### Preparing for Lehigh Valley ARML

Since 2004 I have been assisting Don Davis with coaching the Lehigh Valley ARML team. We have won four national championships and have become one of the dominant high school math teams in the nation.

A Guide for Students

ARML is primarily a team activity. Of the four major rounds, three involve cooperation with your teammates. The Lehigh Valley ARML team consists of students from many different schools throughout our geographic region, most who did not know each other before joining the team. The geographic distances that separate us, the importance of cooperation and teamwork, and the limited time we have at our practice sessions all raise challenges that can only be overcome if every student does his or her part as a member of the team. This page lists some things you can do as a member of the Lehigh Valley ARML team so that together we can have a successful and enjoyable activity.

## 1. Things to Do at Home

At our ARML practices we generally practice former contests as working together with your fellow teammates is something you cannot do at home. Outside of practice, how should an ARML team member study and prepare? Learning the mathematics necessary to do well on mathematics contests like ARML is a very slow and time consuming process – something we could not possibly hope to accomplish completely in our limited practice time. Thus, each student is primarily responsible for practicing and learning the necessary mathematics on their own. But how? Here are some resources and references a student can use outside of ARML practices to improve.

#### Solutions Packets

At the end of most practices you will be given a packet of solutions to the ARML contest we practice on that day. It is important to study these solutions to learn how to do the questions you were unable to answer.

#### Playbooks

My daughter, Maria, and I have compiled two “Playbooks”‘ of mathematical facts and formulas that are typically needed on middle and high school mathematics contests.

The High School Playbook is a continuation of the MATHCOUNTS Playbook, so a student should normally strive to learn all of the topics in the MATHCOUNTS Playbook first, and then move on to the High School Playbook topics. In addition, within each Playbook there is a star symbol next to the topics that should be learned first. The High School Playbook contains many advanced topics that are appropriate for the more difficult Olympiad contests such as USAMO and IMO, but not normally needed for ARML. Thus an ARML student should try to learn the items marked with a star in the MATHCOUNTS Playbook first, then the rest of the items in the MATHCOUNTS Playbook, and finally the items marked with a star in the High School Playbook. But they probably should not worry about the other items in the High School Playbook until they are training for USAMO.

#### Other Math Contests

Preparing for other high school mathematics contests is also good training for ARML. Several of the good high school math contests make both the questions and solutions available from their contests. In particular you might want to obtain and practice the following contests, which are somewhat similar to ARML in difficulty and topics.

- MATHCOUNTS – this middle school contest is a good place for younger ARML students to develop basic skills. It does not cover all of the topics found on ARML contests.
- AMC 8, AMC 10 – also good practice for beginner and intermediate level ARML students.
- Lehigh Contest – where most Lehigh Valley ARML students are born (or discovered)! The problems range from very easy to very challenging and cover similar topics to those found on ARML.
- AMC 12, AIME – ARML covers topics roughly equivalent to those found on AMC 12 and AIME. AIME problems are generally more challenging than AMC 12 problems.
- Mandelbrot – another excellent competition with problems similar to ARML questions in difficulty.
- HMMT – This excellent competition is held each year at either Harvard or MIT and is entirely designed, organized, and operated by undergraduates. Lehigh Valley usually has several teams at this world class competition (contact me for details).
- PUMaC – Princeton’s version of HMMT. Lehigh Valley usually has several teams at this competition (contact Prof. Davis for details).
- USAMTS – This proof-based take-home contest is a good way to hone your skills at writing proofs (see below).

#### Reference Books

There are many good books specifically designed for students who participate in high school contests. The Art of Problem Solving website lists many texts:

- Problem Solving Books at AoPS
- The Art and Craft of Problem Solving by Paul Zeitz
- Math Olympiad Challenges by Titu Andreescu and Razvan Gelca

One natural progression might be as follows:

**Beginners**may want to start with the Art Of Problem Solving Volume 1 by Rusczyk and Lehoczky. It is quite detailed and covers pretty much all of the major basic topics needed for AMC and ARML.**Advanced**students should not miss out on my favorite problem solving book, The Art and Craft of Problem Solving by Paul Zeitz. This book goes beyond the mere exposition of mathematical facts and formulas and instead emphasizes the thought processes used by problem solvers to obtain solutions. While some topics are geared towards Olympiad level contests and are not needed for ARML, the problem solving approach is very useful in ARML and other math contests. This is the textbook I use in my advanced Problem Solving course for math majors at the University of Scranton.**Really Advanced**students who want to participate in Olympiad level contests will benefit a lot by working through Math Olympiad Challenges by Titu Andreescu and Razvan Gelca when preparing for the USAMO in 11th and 12th grade. This probably should be the last book read as it is the most advanced.

It will take a very long time to work through all of the books above, so patience is a must.

#### Learning to Read and Write Proofs

The Power Round in ARML requires that students be able to write mathematical proofs. Learning to read and write mathematical proofs is a bit like learning to play a musical instrument – it is a skill that is usually mastered over a long period of time with a substantial amount of practice. It combines several different skills: knowledge of the rules of logic, experience with mathematical conventions and notation, and a good working knowledge of English and the specific grammatical and punctuation rules that are used in mathematical exposition. That having been said, there are several ways that an ARML student can improve their proof writing and reading skills.

- Prove it! Math Academy is a two week summer math camp that provides an intense, rigorous, introduction to mathematical proofs in the context of problem solving and mathematical research. It was designed specifically to assist students who are trying to make the transition from computational mathematics to proof-based mathematics.
- For a more detailed and careful understanding of mathematical proofs, I teach an Introduction to Mathematical Proof course at the University of Scranton. Students who live in NE Pa are welcome to register for my course. Other students are welcome to try to work through the materials posted on the course web page by themselves.
- I am also the coauthor of Lurch, a word processor that is designed to check the reasoning in mathematical proofs. While it is still in the development stage, it goes hand in hand with the lecture notes and homework assignments in my Introduction to Proof course.
- There are some nice articles with concrete examples of good and bad solution writing styles at the Art of Problem Solving website.
- For a nice intuitive concise introduction to mathematical proof, you might try How to Read and Do Proofs by Daniel Solow. It is very easy to read and approaches proofs from the sort of perspective that problem solvers like ARML students might find useful.
- One way to learn how to write your own proofs is to read other people’s proofs. The solutions packets to ARML contests that we hand out, published solutions to AIME & USAMO contests, the reference books discussed above, and almost any math textbook of sufficiently advanced level will have mathematical proofs of varying degrees of sophistication which can be read to get a sense of what other people consider to be a proof. Also, during our practice Power Rounds, all questions have to be solved, written, and proofread. Less experienced students may also want to proofread the proofs written by more experienced students. Proofreading is an important way to help the team, and also provides the reader with an opportunity to learn some tips about proof writing conventions and style.
- At each ARML practice we usually have a Power Round, during which the teams are supposed to write up and submit mathematical proofs. Prof. Davis grades these and gives the students some feedback on their results. So by attempting some of these proofs in practice you can obtain this valuable feedback.
- ARML students might also want to participate in the USAMTS contest. This contest is open to any student who is interested (i.e. you don’t have to be part of a school or participating team). All of the problems require proofs for their answers and you usually have about a month to work on the problems. The student then faxes, mails, or submits their solutions online, and they are graded. In some cases comments or other feedback is obtained. It is a good way to practice writing mathematical proofs for a student who likes problem solving and math contests.

#### Online Resources, Discussion Forums & Classes

The Art of Problem Solving website is a “must bookmark” site for everyone involved with ARML. It brings together almost every online resource imaginable related to mathematics competitions and problem solving. There are far too many resources at that site to list here. Go to the site and check it out! Here are some ARML specific links.

- Lehigh Valley ARML – home page at Coach Don Davis’s website, containing all of the results, background, and logistical information for our team
- ARML Home Page – the main website for the ARML competition in general
- ARML discussion forum at Art of Problem Solving
- We also have a Lehigh Valley ARML Facebook group that is only open to Lehigh Valley ARML members. Just search for “Lehigh Valley ARML” in Facebook and click the button to request joining the group.

#### Summer Math Programs and Camps

There are numerous summer math programs and training camps available for middle and high school problem solvers. Several of our team members have participated in such camps in the past and have had a very positive experience. The American Math Society maintains a list of links to Summer programs for high school students. Follow the links on that page to learn more about each individual program.

#### And last but not least…

Perhaps the most valuable resource you have for preparing for ARML is … each other! We have the best and friendliest mathematics students and coaches in our area on the team, all sharing a joy and passion for problem solving mathematics. If you have a question, ask a teammate!

Keep in touch with the friends you make at LV ARML by email, online chat, Facebook, text messaging, and so on. The Lehigh Valley ARML team has become a magnet program for advanced mathematics students in our area. Having such a peer group can provide you with tips about scholarships, college applications, other mathematics contests, problem solving tips and tricks, and many other things that students with your interest and aptitude for mathematics may find interesting.

## 2. Things to Do at ARML Practice

OK, you’ve done your studying, participated in other contests, read books, done hundreds… no, thousands… of practice problems to get ready for ARML. What’s next? Naturally, attending our ARML practices is very important in order to meet and interact with your teammates. and to practice the contest itself in a group setting. Here are some ways that you can reap the most benefit from our practices and do the most good for the team.

**Cooperation**: With 15 students trying to work together to solve problems on the Team and Power rounds, cooperation is essential to our success. Every decision you make during these rounds should be made with the good of the team in mind. There is no room for ego and bravado. *How can I help?* should be your mantra. *Do something useful* at all times. If you are not solving a problem, you can be writing a solution. If you are not writing a solution, you can be proofreading a solution. If you are not proofreading, you can be independently verifying an answer or solution. If you are stuck ask for help. If you can offer help to someone else, offer it. If you are good with a calculations you may be able to compute something messy, if you are not, partner up with one of your teammates who is.

**Respect**: Everyone on our team is a superb mathematics student, usually the best math student in their school or local area. It is important to respect your teammates and their mathematical ability, both in terms of trusting their mathematical judgments and also from the ordinary aspect of being collegial to others. Be supportive of each other. Provide encouragement when someone is having a bad day (*everyone* has good days and bad days on math contests). Experienced ARML students should provide leadership and guidance for the newcomers. More advanced students can provide mathematical guidance to students with a weaker math background.

**Communication**: Talk to each other during practice. Make an effort to learn each other’s names. It is difficult to cooperate in a group situation when you can’t refer to each other in practice. We will provide you with name tags to facilitate this. During the Team and Power rounds, talk to each other within your squads, coordinate your activities, keep your team captain informed of essential information, share insights with the entire group. Between rounds and before and after practice get to know each other. Share a cookie and some lemonade.

The Lehigh Valley ARML team has had amazing success in recent years. A large part of that success has been due to our ability to work together as a friendly, cooperative, well organized team.

## 3. Things to Do at the ARML Contest

The trip to Penn State is the highlight of our ARML season. All of the training and practice can now be put to good use. It is also a great opportunity for making new friends and meeting people with a similar passion for mathematics from other places. Creamery ice cream, dinner in downtown State College, games on the bus, ultimate frisbee, math talks, song contest tryouts, what could be better?

During the entire weekend, keep in mind that you *represent the Lehigh Valley ARML team* and that your actions reflect upon both yourself and the rest of the team as well. Treat members of the other teams with respect. Show good sportsmanship. Always follow the rules of the contest, even if there is an opportunity to cheat without getting caught. It is better to lose with honor than to win with no integrity.

We always have a mix of young and old students, and experienced students who have been to ARML at Penn State before and students who are there for their first time. If you are an experienced student, keep an eye out for your younger or less experienced teammates during the trip, especially with regard to navigating around campus. Also everyone should always remember that *safety comes first* when deciding on a game or activity.

In conclusion, as a member of the Lehigh Valley ARML team you are part of a legacy of excellence in mathematics forged by friendships and hard work of our students. By following the suggestions above, you will hopefully find your ARML experience to be equally rewarding and help to continue that legacy in the future.