Dear ,
It's hard to believe that it's already 2020, and ARML is just four months away. Contest season is here: this week and next marks the AMC 10 and 12 contests––good luck, everyone! We'll then have AIME and NYSML at the beginning and end of March, respectively. I hope you enjoy the next set of practice problems (I found them delightful). Thanks again to the Problem Squad for their efforts.
– Japheth Wood
P.S.: To form a robust team, we need to connect with math teachers and their students across New York State. Please help us extend our subscribers list by sharing the sign up form.
P.P.S.: Save your contest scores! We'll soon share the application form for the Upstate New York Math Team.
Problems of the Month - February
Prepared by The Problem Squad (Matthew Babbitt and Brendan Caseria)
Each month we offer a set of problems to help you prepare for ARML. We'll send you the solutions a month later. If you solve a problem and want to share your solution, please email us!
We hope you enjoy these challenge and learn some unexpected mathematics in the process. Solve these with your team or on your own!
NEW! February 2020 Problems
January 2020 Problems
December 2019 Problems
November 2019 Problems
NEW! January 2020 Solutions
December 2019 Solutions
November 2019 Solutions
Please email us if you have helpful feedback for these problem sets!
Did you know that ARML expects your solution to be in a particular format?
Please read the ARML conventions here: ARML Conventions
Bonus Problem of the Month - February
This month's problem is offered by Matthew Babbit.
Prove that for any odd positive integer n, n12 − n8 − n4 + 1 is divisible by 29.
If you are reading this, and you have a math problem that you want to share, send an email to Matthew Babbitt! Include the problem, an answer, and a full solution to your problem.We will include it in a future newsletter, so the other readers have a chance to try it out.
Solution to the Bonus Problem of the Month from January
Submitted by Professor Mukkai Krishnamoorthy of the Albany Area Math Circle
A tetromino is a geometric shape composed of four squares connected orthogonally. Two tetrominoes are the same if they look the same under rotation or by flipping. There are five such tetrominoes as shown above.
The problem we would like to address is whether these five tetrominoes pack or cover a 4×5 rectangle, as shown below.
Solution: We color the 4×5 rectangle in R(ed) and B(lack) so that no two adjacent squares are colored the same. There will be equal number of Red as Black squares, each with 10. Now, we color the tetrominoes with the same constraint, such that adjacent squares are colored differently.
In the first four tetrominos there will be equal number of Red and Black colors. However, the last tetromino will have more Red squares than Black squares (or vice versa). Hence no matter how we color the tetrominoes (with the condition that adjacent tiles are colored different), there will be an unequal number of Red and Black squares. Hence 5 tetrominoes cannot cover the 4×5 rectangle, even though the area matches up.
Get ready for the AMC 10 and AMC 12 contests!
Good luck to those of you taking the AMC 10A and AMC 12A contests this week on Thursday. (The B version of these contests is next week on Wednesday.) We'll ask you for your scores* when you apply to be on the Upstate New York Math Team.
AMC 10/12A: Thursday, January 30, 2020
AMC 10/12B: Wednesday, February 5, 2020
The AMC 10 contests are for students up to and including grade 10, and the AMC 12 contests are for students up to and including grade 12. These timed, multiple-choice exams cover high school mathematics excluding Calculus.
Ideally, your school will offer the AMC contests. Additional locations are listed on the AMC webpage.
*We will also ask you for your scores on NYSML and other contests.
We still need a logo. Can you help?
Perhaps you can start with this silhouette.
Upcoming Contests
To learn what ARML is like, and to be prepared, we recommend looking into other contests in the meantime. Here's a short list of upcoming math contest opportunities nearby. Some are free, but most charge a reasonable fee or request a donation.
Purple Comet (Online). This team-based competition can be done wherever and whenever teams meet, and is held April 21–30, 2020.
CMIMC (Travel). Held in Pittsburgh PA at Carnegie Mellon University, this competition seems to be a bit closer for those on the western side of New York. Contest date: February 1, 2020.
HMMT Online Tournament. This is an unofficial version of the HMMT February tournament, and so if students are unable to participate at the in-place event, they can simulate the rounds at home (or with their team).
SBIMC. The South Brooklyn Invitational Math Competition is a gentle ARML-style contest.
NYSML coaches have been invited to host/participate in NYSMLSeasonal competitions. These local contests include the NYSMLFall (November 16, 2019–December 8, 2019) and the NYSMLWinter (February 24, 2020–March 8, 2020) events. If you know of a NYSMLSeasonal event, please let us spread the word!
NYSML itself is the New York State Math League. Local leagues lead up to the spring championships, this year on March 28, 2020 in Ithaca. We'll ask Upstate New York Math Team applicants for their individual NYSML scores.