HOMEWORK

HOMEWORK MATH 331 (typically due Fri): Solutions available here

Please spend at least 1 hour EVERY night reading the material/looking at the proofs/making sure you understand the details.

Week 10: Apr 17-21, 2017
- Homework \#9: Due April 21, 2017. \#1: Find \(\int_{-\infty}^\infty dx/(1+x^4).\) If you're brave, replace \(4\) with \(2n\). \#2. Let \(f(z) = A(z) B(z)\) where \(A(z) = a_{-3} / z^3 + a_{-2} z^2 + a_{-1} / z + a_0 + a_1 z + \cdots\) and \(B(z) = b_0 + b_1 z + b_2 z^2 + b_3 z^3 + b_4 z^4 + b_5 z^5 + \cdots\). Find the residue of \(f\) at \(z = 0\). What is the residue of \(C(z) = B(z)^4 / (z - 3)^2\) at \(z = 2\)? At \(z = 3\)? \#3 Do not submit, but you are strongly encouraged to write up the Gaussian contour integral.
Week 9: Apr 10-14, 2017
- Homework NO written HW due to exam.
Week 8: Apr 3-7, 2017
- Homework for Monday, April 3: Think about \(\int_0^\infty \sin(x)/x dx\).
- Homework \#8: Due April 7, 2017: \#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: Mar 13-17, 2017
- Read: From the textbook (Famous Puzzles): Find a problem or section you enjoy, read about it, write me a short note by Wednesday on it and let me know if you are willing to present.
- For Monday: 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.
- Homework \#7: Due March 17, 2017: \#1, \#2, \#3, \#4 (counts as four problems): Show that any decomposition of \(N\) as a sum of Fibonacci numbers cannot have fewer summands than the Zeckendorf decomposition. Is there a monovariant that can help?
Week 6: Mar 6-10, 2017 (no class Monday March 6th to make up for remote class).
- Homework \#6: Due Friday, March 10: \#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: Feb 27-Mar 3, 2017
- 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 Monday, February 27 (we will discuss in class): 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.
- Homework: Due Friday, March 3rd: (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 Friday 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: Feb 20-24, 2017
- Project Euler: Make sure you have done the first 15 problems by Friday.
- Read: From the textbook (Famous Puzzles): Pages 11-13, 201-202 (for the brave: derive (7.10)), 296-297. If you want to discuss any parts of these in class, let me know.
- Read: Recurrence relations:
  - Read: Generating Functions Handout: This is from a book I'm writing 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 Friday, Feb 24, 2017: \#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: Feb 13-17, 2017 (no class Fri Feb 17; hand in HW to my mailbox by 3pm Fri)
- 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 Feb 17, 2017: (1) Make sure you have done the first 10 problems on Project Euler. (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 \(1 \times 2\) tiles?
Week 2: Feb 6-10, 2017
- Videos: Mon: https://youtu.be/5grmPRBBbaE
- Videos: Wed: Mathematica code (the pdf) Video 331: https://youtu.be/vPGjly_7QVw
- 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:
- Homework \#2: Due Friday February 10: (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 http://www.math.ucla.edu/~radko/circles/lib/data/Handout-142-159.pdf and do Problem \#1: 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: February 3, 2017
- Welcome slides for first class. Handout for first class.
- Video: Fri: https://youtu.be/VZhnU2OSWAY
- Homework: To be emailed to me by 8pm on Sunday, February 5: 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 (http://www.graduationwisdom.com/speeches/0018-uslan.htm). Full credit, 20/20, so long as you answer both questions on time.
- Homework: To be done by start of class on Monday, February 6:
  - 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/

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