The ICTM State Math Contest is a mathematics competition for Illinois high school students sponsored by the Illinois Council of Teachers of Mathematics. Many Chicago area high schools participate in the North Suburban Math League which forms a "regular season" to which the ICTM regional and state meets are "playoffs". I went to the state meet four times, in 1991 as part of the Naperville North team, then in 1992-94 for the Illinois Mathematics and Science Academy. It was a lot of fun.

A math contest is more like a track meet or swim meet than a tournament in some team sport: in most events no one but you and your teammates can affect your score. (Contrast Scholastic Bowl or Quiz Bowl which I never did in high school.) The ICTM contest is organized into a variety of events so that everyone gets to do several different things. This also means that one or a few very smart people cannot win the contest by themselves without a strong group of reasonably smart teammates backing them up!

Generally the ICTM contest questions cover the U.S. standard high school mathematics curriculum, organized into the nominal "contest grade levels" Freshman Algebra, Sophomore Geometry, Junior Algebra II, and Senior Precalculus. (For non-American readers: these years correspond roughly to ages 14, 15, 16, 17.) Of course your typical math team student is one or more years ahead in this course sequence, but if you are an age-14 freshman, you are allowed to take the freshman test, on which theoretically every question can be answered using only the content of a standard freshman algebra class. (In practice this is nearly true. The ICTM contest is designed to be interesting for kids from all over the state, not challenging for top students from rich suburban schools - although it was fun to try to ace it. As far as I can tell they do a very good job of getting kids involved and excited about math who otherwise might not have been.)

Schools receive invitations to the state meet based on their scores at the regional meet, which follows almost the same lines. Participating teams are grouped into four divisions by school size, and you compete only inside your division. (You can move up: IMSA competes in division 4AA "2000 students and up" despite having an enrollment about 600.) A "full" team contains about 24 kids, six each from each of the four grades; fewer than this puts you at a disadvantage in some events. Some schools that don't field a full team send exceptionally talented individuals to enter one or a few events. There are two different groups of questions: divisions 1A and 2A compete separately on the same (easier) tests, and 3AA and 4AA compete separately on the harder tests.

The events are as follows. All the questions are short answer, that is, not multiple choice, but no proofs and no justification are required. Calculators are allowed in all events except 8-person, but calculated approximations to the correct answer are not accepted except on the Calculating competition!

  • Written competition: at each of the four grade levels, at most six kids take an individual written test. 20 questions, 50 minutes, 2 points per question, top four scores count, for a maximum of 160 points in each grade. Usually there is a handful of perfect scores of 40 in each grade, at least in division 4AA where all my experience is. This event is the great equalizer since it counts for about half the total points. I found that none of the questions were really difficult but it was a challenge to do them all rapidly and correctly.
  • 8-person competition: This one is a blast but I rarely got to do it. Divided into Frosh-Soph (at most 4 sophomores) and Junior-Senior (at most 4 seniors). 8 people, 20 questions, 5 points per question, no calculators, 20 minutes of utter craziness. Organization, strategy and luck are almost as important as math on this event. We usually appointed a slow but careful kid to keep the answer sheet up to date and stay out of the way otherwise, because the answers have to be turned in the instant time runs out. Junior-Senior 8-Person is the one event at this contest that can have some really challenging questions on it; frequently 50 is an excellent score.
  • Calculating competition: like 8-person but focused on calculators. 5 kids, one from each grade level plus one freebie; 20 questions, 5 points per question, 20 minutes, all answers to be given as decimal approximations. This contest is where you put your HP 48 and TI-92 speed programmers; the questions can sometimes be attacked by writing a quick program to converge to the correct answer.
  • Oral competition: This contest is unique in that it covers topics outside the standard high school curriculum. The topics are fixed in advance each year and each team's oralist (usually a senior) studies a pre-announced syllabus. Then at the meet you have 15 minutes to prepare a 7 minute oral presentation answering three questions, then 3 minutes to answer impromptu questions asked by the judges. (In recent years the rules have changed to allow a teammate in the preparation room, but s/he can't talk during the presentation, so this seems more distracting than potentially useful.) 50 points total. The year I did it at state I got 49 - one point off for squeaky chalk.
  • Relays: These are more about having a little tense fun than doing math. Line up in four teams of four, two Frosh-Soph and two Junior-Senior (max two sophomores or seniors). Each team position gets a different question, and seats 2, 3, and 4 need the answer from the seat in front of them to finish their question. 2 points for each question correct (so 8 for the whole sheet), plus 5 bonus if you turn in a perfect sheet at the 3 minute mark (out of 7 allotted), or 3 bonus for a perfect sheet at the 5 minute mark. But once you turn that sheet in, you're committed to it, so you'd better be sure! Total points 13 per round * 2 rounds * 2 teams per class * 2 classes = 104.
  • 2-Person competition: And this one is more about having a LOT of tense "fun" than doing math. Like the other group events this comes in a Frosh-Soph and a Junior-Senior, but you are allowed to have two sophomores or seniors. Teams are grouped into rooms with no more than 13 pairs per room. Each question is flashed on an overhead projector, and you get only one chance per question to submit an answer. Correct in 1 minute gets you 6 points, 2 minutes gets you 4 and 3 minutes gets you 3. In addition, the first correct answer gets 2 bonus points, so it matters who else is in the room with you! Theoretical maximum is thus 80 (per class, i.e. 160 total), but 40 is an excellent score.

    Note that this is the "new" scoring. When I did this event - and I did, every year - the scoring system was much tenser: the first correct answer in the room was worth 7, the second 5 and all other correct answers were worth only 3. I used to sit between questions with knuckles white on my legs, mumbling "I hate 2-person..." to myself over and over. Afterward we usually felt like we'd run a couple of miles. (My senior year Stephen Wang and I made a pretty good run at the theoretical maximum score of 70, but we were foiled at the last question and came out with 68.)

So the total possible score is 1254. If I remember correctly the winning scores in the top division usually run in the high 800s or low 900s. The historically dominant teams, when I was in it, were the suburban "big five" of New Trier (Winnetka), Evanston, Stevenson (Mount Prospect), Naperville North, and IMSA, with the occasional addition of Hinsdale Central. I don't know if this distribution has changed at all since the early 90s.

Math team was a lot of fun in high school, and not just because we won a lot (although that was nice). It was a chance to work together with other bright kids doing something we enjoyed, and then let off a little steam on the bus ride back. Eventually it began to seem a little less interesting, because as you get further into mathematics, you see that timed mathematics competitions measure cleverness and experience more than insight or knowledge, and you want to learn something substantial, rather than new triangle tricks. But I still treasure my math team memories.

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