HOMEWORK MATH 409 (2022):     Solutions  available  here

• Week 14: Prepare for Putnam! Dec 2 - 6 (no class Friday):
  • Videos: Mon: https://youtu.be/bG3CPgrCGv4    Wed:   https://youtu.be/y71SFoiPcqI
  • Homework: Think about A1 problems, try to be ready to present something.
  • Homework: Think about Project Euler problems, try to be ready to present something.
  • Think about Reverse Tic-Tac-Toe (or Toe-Tic-Tac): In standard Tic-Tac-Toe, whomever gets three in a row (vertically, horizontall or diagonally) wins. A simple calculation (though tedious if you don't use symmetry to combine cases) shows that if both play optimally, the game will always end in a tie. Consider now \emph{Reverse Tic-Tac-Toe}, where the object is to force the other person to get three in a row. Does one of the players have a winning strategy (and if so who and what is it), or will the game always end in a tie if each play optimally?

   

• Week 13 and on: Prepare for Putnam! Nov 25:
  • Videos: Mon: https://youtu.be/FV2eWA71nMA
  • Homework: Think about A1 problems, try to be ready to present something.
  • Homework: Think about Project Euler problems, try to be ready to present something.
  • Think about Reverse Tic-Tac-Toe (or Toe-Tic-Tac): In standard Tic-Tac-Toe, whomever gets three in a row (vertically, horizontall or diagonally) wins. A simple calculation (though tedious if you don't use symmetry to combine cases) shows that if both play optimally, the game will always end in a tie. Consider now \emph{Reverse Tic-Tac-Toe}, where the object is to force the other person to get three in a row. Does one of the players have a winning strategy (and if so who and what is it), or will the game always end in a tie if each play optimally?
   
• Week 12 and on: Prepare for Putnam! Nov 18 - 22:
  • Videos: Mon: https://youtu.be/HEdt7OjSqWA   Wed:  https://youtu.be/1SlsUek0GqU  Fri:   https://youtu.be/aL2oOxoKZPY
  • Homework: Think about A1 problems, try to be ready to present something.
  • Homework: Think about Project Euler problems, try to be ready to present something.
  • Think about Reverse Tic-Tac-Toe (or Toe-Tic-Tac): In standard Tic-Tac-Toe, whomever gets three in a row (vertically, horizontall or diagonally) wins. A simple calculation (though tedious if you don't use symmetry to combine cases) shows that if both play optimally, the game will always end in a tie. Consider now \emph{Reverse Tic-Tac-Toe}, where the object is to force the other person to get three in a row. Does one of the players have a winning strategy (and if so who and what is it), or will the game always end in a tie if each play optimally?
   
• Week 11 and on: Prepare for Putnam! Nov 11 - 15:
  • Videos: Mon: No Class: Wed: https://youtu.be/uwvFu57JJBc     Fri:  https://youtu.be/GPX8m8KJBU0
  • Homework: Think about A1 problems, try to be ready to present something.
  • Homework: Think about Project Euler problems, try to be ready to present something.
  • Think about Reverse Tic-Tac-Toe (or Toe-Tic-Tac): In standard Tic-Tac-Toe, whomever gets three in a row (vertically, horizontall or diagonally) wins. A simple calculation (though tedious if you don't use symmetry to combine cases) shows that if both play optimally, the game will always end in a tie. Consider now \emph{Reverse Tic-Tac-Toe}, where the object is to force the other person to get three in a row. Does one of the players have a winning strategy (and if so who and what is it), or will the game always end in a tie if each play optimally?
  • Think about the PRIMEary Colors Problem: We define the coloring number of a graph of V vertices and E edges to be the smallest number of colors such that no two vertices that share an edge are colored the same. Consider the following graph. The vertices are the integers 2, 3, 4, 5, 6, …., 9,998, 9,999, and 10,000 Two vertices are connected by an edge if they share a common factor. So there isn’t an edge between 2 and 3, but there is between 2 and 6, another between 3 and 6, but none between 2 and 5, between 3 and 5, … Prove that the coloring number of this graph is at least 13. Problem created by Steven Miller as a practice test question for Princeton’s Discrete Math 341, Fall ’98.
   
• Week 10: Nov 4 - Nov 8 (no class on Friday: work on project)
  • Videos: Mon:  https://youtu.be/uhP6HusxCdc   Wed:  https://youtu.be/bLTh48U6VTg  Fri: No class, work on project
  • Putnam A1 Problems:
    • list: https://www.math.uh.edu/~torok/Putnam/problems_A1.pdf
    • possible problems: https://math.mit.edu/~rstan/myputnam.pdf
    • problems and solns (try to do without looking at soln first): https://www.math.uci.edu/~krubin/oldcourses/07.194/SolutionsA1B1.pdf
  • Homework for Wednesday: Think about A1 problems, try to be ready to present something.
  • Homework \#8: Due Wednesday November 8: \#1: There are four people who want to cross a bridge. They all begin on the same side. You have 17 minutes to get all of them across to the other side. It is night. There is one flashlight. A maximum of two people can cross at one time. Any party who crosses, either one or two people, must have the flashlight with them. The flashlight must be walked back and forth, it cannot be thrown, etc. Each walks at a different speed. A pair must walk together at the rate of the slower. Their times are 1, 2, 5 and 10 minutes to cross. How do you get them across in 17 minutes? \#2: Generalizing Conway's Checker Problem: Consider the three-dimensional version, where we again have an integer lattice and now have checkers at all points with coordinates (x,y,z) with z non-positive. We jump as before, but can now do north/south, east/west, and up/down. Prove there is a positive number N such that we cannot get a checker to a point (x,y,N) in finitely many moves. (Note: you can probably see how to generalize to arbitrarily many dimensions, the bound N will grow the dimensions, but the bound and the truth could be far apart). \#3: 1989A2 Evaluate \(\int_0^a \int_0^b \exp\left(\max\{b^2 x^2, a^2 y^2\}\right) dy dx\) where \(a, b\) are positive.
  • Due Tuesday November 12: Draft of 'midterm'
  • Homework: do two Project Euler problems you have not done yet and be prepared to present.
  • Homework: do two Project Euler problems you have not done yet and be prepared to present. It will not be before Wednesday November 13th.

   

• Week 9: Oct 28 - Nov 1 (no class Friday, work on paper)
  • Videos: Mon: https://youtu.be/pVV2oe-Ik-E  Wed: https://youtu.be/w_SBDcZY1Ac  Fri: no class, work on paper
  • Homework for Wednesday, October 30: Think about \(\int_0^\infty \sin(x)/x dx\).
  • Homework \#8: Due Friday November 8: \#1: There are four people who want to cross a bridge. They all begin on the same side. You have 17 minutes to get all of them across to the other side. It is night. There is one flashlight. A maximum of two people can cross at one time. Any party who crosses, either one or two people, must have the flashlight with them. The flashlight must be walked back and forth, it cannot be thrown, etc. Each walks at a different speed. A pair must walk together at the rate of the slower. Their times are 1, 2, 5 and 10 minutes to cross. How do you get them across in 17 minutes? \#2: Generalizing Conway's Checker Problem (worth 20 points): Consider the three-dimensional version, where we again have an integer lattice and now have checkers at all points with coordinates (x,y,z) with z non-positive. We jump as before, but can now do north/south, east/west, and up/down. Prove there is a positive number N such that we cannot get a checker to a point (x,y,N) in finitely many moves. (Note: you can probably see how to generalize to arbitrarily many dimensions, the bound N will grow the dimensions, but the bound and the truth could be far apart). \#3: 1989A2 Evaluate \(\int_0^a \int_0^b \exp\left(\max\{b^2 x^2, a^2 y^2\}\right) dy dx\) where \(a, b\) are positive.
  • Homework: do two Project Euler problems you have not done yet and be prepared to present.

   

• Week 8: Oct 21 - 25
  • Video: Mon:  https://youtu.be/-ZujtK5k-fM  Wed:  https://youtu.be/sH66AX3-YhM   (computer program   and pdf version) 
  • Video: Fri: Expanded version of the German Tank Problem (from Math/Stat 341: Probability: Fall 2021):

    • Lecture 06: 9/22/21: German Tank Problem I: Theory: https://youtu.be/APsubcDVl1s  (slides)

    • Lecture 08: 9/27/21: German Tank Problem II: Statistical Inference: https://youtu.be/JnaVkeO9qtc (slides)

  • Homework for Wednesday, October 23: Think about the bridge problem: in each hand of bridge each of four players get 13 cards, all hands are equally likely. How many hands do you expect to need before you see each card at least once?
  • Homework \#8: Due Friday November 8: \#1: There are four people who want to cross a bridge. They all begin on the same side. You have 17 minutes to get all of them across to the other side. It is night. There is one flashlight. A maximum of two people can cross at one time. Any party who crosses, either one or two people, must have the flashlight with them. The flashlight must be walked back and forth, it cannot be thrown, etc. Each walks at a different speed. A pair must walk together at the rate of the slower. Their times are 1, 2, 5 and 10 minutes to cross. How do you get them across in 17 minutes? \#2: Generalizing Conway's Checker Problem (worth 20 points): Consider the three-dimensional version, where we again have an integer lattice and now have checkers at all points with coordinates (x,y,z) with z non-positive. We jump as before, but can now do north/south, east/west, and up/down. Prove there is a positive number N such that we cannot get a checker to a point (x,y,N) in finitely many moves. (Note: you can probably see how to generalize to arbitrarily many dimensions, the bound N will grow the dimensions, but the bound and the truth could be far apart). \#3: 1989A2 Evaluate \(\int_0^a \int_0^b \exp\left(\max\{b^2 x^2, a^2 y^2\}\right) dy dx\) where \(a, b\) are positive.
  • Homework: do two Project Euler problems you have not done yet and be prepared to present.
   
• Week 7: October 14 - 18, 2024
  • Video: Wed: https://youtu.be/weGmZ3YmRqE
  • Readings on monovariants:
    • Theory and examples: http://www.mit.edu/~pengshi/math149/talk2.pdf
    • Examples and problems: http://www.math.kent.edu/~soprunova/64091s15/monovariants15.pdf
    • Problems: http://math.stanford.edu/~vakil/putnam07/07putnam5.pdf
  • For Monday October 21: think about which is larger: \(e^\pi\) or \(\pi^e\). You are NOT allowed to use a computer to calculate anything; try to prove elementarily which wins. Let me know if there are any HW / Project Euler problems you would like to discuss in class on Monday October 21.
  • Remember no class on Friday October 18: Use the time to work on your project.
  • Homework \#6: Due Friday, October 18 (even if mountain day): \#1: Let \(a_1, a_2, \dots, a_n\) be positive integers. Show a subset sums to a multiple of \(n\). \#2: Given any \(n\), show there is a number \(x_n\) whose digits are only 0's and 7's such that \(n\) divides \(x_n\). \#3: Consider the previous problem. Find such a number for \(n=2017\); what is the smallest such number? \#4: Show that if \(n\) divides a Fibonacci number that it divides infinitely many Fibonacci numbers. \#5: For all positive real numbers \(a, b, c\) show that \(a^a b^b c^c >= a^b b^c c^a\).
   
• Week 6: October 10 - 14
  • Mon: https://youtu.be/vWYXyXN6YSE  Wed: https://youtu.be/F2_h4cwwuKg   Fri: No class, watch: 3/1/2017: Irrationality Proofs, Morley's Theorem, Pigeon Hole Problems: https://youtu.be/xx8f8z8zzfQ
  • Readings on monovariants:
    • Theory and examples: http://www.mit.edu/~pengshi/math149/talk2.pdf
    • Examples and problems: http://www.math.kent.edu/~soprunova/64091s15/monovariants15.pdf
    • Problems: http://math.stanford.edu/~vakil/putnam07/07putnam5.pdf
  • Read some of the Geometry handouts:
    • College geometry book: http://teachers.sduhsd.net/mchaker/Chakers_WebSite!!!/Honors_Geometry_files/College%20Geometry.pdf
    • List of useful geometry results for math contests: http://www.geometer.org/mathcircles/geometry.pdf
    • Geometry competition handout and problems: Part I: http://www.mit.edu/~alexrem/MC_Geometry.pdf and Part II: http://www.mit.edu/~alexrem/MC_Geometry2.pdf
    • Some geometry problems: https://sumo.stanford.edu/pdfs/smt2014/geometry-problems.pdf
  • Homework \#5: Due Friday Oct 11 (even if mountain day): (0) Show that no matter what 5 points are chosen on the surface of a unit sphere, there is at least one closed hemisphere containing at least 4 of the points. (1) Prove the law of cosines: if a, b and c are the sides of a triangle and theta is the angle between a and b, then c^2 = a^2 + b^2 - 2 a b cos(theta). (2-21) Complete the first 20 Project Euler Problems, and include in your HW a screenshot showing that you have completed all of these. Note this problem is worth 200 points (20 questions), and is thus giving you credit for all the work you have been doing. We will spend a future class discussing the coding and these problems, so let me know in advance ones you find particularly interesting. Homework (optional): Geometry problems typically invoke extreme reactions: some love, and some hate. If you like geometry problems look at the resources above, and choose 1-2 problems to do and submit. You may use these as HW exemptions for problems in future weeks (i.e., if you get full credit on either of these, you can skip a future problem and receive full credit).
  • Homework \#6: Due Friday, October 18 (even if mountain day): \#1: Let \(a_1, a_2, \dots, a_n\) be positive integers. Show a subset sums to a multiple of \(n\). \#2: Given any \(n\), show there is a number \(x_n\) whose digits are only 0's and 7's such that \(n\) divides \(x_n\). \#3: Consider the previous problem. Find such a number for \(n=2017\); what is the smallest such number? \#4: Show that if \(n\) divides a Fibonacci number that it divides infinitely many Fibonacci numbers. \#5: For all positive real numbers \(a, b, c\) show that \(a^a b^b c^c >= a^b b^c c^a\).
   
• Week 5: Sept 30 - Oct 4 (no class Friday)
  • Videos: Mon: https://youtu.be/vCjXRTuFbFQ  Wed: https://youtu.be/cgpxfVRE-vQ  Fri: No class, watch From M&Ms to Mathematics, or, How I learned to answer questions and help my kids love math,  University of Michigan, pre-100k Step Preparation, October 4, 2024 (video: https://youtu.be/-plcuC64a6E) (slides:  pdf )  
  • Read some of the Geometry handouts:
    • College geometry book: http://teachers.sduhsd.net/mchaker/Chakers_WebSite!!!/Honors_Geometry_files/College%20Geometry.pdf
    • List of useful geometry results for math contests: http://www.geometer.org/mathcircles/geometry.pdf
    • Geometry competition handout and problems: Part I: http://www.mit.edu/~alexrem/MC_Geometry.pdf and Part II: http://www.mit.edu/~alexrem/MC_Geometry2.pdf
    • Some geometry problems: https://sumo.stanford.edu/pdfs/smt2014/geometry-problems.pdf
  • Homework \#5: Due Friday Oct 11 (even if mountain day): (0) Show that no matter what 5 points are chosen on the surface of a unit sphere, there is at least one closed hemisphere containing at least 4 of the points. (1) Prove the law of cosines: if a, b and c are the sides of a triangle and theta is the angle between a and b, then c^2 = a^2 + b^2 - 2 a b cos(theta). (2-21) Complete the first 20 Project Euler Problems, and include in your HW a screenshot showing that you have completed all of these. Note this problem is worth 200 points (20 questions), and is thus giving you credit for all the work you have been doing. We will spend a future class discussing the coding and these problems, so let me know in advance ones you find particularly interesting. Homework (optional): Geometry problems typically invoke extreme reactions: some love, and some hate. If you like geometry problems look at the resources above, and choose 1-2 problems to do and submit. You may use these as HW exemptions for problems in future weeks (i.e., if you get full credit on either of these, you can skip a future problem and receive full credit).

   

• Week 4: Sept 23-27
  • Project Euler: Make sure you have done the first 15 problems by Friday.
  • Videos: Mon: https://youtu.be/u54ss-alhNc   Wed: https://youtu.be/pGixuGkTuQM  Fri:  https://youtu.be/eV1Kd0t0ewI

  • Optional Reading: Recurrences and determining the longest run of heads in tosses of a coin: https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020742.02p0021g.pdf 

  • Homework \#4: Due Friday Oct 4 (even if mountain day): \#1: Prove for \(a_i > 0\) that \((1+a_1) \cdots (1+a_n) \ge 2^n \sqrt{a_1 \cdots a_n}\). \#2: Prove for \(a, b > 0\) that \(a/b + b/a \ge 2\), both by using an inequality approach \emph{and} without using an inequality! \#3: Solve the double recurrence \(f_n = f_{n-1} + 3 g_{n-1}, g_n = -3 f_{n-1} + 9 g_{n-1}\). \#4: Define a set to be selfish if it contains its cardinality (i.e., its number of elements) as an element; thus \(\{1,3,5\}\) is selfish, while \(\{1,2,3,5\}\) is not. Find, with proof, the number of subsets of \(\{1,2,...,n\}\) that are minimal selfish sets (that is, selfish sets none of whose subsets are selfish; thus \(\{1,3,5\}\) is not minimal selfish as \(\{1\}\) is a subset). This is a Putnam problem..... Also, make sure you have done the first 15 Project Euler Problems.

   
• Week 3: Sept 16 - 20, 2024

  • Videos: Mon: https://youtu.be/3LzFI3cfJYI (video failed to record, listen to audio while looking at slides) Wed: https://youtu.be/0fTiXEQua7Q  Fri: https://youtu.be/p_yx7p8UOoc

  • Reading for Monday: read about Recurrence relations.

    • Baby bear introduction to solving recurrences: https://www.cs.duke.edu/~reif/courses/alglectures/skiena.lectures/lecture3.pdf

    • Papa bear: http://www.cs.columbia.edu/~cs4205/files/CM2.pdf

    • Read Appendix A:  The Average Gap Distribution for Generalized Zeckendorf Decompositions (with Olivia Beckwith, Amanda Bower, Louis Gaudet, Rachel Insoft, Shiyu Li and Philip Tosteson), the Fibonacci Quarterly (51 (2013), 13--27). pdf

    • Video I made with OIT about applications of recurrence relations to gambling in Vegas: https://www.youtube.com/watch?v=Esa2TYwDmwA

  • Homework \#3: Due Friday September 27: (1) Make sure you have done the first 10 problems on Project Euler by Monday September 30 (these will not be collected). (2) How many ways are there to cover a \(3 \times n\) board using just \(1 \times 2\) tiles? (3) What if now we have a \(2 \times 2 \times n\) box and just \(1 \times 1 \times 2\) tiles? (For fun, no need to write up, just email me: can you find the exact answer for n = 1000? 10,000? 100,000? 1,000,000? What is the highest you can go in one hour of computing on your device?) (4) Let a, b be positive integers each at least 2. Prove that the number of ways to tile an a x b x n box is given by a recurrence relation of finite depth and constant coefficients; can you bound the size of the coefficients or the depth of the relation?

   

• Week 2: Sept 9-13, 2024

  • Videos Mon: https://youtu.be/1KKfjTTGzys    Wed: https://youtu.be/fDaNheZa33k   Fri:  https://youtu.be/F6ZQFjIzfEQ  (video failed to record, listen to audio while looking at slides)

  • Some suggested coding problems you should try (click here for Mathematica notebook, and click here for a pdf).
  • Readings: Read up on induction. There are lots of good webpages with induction problems:

    • http://math.stackexchange.com/questions/294677/looking-for-induction-problems-that-are-not-formula-based

    • http://www.math.cornell.edu/~mec/2008-2009/ABjorndahl/ppmi.pdf

    • https://faculty.math.illinois.edu/~hildebr/213/inductionsampler.pdf

  • Homework \#2: Due Friday September 16: (1) Go to Project Euler (https://projecteuler.net/) and create an account for yourself, and solve the first problem. You do not need to submit this, just email me when done. (2) Read up on induction and do the following: If n lines are drawn in a plane, and no two lines are parallel, how many regions do they separate the plane into?  (3) Prove that \((1 − 1/4 )(1 − 1/9) \cdots (1 − 1/n^2) = (n+1) / 2n\).

• Week 1: September 6, 2024

  • Video: https://youtu.be/n4nIJ2Si8ho (slides)

  • Homework 1: To be emailed to me by 9am on Monday, September 9: Email me a short note (a paragraph suffices) on what you want to get out of this course, and what lesson  you learned from the graduation speech by Uslan (https://www.bestgraduationspeeches.com/michael-uslan-graduation-speech/). Full credit, 20/20, so long as you answer both questions on time.

  • Homework: To be done by start of class on Monday, September 9:

    • Read this short note on proof techniques: http://web.williams.edu/Mathematics/sjmiller/public_html/handouts/proofs.html 

    • Read the induction section: http://web.williams.edu/Mathematics/sjmiller/public_html/handouts/AppendixI.pdf

    • Create an account at Project Euler: https://projecteuler.net/

HOMEWORK MATH 409 (2022):     Solutions  available  here

• Week 10: Nov 7 - Nov 11
  • Homework for Wednesday: Think about A1 problems, try to be ready to present something.
  • Homework \#8: Due Wednesday November 9: \#1: There are four people who want to cross a bridge. They all begin on the same side. You have 17 minutes to get all of them across to the other side. It is night. There is one flashlight. A maximum of two people can cross at one time. Any party who crosses, either one or two people, must have the flashlight with them. The flashlight must be walked back and forth, it cannot be thrown, etc. Each walks at a different speed. A pair must walk together at the rate of the slower. Their times are 1, 2, 5 and 10 minutes to cross. How do you get them across in 17 minutes? \#2: Generalizing Conway's Checker Problem: Consider the three-dimensional version, where we again have an integer lattice and now have checkers at all points with coordinates (x,y,z) with z non-positive. We jump as before, but can now do north/south, east/west, and up/down. Prove there is a positive number N such that we cannot get a checker to a point (x,y,N) in finitely many moves. (Note: you can probably see how to generalize to arbitrarily many dimensions, the bound N will grow the dimensions, but the bound and the truth could be far apart). \#3: 1989A2 Evaluate \(\int_0^a \int_0^b \exp\left(\max\{b^2 x^2, a^2 y^2\}\right) dy dx\) where \(a, b\) are positive.
  • Homework: do two Project Euler problems you have not done yet and be prepared to present.
  • Homework: do two Project Euler problems you have not done yet and be prepared to present. It will not be before Monday November 14th.

   

• Week 9: Oct 31 - Nov 4
  • Homework for Monday, October 31: Think about \(\int_0^\infty \sin(x)/x dx\).
  • Homework \#8: Due Wednesday November 9: \#1: There are four people who want to cross a bridge. They all begin on the same side. You have 17 minutes to get all of them across to the other side. It is night. There is one flashlight. A maximum of two people can cross at one time. Any party who crosses, either one or two people, must have the flashlight with them. The flashlight must be walked back and forth, it cannot be thrown, etc. Each walks at a different speed. A pair must walk together at the rate of the slower. Their times are 1, 2, 5 and 10 minutes to cross. How do you get them across in 17 minutes? \#2: Generalizing Conway's Checker Problem: Consider the three-dimensional version, where we again have an integer lattice and now have checkers at all points with coordinates (x,y,z) with z non-positive. We jump as before, but can now do north/south, east/west, and up/down. Prove there is a positive number N such that we cannot get a checker to a point (x,y,N) in finitely many moves. (Note: you can probably see how to generalize to arbitrarily many dimensions, the bound N will grow the dimensions, but the bound and the truth could be far apart). \#3: 1989A2 Evaluate \(\int_0^a \int_0^b \exp\left(\max\{b^2 x^2, a^2 y^2\}\right) dy dx\) where \(a, b\) are positive.
  • Homework: do two Project Euler problems you have not done yet and be prepared to present.

   

• Week 8: Oct 24 - 28
  • Homework for Monday, October 31: Think about \(\int_0^\infty \sin(x)/x dx\).
  • Homework \#8: Due Wednesday November 9: \#1: There are four people who want to cross a bridge. They all begin on the same side. You have 17 minutes to get all of them across to the other side. It is night. There is one flashlight. A maximum of two people can cross at one time. Any party who crosses, either one or two people, must have the flashlight with them. The flashlight must be walked back and forth, it cannot be thrown, etc. Each walks at a different speed. A pair must walk together at the rate of the slower. Their times are 1, 2, 5 and 10 minutes to cross. How do you get them across in 17 minutes? \#2: Generalizing Conway's Checker Problem: Consider the three-dimensional version, where we again have an integer lattice and now have checkers at all points with coordinates (x,y,z) with z non-positive. We jump as before, but can now do north/south, east/west, and up/down. Prove there is a positive number N such that we cannot get a checker to a point (x,y,N) in finitely many moves. (Note: you can probably see how to generalize to arbitrarily many dimensions, the bound N will grow the dimensions, but the bound and the truth could be far apart). \#3: 1989A2 Evaluate \(\int_0^a \int_0^b \exp\left(\max\{b^2 x^2, a^2 y^2\}\right) dy dx\) where \(a, b\) are positive.
  • Homework: do two Project Euler problems you have not done yet and be prepared to present.

   

• Week 7: October 17 - 21, 2022
  • Readings on monovariants:
    • Theory and examples: http://www.mit.edu/~pengshi/math149/talk2.pdf
    • Examples and problems: http://www.math.kent.edu/~soprunova/64091s15/monovariants15.pdf
    • Problems: http://math.stanford.edu/~vakil/putnam07/07putnam5.pdf
  • For Wednesday: think about which is larger: \(e^\pi\) or \(\pi^e\). You are NOT allowed to use a computer to calculate anything; try to prove elementarily which wins.
  • For Friday: Prepare two project Euler problems so you can present one at the board. Email me what you want to do.

   

• Week 6: October 10 - 14
  • Mon: No Class;  Wed:    Fri: 
  • Readings on monovariants:
    • Theory and examples: http://www.mit.edu/~pengshi/math149/talk2.pdf
    • Examples and problems: http://www.math.kent.edu/~soprunova/64091s15/monovariants15.pdf
    • Problems: http://math.stanford.edu/~vakil/putnam07/07putnam5.pdf
  • Read some of the Geometry handouts:
    • College geometry book: http://teachers.sduhsd.net/mchaker/Chakers_WebSite!!!/Honors_Geometry_files/College%20Geometry.pdf
    • List of useful geometry results for math contests: http://www.geometer.org/mathcircles/geometry.pdf
    • Geometry competition handout and problems: Part I: http://www.mit.edu/~alexrem/MC_Geometry.pdf and Part II: http://www.mit.edu/~alexrem/MC_Geometry2.pdf
    • Some geometry problems: https://sumo.stanford.edu/pdfs/smt2014/geometry-problems.pdf
  • Homework \#6: Due Wednesday, October 19: \#1: Let \(a_1, a_2, \dots, a_n\) be positive integers. Show a subset sums to a multiple of \(n\). \#2: Given any \(n\), show there is a number \(x_n\) whose digits are only 0's and 7's such that \(n\) divides \(x_n\). \#3: Consider the previous problem. Find such a number for \(n=2017\); what is the smallest such number? \#4: Show that if \(n\) divides a Fibonacci number that it divides infinitely many Fibonacci numbers. \#5: For all positive real numbers \(a, b, c\) show that \(a^a b^b c^c >= a^b b^c c^a\).

   

• Week 5: Oct 3-7 (possibly no class Friday, if class it will be remote)
  • Read some of the Geometry handouts:
    • College geometry book: http://teachers.sduhsd.net/mchaker/Chakers_WebSite!!!/Honors_Geometry_files/College%20Geometry.pdf
    • List of useful geometry results for math contests: http://www.geometer.org/mathcircles/geometry.pdf
    • Geometry competition handout and problems: Part I: http://www.mit.edu/~alexrem/MC_Geometry.pdf and Part II: http://www.mit.edu/~alexrem/MC_Geometry2.pdf
    • Some geometry problems: https://sumo.stanford.edu/pdfs/smt2014/geometry-problems.pdf
  • Homework \#5: Due Wednesday Oct 12: (0) Show that no matter what 5 points are chosen on the surface of a unit sphere, there is at least one closed hemisphere containing at least 4 of the points. (1) Prove the law of cosines: if a, b and c are the sides of a triangle and theta is the angle between a and b, then c^2 = a^2 + b^2 - 2 a b cos(theta). (2-21) Complete the first 20 Project Euler Problems, and include in your HW a screenshot showing that you have completed all of these. Note this problem is worth 200 points (20 questions), and is thus giving you credit for all the work you have been doing. We will spend a future class discussing the coding and these problems, so let me know in advance ones you find particularly interesting.
  • Homework (optional): Geometry problems typically invoke extreme reactions: some love, and some hate. If you like geometry problems look at the resources above, and choose 1-2 problems to do and submit. You may use these are HW exemptions for problems in future weeks (i.e., if you get full credit on either of these, you can skip a future problem and receive full credit).

   

• Week 4: Sept 26-30
  • Project Euler: Make sure you have done the first 15 problems by Friday.

  • Read: Recurrence relations:

    • Read: Generating Functions Handout: This is from a book I wrote on probability. The first section is motivation, feel free to skim. Section 19.2 is the most important. Section 19.3  is more technical and included for completeness; we won't cover. Section 19.4 talks about convolutions -- we'll need the very beginning to analyze the Catalan numbers.

    • Read: Recurrence Relations Handout: Goes through the analysis of the double plus one strategy from roulette (this was the video I made with students from OIT).

    • Read: Recurrence Relations Handout Part 2: Goes through the algebra to solve recurrences.

    • Optional Reading: Recurrences and determining the longest run of heads in tosses of a coin: https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020742.02p0021g.pdf 

  • Homework \#4: Due Wednesday October 5: \#1: Prove for \(a_i > 0\) that \((1+a_1) \cdots (1+a_n) \ge 2^n \sqrt{a_1 \cdots a_n}\). \#2: Prove for \(a, b > 0\) that \(a/b + b/a \ge 2\), both by using an inequality approach \emph{and} without using an inequality! \#3: Solve the double recurrence \(f_n = f_{n-1} + 3 g_{n-1}, g_n = -3 f_{n-1} + 9 g_{n-1}\). \#4: Define a set to be selfish if it contains its cardinality (i.e., its number of elements) as an element; thus \(\{1,3,5\}\) is selfish, while \(\{1,2,3,5\}\) is not. Find, with proof, the number of subsets of \(\{1,2,...,n\}\) that are minimal selfish sets (that is, selfish sets none of whose subsets are selfish; thus \(\{1,3,5\}\) is not minimal selfish as \(\{1\}\) is a subset). This is a Putnam problem..... Also, make sure you have done the first 15 Project Euler Problems.

   

• Week 3: Sept 19 - 23, 2022

  • Reading for Monday: read about Recurrence relations.

    • Baby bear introduction to solving recurrences: https://www.cs.duke.edu/~reif/courses/alglectures/skiena.lectures/lecture3.pdf

    • Mama bear: http://www.webpages.uidaho.edu/~markn/395/pdf/rec-eq.pdf  (not working...)

    • Papa bear: http://www.cs.columbia.edu/~cs4205/files/CM2.pdf

    • Video I made with OIT about applications of recurrence relations to gambling in Vegas: https://www.youtube.com/watch?v=Esa2TYwDmwA

  • Homework \#3: Due Wednesday September 21: (1) Make sure you have done the first 10 problems on Project Euler by Monday September 26 (these will not be collected). (2) How many ways are there to cover a \(3 \times n\) board using just \(1 \times 2\) tiles? (3) What if now we have a \(2 \times 2 \times n\) box and just \(1 \times 1 \times 2\) tiles?

   

• Week 2: Sept 12-16, 2018
  • Videos Mon:  https://youtu.be/tax8BkrPB6A    Wed:  https://youtu.be/gUvX8r-nnbc   Fri:  
  • Some suggested coding problems you should try (click here for Mathematica notebook, and click here for a pdf).
  • Readings: Read up on induction. There are lots of good webpages with induction problems:

    • http://math.stackexchange.com/questions/294677/looking-for-induction-problems-that-are-not-formula-based

    • http://www.math.cornell.edu/~mec/2008-2009/ABjorndahl/ppmi.pdf

    • https://faculty.math.illinois.edu/~hildebr/213/inductionsampler.pdf

  • Homework \#2: Due Friday September 16: (1) Go to Project Euler (https://projecteuler.net/) and create an account for yourself, and solve the first problem. You do not need to submit this, just email me when done. (2) Read up on induction and do the following: If n lines are drawn in a plane, and no two lines are parallel, how many regions do they separate the plane into?  (3) Prove that \((1 − 1/4 )(1 − 1/9) \cdots (1 − 1/n^2) = (n+1) / 2n\).

• Week 1: September 9, 2017
  • Welcome slides for first class. Handout for 2017's first class.  Video: https://youtu.be/FXLv9yhxA20

  • Homework 1: To be emailed to me by 8pm on Sunday, September 11: Email me a short note (a paragraph suffices) on what you want to get out of this course, and what lesson  you learned from the graduation speech by Uslan (https://www.bestgraduationspeeches.com/michael-uslan-graduation-speech/). Full credit, 20/20, so long as you answer both questions on time.

  • Homework: To be done by start of class on Monday, September 12
    • Read this short note on proof techniques: http://web.williams.edu/Mathematics/sjmiller/public_html/handouts/proofs.html 
    • Read the induction section: http://web.williams.edu/Mathematics/sjmiller/public_html/handouts/AppendixI.pdf
    • Create an account at Project Euler: https://projecteuler.net/

HOMEWORK MATH 331 (2018):     Solutions  available  here

Kansas State Math Competition: http://www.math.ksu.edu/main/events/parker-mathcomp/