Dear ,
As the year is coming to a close, I hope that you'll have a chance to think forward to the end of May, when we'll be traveling to ARML with a brand new Upstate New York Math Team. It's been long enough since the team has been active, that it's quite likely that no member of the 2020 team will have ever participated in ARML before.
If you are a potential team member, the next few months is critical for your preparation. The AMC 10 and 12 contests are coming up soon (see below for more information), and NYSML will follow soon after, at the end of March. We'll ask you for your scores on these contests and others, to help us select the best possible team for ARML.
We hope that you'll enjoy this latest collection of practice problems. You might find that the January problems fight back a little harder than you'd like, but I know that you'll have a real sense of satisfaction and accomplishment when you finally do solve them. Solve the problems or not, please be sure to read our solutions in the next newsletter. Seeing others' solutions can help fill in gaps in your mathematical knowledge, and help you develop your writing style.
I'm very grateful to the Problem Squad for preparing this set of practice problems. And a big thanks again to Professor Mukkai Krishnamoorthy of the Albany Area Math Circle, for sharing a very special math challenge problem with us.
Thanks again for doing your part to rebuild the Upstate New York Math Team, and a very happy holiday season to you and your family!
– Japheth Wood
Problems of the Month - January
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! January 2020 Problems
December 2019 Problems
November 2019 Problems
NEW! 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 - January
Our thanks to Professor Mukkai Krishnamoorthy of the Albany Area Math Circle for submitting another great problem!
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.
Have you a bonus problem that you'd like to share? Send an email to Matthew Babbitt! Include the problem, an answer, and a full solution to your problem.
Solution to the Bonus Problem from December
Submitted by Professor Mukkai Krishnamoorthy, Albany Area Math Circle
Problem: Solve for x (as a closed form expression. You may use logarithms, radicals, as well as algebraic numbers).
12x + 9x = 16x
(Adapted from IIT Entrance Examination)
Solution: Divide the above equation by 9x to get
(12/9)x + (9/9)x = (16/9)x
which simplifies to
(4/3)x + 1 = (4/3)2x
By substituting p for (4/3)x, we get p + 1 = p2.
The solutions of this quadratic equation are φ and 1-φ, where φ is the golden ratio (1+√5)/2.
Therefore, p = (4/3)x = φ and 1-φ.
Taking logarithms, we get x = log(φ)/log(4/3).
Since 1-φ is negative, that solution will not appear (because logarithm functions are not defined on negative numbers).
Get ready for the AMC 10 and AMC 12 contests!
If you are a student, we hope you'll go above and beyond our problem sets as you prepare for the upcoming AMC contests. 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.
New Award! The MAA has just announced the Maryam Mirzakhani AMC 10 Prize for top-scoring females on the AMC 10A. The top female scorer will receive the Maryam Mirzakhani AMC 10 Prize of $5,000. Also, the 5 top scoring females in each of the MAA sections will receive a Maryam Mirzakhani AMC 10 certificate. Please let us know if you win one of these honors! New York State includes the Metro New York section and also part of the Seaway section of the MAA. Learn more about Maryam Mirzakhani and her mathematics on the Tribute Page hosted by the American Math Society. This new prize is funded by Awesome Math Girls!
photo source: https://www.maa.org/sites/default/files/MaryamMirzakhani_Headshot.jpg
What is ARML?
The first ARML was in 1976, and was created out of the popularity and success of NYSML, the New York State Math League.
The American Regions Math League is an annual math competition that takes place on the first Saturday after Memorial Day. Teams of fifteen students (in grades 12 and below) compete in this day-long, multi-format event. There are four ARML sites across the USA, the closest being at Penn State. Considered by many as the "World Series of Math Competitions", ARML offers motivated students excellent exposure to math enrichment topics like Algebra, Combinatorics, Geometry and Number Theory, and camaraderie with math-loving peers across the region.
There has not been an Upstate NY Math Team at ARML for about five years, so re-establishing this tradition will re-invigorate the math education of Upstate NY students who are curious about math and in search of deeper mathematical challenges.
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.