2015-2016 Putnam preparation

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2016 Spring

During the Spring semester we plan to meet Fridays 1-3 pm, most likely starting with the second week of classes.

For details, please see the Spring 2016 activities link on the MENTOR site.

BTW, a place you can use anytime you have some time to spend on campus is MUSL, the Mathematics Undergraduate Student Lounge.

2015 Fall

Nov. 20, noon-2pm, AH 10 (underground)

As usual, two sessions, one from noon and one from 1pm.
Discussed the problems on the test and applications of the arithmetic/geometric/harmonic mean inequality (see this Wikipedia page). Among the proofs on that page, there is one by forward-backward induction, due to Cauchy.
For the harmonic mean, see http://www.math.washington.edu/~dumitriu/Inequalities (which also contains the Generalized Means Inequality). The inequality between the geometric and harmonic mean is obtained by applying the inequality between the arithmetic and geometric means to the inverses: for \(x_i \gt 0\) \[ \frac{ \frac{1}{x_1} + \frac{1}{x_2} + \dots +\frac{1}{x_n} }{n} \ge \left( \frac{1}{x_1} \cdot \ldots \cdot \frac{1}{x_n} \right)^{1/n} \] In all these, equality holds if and only if all the variables are equal.

Nov. 13, noon-2pm, AH 10 (underground)

We will have a short practice test (2-3 problems).
The test lasts one hour and will be administered twice, once from noon and once from 1pm. If you cannot make either of these times, please let us know and we will try to find another option.
See the e-mail you received on Monday about more details.

Nov. 6, noon-2pm, AH 10 (underground)
   Undergraduate Colloquium talk & discussion

Krešimir Josić, UH: What can you believe when the truth keeps changing?

Oct. 30, noon-2pm, AH 10 (underground)

As usual, two sessions, one from noon and one from 1pm.
We discussed problems 95 and 96 from the Putnam and Beyond book.
Look at the other problems from section 2.1.2, Inequalities.

Oct. 23, noon-2pm, AH 10 (underground)

We discussed proofs by contradiction. Here are some details:

• problems
• solution to #2
• rational roots of polynomials with integer coefficients,
  solution to #3 (using the above; or, prove that again for this case)
• solution to #4

Oct. 16, noon-2pm, AH 10 (underground)

• There will be two one-hour sessions, one from noon, the second from 1pm. Come to the one you prefer.
• Continued with induction.

Oct. 9, noon-2pm, AH 10 (underground)

• There will be two one-hour sessions, one from noon, the second from 1pm. Come to the one you prefer.
• We discussed induction.
  Here is a snapshot of the problems (some actually can be solved more easily by other methods).

Oct. 2, noon-1:30pm, AH 10 (underground)

Continued with the Pigeon Hole Principle; see also more detailed proofs.
What was done, and proposed problems from the Putnam and Beyond book, are here.

Sep. 25, noon-2pm, 646 PGH.

Since for now there is not a time good for everyone, this Friday there will be two one-hour sessions, one from noon, the second from 1pm. Come to the one you prefer.

• The first book is about problem solving; for now we solve problems from the second.
  • Loren C. Larson: Problem solving through problems, Springer-Verlag 1983 or 1992, 332 pages.
    See a selection of problems from the first chapter of the 1983 edition.
  • Titu Andreescu, Razvan Gelca: Putnam and beyond, 2010, 814 pages; this is the library copy, there is an electronic version at UHD.
    Here is the list of corrections.
• Done in class: The Pigeon Hole Principle; solved the following problems (from the Putnam and Beyond book and other sources).
  • Example. Prove that every set of 10 two-digit (positive) integer numbers has two disjoint subsets with the same sum. [IMO 1972]
  • If 5 points are inside a circle of radius 1, then there are two of them at most \(\sqrt{2}\) apart.
    [Actually, this also is true for only 4 points in the circle.]
  • Example. Given nine points inside the unit square, prove that some three of them form a triangle whose area does not exceed \(\frac{1}{8}\).
  • 33. Given 50 distinct positive integers strictly less than 100, prove that some two of them sum up to 99.
    [This is actually incorrect, the problem meant to say "sum of at most two", instead of "sum of two"; or, replace 100 by 99.]
• Suggested problems
  • Let \(n>1\) and let \(P(x)\) be a polynomial with integer coefficients and degree at most \(n\). Suppose that \(|P(x)| \lt n\) for all \(|x| \lt n^2\). Show that \(P\) is constant.
  • Let \(A\) be a set of 20 distinct integers chosen from the arithmetic progression \(1, 4, 7, \dots, 100\). Prove that there must be two distinct integers in \(A\) whose sum is 104. [Putnam 1978, A1]
  • Given any \(n+2\) integers, show that there exist two of them whose sum, or else whose difference, is divisible by \(2n\).

Sep. 18, noon, 646 PGH. Pizza will be provided.

Orientation meeting. We will discuss possible times for the weekly Putnam preparation, and the EURE/REU option for those who participate in both the Putnam preparation/competiion and the Undergraduate Colloquium.

See here more details and a link to the slides from the meeting.

Putnam problems and solutions

• Here is the old "official" link with problems, solutions and scores; http://kskedlaya.org/putnam-archive/ and comments about its history (and future).
• For the problems with comments/solutions/etc., see this link Art of problem solving. Search the web for "art of problem solving putnam".
  Thanks to Nikhil for this link.

UH problem of the week (no new problems since Fall 2014)

• You can look at the UH Problem of the week (also http://www.facebook.com/uhmpow), which is aimed at graduate students but often suitable for Putnam as well.
  Undergraduate solvers may be eligible for prize money (see the web-site).