# Rambling Thoughts

## ARML

ARML (American Regions Mathematics League) is an annual math contest that takes place every year at four sites around the country. It’s one of my favorite math contests.

ARML is different from most other math contests because it requires teams to travel to one of four central sites. This means that teams have to travel (some times for very long distances) to participate. Inevitably, this means that participation at ARML is not especially high; only 124 teams of fifteen people (in 2010), or less than one percent of the more than 200000 people who participated in the AMC. Teams that do participate at ARML, however, are much more organized; most teams organize several practices before the contest, while some teams prepare for ARML year-round. The team aspect of the contest (3 out of 4 rounds have a team component, together constituting 50% of the total team score) also makes ARML stand out. Indeed, ARML is primarily a team contest, and the team results every year are watched much more closely than the individual high scorers. As a result, ARML is much more social than most other math contests, giving participants a chance to work closely with their team and to meet people from other teams. This social aspect of ARML is what makes ARML enjoyable.

Mathematically, ARML feels different from most other contests. The ARML style is to have problems that look hard but are made much easier through clever tricks. Often, these problems can be solved without being clever, but cleverness can often yield very simple solutions. As an example, consider this year’s coffee mug problem: [EDIT (17 September 2010): Updated with the original wording.]

Let $P(x) = x^2 + 2010 x + 2010$, and let $r$ and $s$ be the roots of $P$. If $Q$ is a quadratic polynomial with leading coefficient 1 and roots $r + 1$ and $s + 1$, compute the sum of the coefficients of $Q(x)$.

This can be bashed out without the quadratic formula or with Vieta’s formulas, but the clever solution is much simpler:

Because $Q$ is monic,
$Q(x) = (x - (r+1))(x - (s+1))$
$= ((x - 1) - r)((x - 1) - s) = P(x-1)$. The sum of the coefficients of $Q(x)$ is $Q(1) = P(0) = 2010$.

Together with the clever tricks, ARML imposes a short time limit on its tests, making it difficult for contestants to finish the team and individual tests. The relay round’s three-minute and six-minute time limits provides a further emphasis on speed. As a result, ARML problems (with the exception of the proof-based Power Question) are usually not very hard mathematically; they’re just hard to do under severe time pressure. This short time limit, along with the small number of questions on ARML, make the results of the contest subject to a lot of random noise. The top teams are usually within a few points (out of 300 points total) of each other, so a five-point team round problem could easily change a team’s ranking by several places. The individual round has only ten problems, so achieving a national high score (usually at least eight problems) requires making no computational errors. Thus, the list of individual high scorers is somewhat meaningless; high scorers often do not repeat as high scorers when they return, and high individual scorers are often as much a result of luck as of mathematical ability.

Evaluating ARML based on the criteria that I listed in my post on Contest Math, we obtain mixed results. ARML, perhaps more than any other contest, highlights the social side of mathematics and hooks people into liking and doing math; though it’s hard to start doing ARML, it’s even harder to stop. Mathematically, though ARML does reinforce the importance of creativity, it does not give particularly challenging problems, and ARML results do not really constitute a good measure of mathematical achievement.

I enjoy ARML primarily based on the social part of the contest, and many people agree with me; the highlight of ARML for some people is the long bus ride to the contest. Indeed, I think of ARML primarily as a social event, with some mathematics to make it look like the participants are actually doing something important.