2011 Preparation

• For other details see this link.
• Until further notice
  • Unless there is a special reason, we will meet each Friday at 1pm, in 345 PGH (or, if not there, in another room on the same floor).
  • Since Nov. we meet 1-3pm.
• December 2
  • last meeting before the competition
• November 25
  • Thanksgiving, no meeting
  • HOWEVER, if you have questions we can talk between Nov. 28-Dec. 2, before Friday.
• November 18
  • We will meet 1-3 pm.
  • Here are the problems from the test last Friday.
    Try to solve the problems and write the solutions!
    Solutions will be posted later.
  • The competition is Saturday, December 3: part A 9-noon, part B 2-5.
    Location TBA (most likely in PGH). Lunch will be provided.
    One can attend only one of the sessions.
  • Note that you should only submit those notes that are useful for the solutions - you may lose points for incorrect statements.
  • We discussed more about polynomials: criteria for a polynomial with integer coefficients to have a rational root, Section 2.2.5 (the Eisenstein/Schönemann theorem - a sufficient condition for a polynomial with integer coefficients to be irreducible over the integers), and a few exercises.
• November 11
  • There will a test (about one-hour long, starting at 1pm) to select the team/substitutes and to practice for the real exam (which is December 3).
    [Being in the UH team makes no difference for the individual competition.]
  • If 1pm is not good for you, let us know.
• November 4
  • More about Viete's relations, problems involving polynomials.
  • A few problems:
    • for three real numbers a, b, c, TFAE:
      • a, b, c greater or equal to zero
      • a+b+c, a b + b c + a c, a b c greater or equal to zero
    • The above problem easily extends to more variables. The expressions in the second condition are the fundamental symmetric polynomials in three variables.
    • Theorem: any symmetric polynomial (in n variables) can be expressed (as a polynomial) with the fundamental symmetric polynomials (there are n of them).
    • B1 problems from 2004, 2007
    • polynomials in more variables: B1 problems from 2005, 2008
• October 21-28
  • More inequalities, formula for the sum of cubes of three numbers.
  • Viete's identities.
• October 21
  • More inequalities, formula for the sum of cubes of three numbers.
• October 14
  • We discussed a few problems from the Oct. 7 list.
  • We began solving a more complicated inequality.
    • The details are here.
    • Try to figure out the unfinished step (see the "Assignment" at the end).
    • The message is that for inequalities, calculus is a more powerful tool than algebra. Might not be always nice, but works.
  • Look at the proof of AM-GM in the book, it is also done with calculus.
• October 7
  • Unless there is a special reason, we will meet each Friday at 1pm, in 345 PGH (or, if not there, in another room on the same floor).
  • In case you did not do that already, please let us know by October 10 if you want to participate in the competition.
  • Whether you register or not for Putnam, you are welcome to the preparation.
  • We will continue with other topics from the Putnam and beyond book. [Unfortunately the library does not have a copy.]
  • Topics discussed today:
    The book is at http://www.springer.com/mathematics/book/978-0-387-25765-5; can download the "Front Matter" and read some of if for free.
    • Chapter 1: proof by contradiction, induction; forgot to mention the pigeon-hole principle
    • Section 2.1: inequalities (geometric mean vs. arithmetic mean - for short AM-GM, Cauchy-(Bunyakovski)-Schwarz)
      Note: there is indeed a proof of AG-GM using that the log function is concave down.
  • A few problems to apply the material (section, problems).
    The hints are written in white, to see them highlight that area.
    Notation: _ for subscripts, ^ for superscripts

    1.1:

    1 (try sqrt(2)+sqrt(3)), 2 (cannot be too complicated, since so few constraints are given)

    1.2:

    11, 12, 13 (hint: replace x_n with a+b), 18, 24 (it is "standard" to find the formula for F_n, can discuss this next time).

    1.3:

    33 (hint: how would you choose the numbers to avoid a sum of 99?)

    2.1.3:

    the proof discussed in class is more general; when does equality hold?
    103, 104 (hint: show first that sum of the squares is at least n)

    2.1.5:

    read the proof of AM-GM in the book, it is longer but more generally applicable;
    prove that "arithmetic mean ≥ harmonic mean" using Cauchy-Schwarz
    121 (this is a calculus problem);
    more to come

• September 30
  • The next meeting is Friday, 9/30, at 1pm, in 345 PGH (or, if not there, in another room on the same floor).
• September 26
  • The Loren Larson book, Problem solving through problems, 1983 edition, is now on reserve.
    • See http://library.uh.edu/record=b1353322~S11, or search Course Reserves for "Instructor name: Torok" (it is listed at a course I teach, the number should be Math 6397).
    • "This item can be found in Access Services on the first floor of the MD Anderson Library".
      It can be checked out for two days.
• September 23
  • As of 9/16, it looks that the next meeting is Friday, 9/23, at 1pm, on the third floor of PGH.
  • If this is not fine with you let us know; enter the information in the google spreadsheet (see below).
  • Material related to what was discussed in class, functional equations:
    • There are two files below. The first is with some problems about finding functions as we discussed in class on Sep. 23. The second file is with the solutions for those problems. Try to work on the problem for about 10-15 minutes and brainstorm see what ideas you have on it ... then if you feel it is going to work out please write it down and check the solution afterward. If the solution is different from yours then see what trick it used and try to understand it. If in 10-15 minutes you made no progress on the problem than read the solution carefully and try to get their technique. Please forward to Prof. Onofrei any possible questions you might have.
    • problems and solutions
• September 16
  • If you attended the Sep. 16 meeting, please mark this on the google spreadsheet.
  • If you don't know where the spreadsheet is, send an e-mail.
  • If you did not get the handout please send an e-mail; we'll find a way to get it to you.
  • Registering for the competition:
    • The registration sheet has to reach the organizers (at Santa Clara Univ., CA) by Oct. 13.
    • Only your name is needed.
    • The competition is administered here at UH.
    • There is no penalty if you are registered but cannot make one or both sessions of the competition. However, if you are not registered you might not be able to participate.
  • Tips for success:
    • Solving many problems is quite difficult. Therefore, start with the problem that looks most accessible.
      If you solve one problem in the morning and one in the afternoon (so 20 pts out of 120) you are already in a very respectable group.
    • Just as important as solving a problem is to write a complete solution. Practice this! Bring the solutions that you wrote so that we can comment.
    • Partial credit is given only for substantial progress toward a correct solution.
  • We discussed the A1 problems from 2010 to 2007. See all A1 problems from recent competitions.
    These are the first problems of the morning session, it is said that in general they are simpler.
  • Please write a full solution to the problems that you can, and bring (or e-mail) it, so that we can comment.
  • More about what was discussed on Sep. 16 could be posted later.
• September 13
  • To kick things off, let us meet this Friday, Sep. 16, at 1pm, on the 3rd floor of PGH (there should be a few free rooms there). If we are not there, please come to my office (672 PGH), I will leave a note on the door.
  • It is OK if you cannot make it at 1pm, we will be there for a while. If you cannot come at all, let us know and we'll arrange something else.
  • There is no need to "register" in any way, just show up on Friday.
• The rules (see http://math.scu.edu/putnam/rulescJan.html):

  RULES
  
  The competition is open only to regularly enrolled undergraduates, in colleges and universities of the United States and Canada, who have not yet received a college degree. No individual may participate in the competition more than four times. An eligible entrant who is also a high school student must be informed of this four time limit.
  A college or university with at least three registered entrants obtains a team rank through the positions achieved by three designated individual contestants.
  No collaboration or outside assistance is permitted during the examination. Each contestant, even if designated as a team member, must work independently on the examination questions.
  
  Please note that there are no provisions for "unofficial" entrants.
  
  The local supervisor must be a regular faculty member.