about a concept introduced for the round.
  • The individual round: Each team member independently answers a set of 8 numerical-answer questions, given in groups of 2.
  • The relay round: Each team splits into 5 3-member subgroups. Each subgroup is given 3 questions, 1 per member, such that the answer to the first is a value in the second, the answer in the second is a value in the third, and only the answer to the third is handed in for a score.

Points are awarded to each team for all of the rounds, and to the individual members for their performance on the individual round. Competition is in two divisions: Division A, with about 25 of the best teams, and Division B, with about 75 other teams.

The Power Contest is a newer affair, started in 1995. There are two problems during the school year with a format similar to the power round of the national competition, except that it is geared toward being done in a classroom-like situation, with no limit on the number of people and a shorter time limit (45 minutes instead of 1 hour).

The experience of competing in ARML

Disclaimer: I last did this in 2000, so the facts might not be all straight. I'll do the best I can to capture the feel of what my experience was like, though the experiences would most likely have been different for the other participants.

In my home state, the teams were treated as all-star teams for the statewide math league. As our coach was interested in building up a solid base from which to draw from, I was invited in 9th grade to practice with them. That meant that I would spend 3 May Saturdays practicing for the competition for about 8 hours a day.

There were about 35 of us, many of whom already knew each other from having been there before. 30 of us would be picked to be on one of our teams, and another 2 would go along to Iowa City as alternates. Of course, I wanted to be one of the ones to make the team, but I would still have a few years still to go, and my invitation was as a student in training. In any case, I couldn't help but to get my hopes up some.

We gathered in a classroom at Macalester College and introductions were made. We were given 4 pages of formulas that would prove helpful and told that we should know them well enough to use from memory. Then we started working.

It certainly felt very different than it did practicing for my school team. There, practices simply consisted of the coach giving us packets of problems and us being told to do them before the meet. Here, we sat down in teams and learned as much about how to organize the team for greatest efficiency as how to do the problems themselves. It was far different from the chaos that my school team encountered working together. Here, everything was recorded publicly: who was working on what, what answers they had gotten, and who had checked to make sure the answers were right. While it wasn't perfect order, it was quite a change from what I was used to.

Answering the questions was a different matter. Instead of being one of 100 teams in the state, we were part of 100 teams in the country, and there had to be serious competition. I wasn't expecting it. The 4 pages of formulas proved to be necessary to getting through the problems, while I hadn't heard of many of them before coming here. The hours of practice would be needed, not only to build ourselves as a team, but to be able to do the problems that they threw at us in a short enough time.

The second weekend showed me my first power question. I had been used to doing proofs in my math classes, but nothing like this, with topics resembling topology and number theory more than the calculus which my high school offered at the time. It was another new thing that would take practice, but once I had been through a couple, I knew the procedure and where I fit in the team.

The final practice had us going through a mock competition, so we would have an idea of the schedule and finalize our ideas of where we fit best in the relay. I knew that I wasn't the best (then again, we were all the best; I just wasn't the best of the best), but I still hoped for a spot on the team.

It didn't happen. The next year, maybe.

The next year, I was invited to be an actual team member, not just a student in training. I went through the same 3 weekends of training again, with many of the same people. It didn't feel nearly as strange this time around, mostly because I had been through it before and knew what was going on. And this time, I got to be on one of the teams.

The competition would be held on Saturday, so early Friday morning, we piled on a bus and got our official team T-shirts. We would drive all day, stopping for lunch at a strip mall somewhere in Iowa, where about half of the team went to the Burger King, even if only to get one of their crowns, a mark of pride among us math geeks. We finally got to the dorm where we were staying, where we just hung out and unwound. We were ordered to get a good night's sleep that night, but some of us stayed up later playing card games.

In the morning, we were gotten up bright and early and told to make sure we ate breakfast before we met to go to the test. The team and power rounds went pretty well; it was much as we had practiced, and everything went smoothly. My first major shock of the day came when everybody came together for the individual event. We had been warned about this, but I was still unprepared. It was held in a gym, with the teams lined up and several hundred people sitting in a hot, noisy room. It was, after all, the beginning of June, and there wasn't any air conditioning for us. We had practiced in far more comfortable conditions.

That didn't keep me from doing ok, though. It was probably my best moment there.

Then came lunch outside, to the sights and sounds of the various teams playing pickup ultimate, and it was a time to relax. Then we went back in, did the relay round in the same extreme heat, and sat through the awards being given out. Nothing special for us this time.

The bus ride back was just as long. We stopped in Rochester to drop off a few people, and while we were there, I learned of a new tradition: The new captains were named and threw the first frisbee for a game of 500 at dusk. It was a fun time. We eventually had to hit the road again, and we made it back to Macalester fairly late.

The next year, I would go back, this time as part of our better team. While I didn't do as well, probably in part due to the late-night card games, our teams did better, well enough to walk away with a cool little trinket. I wanted to go back the next year, my senior year, but since Graduation was that same weekend, I joined the ranks of those who had to skip ARML their senior year.

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