Math 479 – Fall 2022: Training Workouts

Welcome to Math 479, where great mathematicians are born!  I will post intriguing activities, course announcements, and other exciting information here throughout the semester.  Check back frequently. Below are links to some resources we will be using in the course.

• Work the 10 problems In Training Workout #1 in AiM. No calculators or googling are allowed. If there is a math term or problem statement that you don’t understand you can ask me about it. You can ask questions about the problem statement, but not about the solution. Focus not only on getting the correct answers, but also coming up with the most clever elegant way to obtain those answers. This is Problem Solving after all!
• Read: both of the Prefaces in the textbook (pg ix-xiii)

• USAMTS!

  Start Working on USAMTS Year 34 Round 1! This is it, team! The first actual official math competition of the season!

  • Get the USAMTS Year 34 Round 1 Problems!
  • Read all of the problems carefully now! That will load them into your brain and allow ideas to simmer to the surface.
  • Start working on them now.
  • Write up a brief (one or two paragraphs per problem) initial progress report to hand in on Tuesday in class. For each of the five problems you should give your initial impressions, and discuss which strategies you think might be helpful as a possible initial approach. If you solve any, don’t hand in your solution yet, just discuss what strategies, if any, you used.
  • Be patient! Be confident! Don’t give up easily!!
  
  

  Rules: This will be a major consideration in determining your grade in this course, similar to a take-home exam (except that you have more than five weeks to work on it!). We will follow the contest rules stated at the USAMTS website. Read them carefully! You can use reference books, online math references (that don’t involve asking another human being), calculators, Maple, spreadsheets, Geogebra, mechanical models, rulers, protractors, etc. to solve these problems. What you cannot do is discuss the problem with anyone either directly or indirectly except for each other and me. You can ask me to explain Playbook facts, general mathematical terminology, theorems, and any questions about LaTeX. I probably won’t say much about the problems themselves.

  These five problems will be due on Tuesday, October 11, 2020 at 10pm Eastern time by placing a pdf of your solutions in your Dropbox folder (we don’t have class that day because it is Fall Break). Note that you should hand these solutions in to me, not submit them to the USAMTS competition! Undergraduates are not eligible for this contest, only high school students. Neatness counts for this training workout. LaTeX solutions are required. You should use the official LaTeX template provided at the bottom of that page at the USAMTS website. I will answer any LaTeX or Overleaf questions you have. I will try to provide you with a LaTeX template for answering Problem number one. Including figures or diagrams from Geogebra or Asymptote to help your exposition is highly encouraged.

  I will grade it according to the rules of the USAMTS contest. In particular, rule #14 says “Each solution will be given a score of 0, 1, 2, 3, 4, or 5. Solutions will be graded both on mathematical correctness and on writing style.” No solutions will be graded before the due date of October 11.

  Game on!! Go Scranton Mathletes!

• Work the 10 Problems in Training Workout #2 on AiM. No calculator, googling, or other software is allowed. If there is a math term or problem statement that you don’t understand you can ask me about it. Focus not only on getting the correct answers, but also coming up with the most clever elegant way to obtain those answers. This is Problem Solving after all!
• Read Section 1.1 in the textbook.
• Continue to work on USAMTS!.

• Work the 8 Problems in Training Workout #3 on AiM. No calculator, googling, or other software is allowed.

  This is from a high school contest, not a middle school contest, so the problems are a little more advanced than those you have done so far.
• Read Chapter 1.2 in Zeitz. In particular, read carefully Example 1.2.1. This is a very instructive and enlightening example!
• Type up all of the Must Know facts in an Overleaf document using our course LaTeX template linked to below. If you want to include geometric figures you can draw the by hand and just upload a scan, or plot them in geogebra, or use any software you like to draw them. They don’t have to be perfect. Put the result in Dropbox.
• Keep working on USAMTS.

• Work the 10 Problems in Training Workout #4 on AiM. No calculator, googling, or other software is allowed.

• Continue working on USAMTS.
• Read Chapter 2.1 in Zeitz.

• Work the 25 Problems in Training Workout #5 on AiM. No calculator, googling, or other software is allowed. These are from a tenth grade contest.
• Work on USAMTS. Put a new brief progress report in Dropbox explaining what additional progress you have made since your last progress report.
• Read Chapter 2.1 in Zeitz (if you haven’t already).

• Work the 8 Problems in Training Workout #6 on AiM. No calculator, googling, or other software is allowed. These are from a high school contest.
• Start reading section 2.2 in Zeitz.
• Continue working on USAMTS.

• Work the 25 Problems in Training Workout #7 on AiM. No calculator, googling, or other software is allowed. These are from a high school contest.
• Continue reading section 2.2 in Zeitz if you haven’t finished it yet.
• Continue working on USAMTS.
• Update your Must Know facts Overleaf document. – don’t make a new Overleaf document, just update the last one you shared with me.

• Work the 8 Problems in Training Workout #8 on AiM. No calculator, googling, or other software is allowed. These are from a high school contest.
• Finish reading section 2.2 in Zeitz if you haven’t finished it yet.
• Continue working on USAMTS.

Proof time!

• Answer the Logical Prairie Dogs question that I put in your Dropbox folder. Write up your solutions in Overleaf and place them in Dropbox as usual (no need to share your Overleaf project with me). As usual, no calculator, googling, or other software is allowed.
• Finish solving the Alice-Bob game we started in class – if you get a solution you can present it to the class on Tuesday. I also put a copy of that question in Dropbox.
• Continue working on USAMTS. – we are getting close to the due date, so you should really try to write up and solve as many as you can. This is the equivalent of a ‘term paper’ project, or take-home exam, and will be counted as such.

• Work the 8 Problems in Training Workout #10 on AiM. No calculator, googling, or other software is allowed. These are from a high school contest.
• Finish solving the signed sum of the first $m$ squares problem that you worked on in class. You can collaborate with your teammate.
• Start reading section 2.3 in Zeitz.
• Continue working on USAMTS.

• USAMTS!

  Hand in USAMTS Year 34 Round 1! This is it, team! The first actual official math competition of the season!

  
  

  Rules: This will be a major consideration in determining your grade in this course, similar to a take-home exam. We will follow the contest rules stated at the USAMTS website. Read them carefully! You can use reference books, online math references (that don’t involve asking another human being), calculators, Maple, spreadsheets, Geogebra, mechanical models, rulers, protractors, etc. to solve these problems. What you cannot do is discuss the problem with anyone either directly or indirectly except for each other and me. If you collaborate with another student, attribute them at the start of your solutions to the relevant problem.

  These five problems are due on Tuesday, October 11, 2022 at 10pm Eastern time by placing a pdf of your solutions in your Dropbox folder (we don’t have class that day because it is Fall Break). Name your file: $$\mathtt{USAMTS~Round~1~Final~\text{-}~yourname.pdf}$$ You can share your Overleaf project with me too, but to be sure I’m seeing the correct version, put a copy of the pdf in Dropbox too. Note that you should hand these solutions in to me, not submit them to the USAMTS competition! Undergraduates are not eligible for this contest, only high school students. Neatness counts for this training workout. LaTeX solutions are required. You should use the official LaTeX template I provided to you by email. Neatness counts. Including figures or diagrams from Geogebra or Asymptote to help your exposition is highly encouraged but not required.

  I will grade it according to the rules of the USAMTS contest. In particular, rule #14 says “Each solution will be given a score of 0, 1, 2, 3, 4, or 5. Solutions will be graded both on mathematical correctness and on writing style.” No solutions will be graded before the due date of October 11 at 10pm.

  Game on!! Go Scranton Mathletes!

• Work the 10 Problems in Training Workout #11 on AiM. No calculator, googling, or other software is allowed. These are from a middle school contest.
• Finish reading section 2.3 and start reading Section 3.1 in Zeitz.

• USAMTS!

  Start Working on USAMTS Year 34 Round 2! This is it, team! The second round, featuring even more Grogg and cookies!

  • Get the USAMTS Year 34 Round 2 Problems!
  • Read all of the problems carefully now! That will load them into your brain and allow ideas to simmer to the surface.
  • Start working on them now.
  • Write up a brief (one or two paragraphs per problem) initial progress report to hand in on Thursday before class. For each of the five problems you should give your initial impressions, and discuss which strategies you think might be helpful as a possible initial approach. If you solve any, don’t hand in your solution yet, just discuss what strategies, if any, you used.
  • Be patient! Be confident! Don’t give up easily!!
  
  

  Rules: Like the last round, this will be a major consideration in determining your grade in this course, similar to a take-home exam (except that you have more than five weeks to work on it!). We will follow the contest rules stated at the USAMTS website. Read them carefully! You can use reference books, online math references (that don’t involve asking another human being), calculators, Maple, spreadsheets, Geogebra, mechanical models, rulers, protractors, etc. to solve these problems. What you cannot do is discuss the problem with anyone either directly or indirectly except for each other and me. You can ask me to explain Playbook facts, general mathematical terminology, theorems, and any questions about LaTeX. I probably won’t say much about the problems themselves.

  These five problems will be due on Monday, November 28, 2022 at 10pm Eastern time by placing a pdf of your solutions in your Dropbox folder. Note that you should hand these solutions in to me, not submit them to the USAMTS competition! Undergraduates are not eligible for this contest, only high school students. Neatness counts for this training workout. LaTeX solutions are required. You should use the LaTeX template at the link that I sent you by email since that has a diagram for the first problem. I will answer any LaTeX or Overleaf questions you have. Including figures or diagrams from Geogebra or Asymptote to help your exposition is highly encouraged.

  I will grade it according to the rules of the USAMTS contest. In particular, rule #14 says “Each solution will be given a score of 0, 1, 2, 3, 4, or 5. Solutions will be graded both on mathematical correctness and on writing style.” No solutions will be graded before the due date of November 28.

  Game on!! Go Scranton Mathletes!

VTRMC!

• Participate in this year’s VTRMC as a member of this year Scranton Math Team! See you Saturday, October 22, in LSC 233 from 8:45am-11:30am (plus lunch!). Game on!

• Continue working on USAMTS Round 2!

• Work the 10 Problems in Training Workout #12 on AiM. No calculator, googling, or other software is allowed. These are from a high school contest.
• Finish reading Section 3.1 in Zeitz.
• Continue working on USAMTS Round 2.
• Continue saving the Gnomes from their latest predicament.

• Work the five Problems in Training Workout #13 on AiM. No calculator, googling, or other software is allowed. These are from a high school contest.
• Start reading Section 3.2 in Zeitz.
• Continue working on USAMTS Round 2. Try to solve at least one of the five problems and put a draft of your progress in Dropbox.

• Work the eight Problems in Training Workout #14 on AiM. No calculator, googling, or other software is allowed. These are from a high school contest.
• Finish reading Section 3.2 in Zeitz.
• Continue working on USAMTS Round 2. No, seriously, work on USAMTS Round 2!

• Work the twelve Problems in Training Workout #15 on AiM. No calculator, googling, or other software is allowed. These are from an undergraduate contest.
• Start reading Section 3.3 in Zeitz.
• Continue working on USAMTS Round 2.

• Work the eight Problems in Training Workout #16 on AiM. No calculator, googling, or other software is allowed. These are from a high school contest.
• Finish reading Section 3.3 and start Section 3.4 in Zeitz.
• Continue working on USAMTS Round 2. No, seriously, work more on USAMTS Round 2!

• Answer Pima’s Broken Calculator Problem – problem #4 on the 2020 Prove it! Admissions test. Put your write-ups in Dropbox in pdf format.
• Finish reading Section 3.4 in Zeitz.
• Continue working on USAMTS Round 2.

• Putnam Practice! Answer (with proof!) the following question. No calculator, googling, or other software is allowed. This is a Putnam problem. Seven University of Scranton mathletes earned 10 points on the Putnam that year!
A1: Given a positive integer $n$, what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same?
[When $n=8$, the example $\{1,2,3,6\}, \{4,8\}, \{5,7\}$ shows that the largest $k$ is at least 3.]
• Finish reading Section 4.1 in Zeitz.
• Continue working on USAMTS Round 2. No, seriously, work more on USAMTS Round 2! Time’s almost up!

• USAMTS!

  Hand in USAMTS Year 34 Round 2! This is it, team! The third actual official math competition of the season!

  
  

  These five problems are due on Monday, November 28, 2022 at 10pm Eastern time by placing a pdf of your solutions in your Dropbox folder (we don’t have class that day because it is a Monday). Name your file: $$\mathtt{USAMTS~Round~2~Final~\text{-}~yourname.pdf}$$ You can share your Overleaf project with me too, but to be sure I’m seeing the correct version, put a copy of the pdf in Dropbox too. Note that you should hand these solutions in to me, not submit them to the USAMTS competition! Undergraduates are not eligible for this contest, only high school students. Neatness counts for this training workout. LaTeX solutions are required. You should use the official LaTeX template I provided to you by email. Including figures or diagrams from Geogebra or Asymptote to help your exposition is highly encouraged but not required.

  I will grade it according to the rules of the USAMTS contest. In particular, rule #14 says “Each solution will be given a score of 0, 1, 2, 3, 4, or 5. Solutions will be graded both on mathematical correctness and on writing style.” No solutions will be graded before the due date of November 28 at 10pm.

  Game on!! Go Scranton Mathletes!

• Putnam Practice! Answer the following Putnam problem. No calculator, googling, or other software is allowed. No Scranton student earned any points on this problem when the contest was offered, so you can be the first!
A2: Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many $n$ such that Bob has a winning strategy.
[For example, if $n = 17$, then Alice might take $6$ leaving $11$; then Bob might take $1$ leaving $10$; then Alice can take the remaining stones to win.]
• Read Section 4.2 in Zeitz.

• Take the 2022 Putnam contest from 10:00am-6:00pm, Saturday, Dec 3, 2022 in the LSC 233. Just bring pencils, pens, and erasers. Get sleep the night before. Hydrate. Don’t eat anything strange for breakfast. Study your Must-Know facts. Study your problem solving strategies and plan a mental checklist for using them if you get stuck. Lunch is on the math department at a local restaurant. This counts as Part I of your Final Exam. Game on!

• Finalize your Must Know Facts document and place a final copy of it in pdf form in Dropbox by the start of class on Thursday.
• Know the Must-Know Facts and be prepared to solve problems quickly using them in class on Thursday.
• Finish Reading Section 4.3 in Zeitz.