An Introduction to ARML

1. An Introduction to ARML Paul Dreyer ARML Western Site Coordinator pauldreyer@gmail.com

2. Outline • A Brief History of Time ARML • ARML Round-by-Round • Team Round • Power Round • Individual Round • Relay Round XXXX

3. A Brief History of ARML • The New York State Mathematics League (NYSML) held its first contest in 1973 • Massachusetts competed in 1974 (and won) • 1976: Creation of Atlantic Regions Math League • 1984: Renamed American Regions Math League • Today: • 4 sites (UNLV, Penn State, Univ. of Iowa, Univ of GA) • 120+ teams from 38 states and Toronto • International teams from Taiwan, Hong Kong, the Philippines, Columbia, Turkey, Bulgaria, and Vietnam

4. Team Round • 10 questions, 20 minutes, one answer sheet per team • SAT approved calculators allowed • Last year for calculators, no question requires them • Strategies: • Organization is key! • Make sure each question is covered at least twice (or has two people working together) • Questions tend to increase in difficulty from #1 to #10 • No penalty for guessing • You will get three minute and one minute warnings from the proctor, do not wait until the last second to fill out the answer sheet!

5. Power Round • 60 minutes, long-form solutions, calculators allowed • Some problems are computational, some are proofs • From the instructions: To receive full credit the presentation must be legible, orderly, clear, and concise. Even if not proved, earlier numbered items may be used in solutions to later numbered items but not vice-versa. • Key points: • There is partial credit for some problems. Even if you do not finish a proof, clear, correct progress may pick up a point or two. • Read the entire round, even if you are only working on later problems. Earlier results often are used to prove later ones. • Same comments as in team round regarding organization. • Be sure you submit only one solution per problem.

6. Power Round Proof Methods • If a question asks to prove an “A if and only if B” statement, be sure to prove both directions of the implication (if A, then B and if B, then A) • Proof by Contradiction (proving “if A, then B” by showing “if not B, then not A”) • Example: Infinitude of primes. Assume there are a finite number of prime numbers p1, …, pn, with pn the largest prime.Consider the number: None of the primes divide N evenly, so N must also be prime. This contradicts the assumption that there are a finite number of prime numbers.

7. Power Round Proof Methods • Induction: Proving a statement true for all positive (or non-negative) integers: • Show the statement is true for k = 0 or 1 (depending) • Induction: Assume the statement is true for k = n – 1, show the statement is true for k = n. • Example:

8. Power Round Proof Methods • Strong induction: Assume the statement is true for all k < n. Show the statement is true for k = n. • Claim: All positive integers can be written as the sum of distinct, non-consecutive Fibonacci numbers. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …, Fm-1 + Fm = Fm+1 • True for k = 1? (Yep!) • Assume true for k < n. Let Fm be the largest Fibonacci number less than n. Then n = Fm + (n – Fm). (n – Fm) can be written as the sum of non-consecutive Fibonacci numbers, the largest of which is notFm-1 (otherwise, Fm + Fm-1 = Fm+1is not larger than n,contradicting the definition of Fm.

9. Individual Round • Four pairs of questions, 10 minutes per pair • General difficulty scale (by pair): easy-easy, easy-medium, medium-hard, medium-very hard • On later rounds, it may be best to concentrate on just one question

10. Relay Round • 5 squads of 3, six minutes per round, two rounds. • Each person gets a question, the second person in the squad uses the answer from the first question in their solution, which they pass to the third person to use in their problem. • Persons 2 & 3 should solve their problems (if possible) in terms of T, the number they will receive (TNYWR). • Only the third person’s answer matters in the scoring. • Two times for Person 3 to submit answers: at 3 minutes (worth 4 points) and at 6 minutes (worth 2 points). • ALWAYS PASS AN ANSWER AT 3 MINUTES! • Only pass an answer at 6 minutes if it has changed from your 3 minute answer.

11. Relay Round Example

12. Relay Round Example

13. Relay Round Example

14. Relay Round Example

15. Relay Answer Passing • Each sheet you pass may contain one answer, underlined if needed to clarify (e.g. 6 vs. 9) • You may pass one sheet at a time, but you may pass multiple times. • Therefore, if you have narrowed your answer down to a set of possible answers, pass each one (it is possible that only one of them makes sense in the next problem) • No other communication is allowed (e.g. do not crumple passed answers that cannot be used in your problem, or pass notes with answers, etc.)

16. Picking Your Relay Teams • Best (?) solution: • Order your teammates from 1 to 15. • Squad 1: 3, 2, 1. Squad 2: 6, 5, 4. Etc. • You want your top students together to maximize the chance of getting points on the round. • Inside each squad, putting the stronger students later gives them the opportunity to solve problems in terms of T and make better (?) educated guesses if needed.

17. Final Notes • Read the ARML rules, particularly “Some Mathematical Ideas Used in the Competition” • Work up a study sheet of formulas, theorems, etc. • The best practice for every round (particularly the power and relay rounds) is to do as many power and relay rounds as you can find. • Have fun!