2019
Putnam Competition

2019

- There
will be one final Putnam prep session before the contest: Tuesday,
Dec. 3 at 11:30 in HH 410. Dr. van Tuyl will go over the
McPutnam which was written this week.

- For those who couldn't make it, here is a copy of this year's McPutnam.

- The 2019 Putnam contest will be written on Dec. 7 from 10 - 1 and 3 - 6 in HH 104.
- The mini Putnam preparation, the McPutnam, will be held on Nov. 26
at 2:30 and 3:30 in HH 207. This is a one-hour test in the
style of the Putnam contest.

- There will be a Putnam session 11:30 - 12:20 on Tuesday, Nov. 5 in HH 207. We will look at problems on geometry. Take a look at 1998 B2, 2002 A2, 2008 B1, 2012 A1 and for extra fun, 2015 B4.

- There is a
Putnam session at 11:30 - 12:20 on Tuesday, Oct. 8 in HH 207.
We will look at the solutions to the two problems I posted
at the information session:

1. Given a 1 x 3 rectangle, make two straight cuts and reassemble as a square.

2. What is the probability that three random numbers chosen from [0,1] represent the side lengths of a triangle? Assume a uniform distribution on [0,1].

We will also look at some Putnam problems related to probability calculations. Take a look at: 2001 A2, 2002 B1 and 1993 B2. For extra fun, look at 1993 B3.

- Dr. van
Tuyl will lead a Putnam training class on this coming Wednesday
(Sept. 25) at 2:30 in HH 410. We will look at some counting and
combinatorial techniques that might by useful for the Putnam
competition. We'll go over some of the ideas in a work sheet
put together by Ravi Vakil found here:
http://math.stanford.edu/~vakil/putnam05/05putnam3.pdf
As well, we'll work on the following Putnam A1 problem from 2013:
*Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all 20 integers is 39. Show that there are two faces that share a vertex and have the same integer written on them.*

- Welcome back for another Putnam season! Our first meeting, an organizational meeting to answer some questions and explain what is going to happen to everyone, is Thursday. Sept. 12 at 11:30 in HH 312. I have been receiving a lot of email questions so let me answer some of the most common here. See you on Thursday!
- The Putnam is a North America-wide mathematics contest for undergraduates and has run approximately every year since 1938. The contest is 6 hours long and is written in two 3 hour sessions on the first Saturday in December. This year, that is Dec. 7 from 10 am - 1 pm and 3 pm- 6 pm. To write the entire Putnam exam you sit both sessions.
- Anyone who is eligible i.e. someone without an undergraduate degree who hasn't written the contest 4 times already, is welcome to write the contest and there is no fee for doing so. It is best if you know that you want to write to let us know so that we can put your name on the list of participants but even if your name is not on the list, if there are extra exams, you can write.
- There is no hiding the fact that this is a hard exam. Roughly half of participants get 0. The exam focuses on clever, math competition problems. There are 6 problems in each sitting and there are usually enough problems that are accessible to students even with only a first year background. The trick is that although the material may be elementary it often requires applying elementary techniques in clever ways.
- There are many websites where you can see examples of Putnam problems and their solutions; here is one: http://kskedlaya.org/putnam-archive/.
- At McMaster, we will run preparation sessions; more details at the organizational meeting. You do not have to attend the prep sessions in order to write the contest but it is a good idea to do some preparation. You can always talk to one of the people helping to organize the contest: they are me (Bradd Hart, hartb@mcmaster.ca), David Earn (earn@mcmaster.ca), Jonathan Dushoff and Adam van Tuyl.

2018

2017

Archive: 2016

2015

Archive: 2014

Archive: 2012

Archive: 2011

- The Putnam
contest will be written on Saturday, Dec. 1 in HH 104. The
test is in two parts: 10 am - 1pm and 3 pm - 6 pm. You write
both parts. Good luck!

- On Nov. 23,
we will take up the mini-contest; here is a copy
of that test.

- On Nov. 16,
we will hold a small, in-house Putnam-like contest to help choose
this year's Putnam team. More details soon.

- The third
problem session will occur on Nov. 9 in HH 312 with Drs. Dushoff
and Earn. Please look at 2003A1, 2010A1, 2005A2. If you are
interested in harder problems, you can also look at 2003A5 and
2007A3.

Here are two other problems for this session:

- You are
sent to the market to buy exactly f fruits. The market has t
types of fruit on offer (and at least f of each type). How many
ways can you choose how many of each fruit to buy?

- p points are placed randomly on a circle. All of the points are connected by lines. What is the largest number of regions that these lines could divide the interior of the circle into?
- The second
problem session will be Friday, Oct. 19 at 10:30 in HH 312. Drs.
Valeriote and van Tuyl will hold a problem session dedicated to
the pigeonhole principle and its uses. Please have a look at the
following problems: 1971 -A1, 1978 - A1, 2000- B1 and 2002 -A2 as
well as these additional problems.

- The first problem session will be Friday, Sept. 28 at 10:30 in HH 312. I will discuss some basic inequalities that are useful for the Putnam and we will look at three problems (time allowing):
- Show that
if x1,x2,…,xn are positive numbers then

(x1 + x2 + … xn)(1/x1 + 1/x2 + … + 1/xn) \geq n^2. - B2 from 2004
- B6 (!)
from 1975; if you can't find it, you can find it here.

- Welcome back for another Putnam season! Our first meeting, an organizational meeting to answer some questions and explain what is going to happen to everyone, is Friday. Sept. 14 at 10:30 in HH 312. I have been receiving a lot of email questions so let me answer some of the most common here. See you on Friday!
- The Putnam is a North America-wide mathematics contest for undergraduates and has run approximately every year since 1938. The contest is 6 hours long and is written in two 3 hour sessions on the first Saturday in December. This year, that is Dec. 1 from 10 am - 1 pm and 3 pm - 6 pm. To write the entire Putnam exam you sit both sessions.
- Anyone who is eligible i.e. someone without an undergraduate degree who hasn't written the contest 4 times already, is welcome to write the contest and there is no fee for doing so. It is best if you know that you want to write to let us know so that we can put your name on the list of participants but even if your name is not on the list, if there are extra exams, you can write.
- There is no hiding the fact that this is a hard exam. Roughly half of participants get 0. The exam focuses on clever, math competition problems. There are 6 problems in each sitting and there are usually enough problems that are accessible to students even with only a first year background. The trick is that although the material may be elementary it often requires applying elementary techniques in clever ways.
- There are many websites where you can see examples of Putnam problems and their solutions; here is one: http://kskedlaya.org/putnam-archive/.
- At McMaster, we will runpreparation sessions; more details at the organizational meeting. You do not have to attend the prep sessions in order to write the contest but it is a good idea to do some preparation. You can always talk to one of the people helping to organize the contest: they are me (Bradd Hart, hartb@mcmaster.ca), David Earn (earn@mcmaster.ca), Jonathan Dushoff, Adam van Tuyl and Matt Valeriote.

2017

- The Putnam contest is written on Saturday, Dec. 2 in two seatings: 10 am - 1 pm and 3 pm - 6 pm. If you are writing the contest, you write during both sessions. This year we are writing in BSB B155 and Dr. Earn will be officiating.
- In the week
of Nov. 20 we will have a small pre-Putnam contest at
McMaster. It will be only one hour long and will be roughly
the difficulty of standard A1 questions. If you are
available to write it on Wednesday, Nov. 22 at 11:30 in HH 410,
let me know. Otherwise, contact me and we will try to
arrange another time. Send me email at hartb@mcmaster.ca

- Fourth
session: Nov. 15 at 11:30 in HH 410. Take a look at B1, B2 and B3
from 2011.

- Third
session: Nov. 1 at 11:30 in HH 410. Take a look at A1, A2,
B1 and B2 from 2014.

- Second
session: Oct. 18 at 11:30 in HH 410. Given the roaring
success of the last session, we will go back one year in time and
look at the 2015 contest. Again, let's concentrate on A1,
A2, B1 and B2.

- First
Putnam session: Oct. 4 at 11:30 in HH 410. Many peope have
been asking what a Putnam test looks like so let's start by
looking at last
year's contest. We will concentrate on A1, A2, B1 and
B2 if you want to look at them in advance. See you there!

- Welcome back for another Putnam season! Our first meeting, an organizational meeting to answer some questions and explain what is going to happen to everyone, is Friday. Sept. 15 at 11:30 in HH 312. I have been receiving a lot of email questions so let me answer some of the most common here. See you on Friday!
- The Putnam is a North America-wide mathematics contest for undergraduates and has run approximately every year since 1938. The contest is 6 hours long and is written in two 3 hour sessions on the first Saturday in December. This year, that is Dec. 2 from 10 am - 1 pm and 3 pm - 6 pm. To write the entire Putnam exam you sit both sessions.
- Anyone who is eligible i.e. someone without an undergraduate degree who hasn't written the contest 4 times already, is welcome to write the contest and there is no fee for doing so. It is best if you know that you want to write to let us know so that we can put your name on the list of participants but even if your name is not on the list, if there are extra exams, you can write.
- There is no hiding the fact that this is a hard exam. Roughly half of participants get 0. The exam focuses on clever, math competition problems. There are 6 problems in each sitting and there are usually enough problems that are accessible to students even with only a first year background. The trick is that although the material may be elementary it often requires applying elementary techniques in clever ways.
- There are many websites where you can see examples of Putnam problems and their solutions; here are a couple: http://kskedlaya.org/putnam-archive/ and http://www.nepalimath.com/Pages/olympiad.aspx
- At McMaster, we will run preparation sessions this fall roughly every other week on Wednesdays at 11:30 in HH 410. The tentative dates of these sessions are: Oct. 4, Oct. 18, Nov. 1, Nov. 15 and Nov. 29. You do not have to attend the prep sessions in order to write the contest but it is a good idea to do some preparation. You can always talk to one of the people helping to organize the contest: they are me (Bradd Hart, hartb@mcmaster.ca), David Earn earn@mcmaster.ca, Jonathan Dushoff, Adam van Tuyl and Matt Valeriote.
- For the preparation sessions, two problems will be posted here in advance and students are encouraged to look at the problems and see what headway they can make. At the session itself, students will work in groups for 15-20 minutes in order to pool their approaches to one or both of the problems and then groups can volunteer attacks on the problems or potential solutions. The sessions will be facilitated by two of the organizers.

Archive: 2016

- Here are
the Putnam exam details: The exam will be held on Dec. 3 from 10
am to 1 pm and 3 pm until 6 pm. We have HH 305 reserved all
day for the contest. Please let me know if you are still
intending to write.

- Final
practice session (led by me): Wednesday, Nov. 30 at 11:30 in HH
410. Look at 1995, A1; 1997, A3 and B1; 1998, B2.

- Fourth
practice session (led by Drs. Earn and Dushoff): Wednesday, Nov.
16 at 11:30 in HH 410. The year is 2012 - look at the whole
contest but focus on A1, A2 and B1.

- Third
practice session (led by Dr. Earn and Dr. Dushoff): Wednesday,
Nov. 2 at 11:30 in HH 410. Please look at 2011 A1, 2010 A1,
and then 2002 B1 and 2008 B1 for extra practice. Since Nov.
2 is "Take your child to work day", the theme this week is
"Problems Dr. Earn's daughter can understand."

- Second
practice session (led by Dr. Valeriote and Dr. Van Tuyl):
Wednesday, Oct. 19 at 11:30 in HH 410. Please look at
1992-B1, 1995-B1, 1996-B1, and 2013-A1.

- First
practice seesion (led by Dr. Van Tuyl and me): Wednesday, Oct. 5
at 11:30 in HH 410. Please look at 1989 A1 and 1994 B.
1993 B1 and 1995 A4 are also good problems to look at although we
won't discuss them at the session.

- Welcome back for another Putnam season! Our first meeting, an organizational meeting to answer some questions and explain what is going to happen to everyone, is Wed. Sept. 21 at 11:30 in HH 410. I have been receiving a lot of email questions so let me answer some of the most common here. See you on Wednesday!
- The Putnam is a North America-wide mathematics contest for undergraduates and has run approximately every year since 1938. The contest is 6 hours long and is written in two 3 hour sessions on the first Saturday in December. This year, that is Dec. 3 from 10 am - 1 pm and 3 pm - 6 pm. To write the entire Putnam exam you sit both sessions.
- Anyone who is eligible i.e. someone without an undergraduate degree who hasn't written the contest 4 times already, is welcome to write the contest and there is no fee for doing so. It is best if you know that you want to write to let us know so that we can put your name on the list of participants but even if your name is not on the list, if there are extra exams you can write.
- There are many websites where you can see examples of Putnam problems and their solutions; here are a couple: http://kskedlaya.org/putnam-archive/ and http://www.nepalimath.com/Pages/olympiad.aspx
- At McMaster, we will run 5 preparation sessions this fall roughly every other week on Wednesdays at 11:30 in HH 410. The dates of these sessions are: Oct. 5, Oct. 19, Nov. 2, Nov. 16 and Nov. 30. You do not have to attend the prep sessions in order to write the contest but it is a good idea to do some preparation. You can always talk to one of the people helping to organize the contest: they are me (Bradd Hart, hartb@mcmaster.ca), David Earn earn@mcmaster.ca, Jonathan Dushoff, Adam van Tuyl and Matt Valeriote.
- For the
preparation sessions, two problems will be posted here in advance
and students are encouraged to look at the problems and see what
headway they can make. At the session itself, students will
work in groups for 15-20 minutes in order to pool their approaches
to one or both of the problems and then groups can volunteer
attacks on the problems or potential solutions. The sessions
will be facilitated by two of the organizers.

2015

- I am on leave for 2015 and Dr. David Earn will be in charge this fall. If you are interested in writing this year's Putnam exam, please send email to me, hartb@mcmaster.ca or Dr. Earn, earn@math.mcmaster.ca You can find out more about the Putnam exam at the official website To see copies of old exams, google "putnam exam problems" to find a number of archives of old problems and solutions.
- This year
the exam is on Saturday, Dec. 5 from 10 - 1 and 3 - 6. We
will be writing in HH 104.

- Preparation
sessions will be held in HH 410 on Monday Nov 16, 23, 30 from
10:30 - 12:30.

- For the session, Nov. 16, first hour, look at A1 from the contests from 1989, 1992, 1993, 1995, 1998, 2000 and second hour, 1985, A2; 1986, B1; 1999, B1; 2000, A3.
- Nov. 23: 1985 A1, (A2), B1 briefly,1986 A1 briefly, A2, (B1), 1989 (revisit A1), B1, 1991 A2, B1; also try A1 and check the solution (no insight is needed). Brackets represent questions we didn't get to last week.
- Nov. 30: In
this session we will look at 1990 A5, B1, 1996 A1, A3, and 1997
A2, A3, A4, B1.

Archive: 2014

- We will meet weekly at 12:30 on Monday in HH 410 for an hour. I will also have office hours MW at 9:30 or by appointment. My email is hartb@mcmaster.ca. The first meeting is Sept. 22.
- For Sept 29, we will finish the second problem from last time: Does there exist a continuous function from [0,1] to [0,1] which takes on every value in [0,1] infinitely many times? We will also look at A1 and B1 from last year; here is a link.
- For Oct. 6, as was suggested in the last session, let's look at some of the more easy problems from recent Putnam's; here is a link from Northwestern. We looked at A1 and B1 from 2013 so let's try A1 and B1 from 2012 (and maybe those from 2011 if we get the chance).
- Oct. 20: We have been making such good progress on problems, let's branch out a bit. We'll finish B1, 2012 and look at 5 (!) problems from 2010 - A1 - A4 and B2.
- Oct. 27: Continuing with 2010 A3, A4 and 2009, B1 and 2008, B4. We will also talk a little about modular arithmetic.
- Nov. 3:
Cancelled; sorry.

- Nov. 10: We have B4 2008 and B1 2009 left over and we'll look at 2005, A1, B1 and B2 - if time permits maybe a little more on modular arithmetic and basic group theory.
- Nov. 17: We still have B4, 2008 and B2, 2005 left to look at (last week was pretty midterm intense). We will also try to look at A2 and B3 from 2001.
- Nov. 24: We will look at problems we didn't address last week - the format will either be a free-for-all or more organized depending on how many people show up!
- Dec. 1:
There will be no formal problem session today but if people want
to drop by my office to ask questions, feel free. Here are
the details for the exam: Saturday, Dec. 6 in two seatings 10 am -
1 pm and 3 pm - 6 pm in HH 305.

- We will meet weekly at 11:30 on Thursday in HH 410 for an hour. I will also have office hours M Th at 10:30 or by appointment. My email is hartb@mcmaster.ca. The first meeting will be Sept. 19.
- If you are looking here before the first meeting, you might try problems A1 and A6 from 1938 and then A1 from 2012.
- Sept. 26: let's finish A1 from 2012, look at A6 from 1938 and B1 from 2012. One can find the contest problems up until 2003 here and here is a link for 2012.
- Oct. 3: We will look at Sean's suggested question which was A3 in 1998. The general theme will be uses of the completeness of the reals in solving some problems; look at A2 from 2011and 1997 B2. Sorry for the late posting but I had connectivity issues.
- Oct. 10: Sorry but I am going to have to cancel this week's meeting. Please let me know by the end of the week if you are interested in registering for this year's Putnam. BTW, it will be held on Saturday, Dec. 7 from 10 - 1 and 3 - 6.
- Oct. 17:
Let's finish B2 from 1997 and look at three other problems (thanks
to the Wisconsin Putnam club!): 1. Suppose that \sum_{n=1}^infty
a_n is convergent and all the a_n's are positive. Let b_k =
1/ka^2_k. Show that sum_{n=1}^\infty n/(b_1 + ... + b_n) is
convergent. 2. Does there exist a polyhedron with an odd
number of faces and each face has an odd number of sides?
This would be a natural time to mention Euler characteristic which
I will do if there is time. Here is a final problem from the
1990 Putnam - A4: Consider a paper punch that can be centered at
any point of the plane and that, when operated, removes from the
plane precisely those points

whose distance from the center is irrational. How many punches are needed to remove every point? - Oct. 24: We'll finish 1990, A4 (quoted above) as well as look at 2000, A1 and B6 and 1978, A1, A2. Here is a site with many of the contests and solutions.
- Nov. 7: We still need to finish 2000 B6, 1978 A1 and A2. We'll also discuss the pigeonhole principle and variants.
- Nov. 14: Since many of you were born in approximately 1995, let's see what the Putnam was up to that year: Let's look at 1995, #A1, A4, B1 and B5.
- Nov. 21: 50
years ago, 1963 - let's look at A1, A2, B2 and B6; the Putnam exam
will be written in HH 104 on Dec. 7 from 10 - 1 and 3 - 6.

Archive: 2012

- We will meet weekly at 11:30 on Friday in HH 410 for an hour. I have office hours by appointment; send me email at hartb@mcmaster.ca
- First
meeting: Sept. 21: we looked at last year's exam. In
particular we looked at questions A1 and A4. The details of
A4 were left for this coming week and I also suggested looking at
B4 - another problem that involves linear algebra.

- Sept. 28: Look at A4, B4 as well as B3 from 2011. Here's a good problem to think about: Does there exist a continuous function from [0,1] to [0,1] which takes on every value in [0,1] infinitely many times?
- Oct. 5: We never got to B3 or the extra problem from last week. We'll look at both of those and B3 from 1994.
- Oct. 12: We are accummulating a big backlog: We will finish B3 from 1994, look at B3 from last year and if we have time, look at B5 from 2006.
- Oct. 19: Well we cleared the backlog last week; let's look at some geometry problems this week: 2002, A2, 1999, B1 and 2003, B5. There will be no training session on Oct. 26.
- Nov. 2: We will finish our discussion of 2003, B5, look at A5, B4 from 2002 and A3 from 2004. If we get a chance, I'd like to say a word about Morley's theorem
- Nov. 9: Let's look at B2 from 1996, A4 from 1998 and A1 from 2000.
- Nov. 16: Test simulation - look at the Putnam exam from 2001. We'll talk through the exam on Friday and try to do some of the problems focusing on A1, A2, B1 and B2.
- Nov. 23: Under the assumption that question 1 is always the easy question, let's look at some question 2's: A2 and B2 from 2008 as well as A2 and B2 from 2003;there will be no practice session on Nov. 30.
- The Putnam
exam takes place on Dec. 1 in two sessions: 10 - 1 and 3 - 6 in HH
305.

Archive: 2011

- We will meet weekly at 10:30 on Friday in HH 312 for an hour. I also have office hours T 10:30 and F 9:30. You can also contact me by email hartb@mcmaster.ca
- Friday, Sept. 30:
- First problem: Suppose you choose two natural numbers at random; what is the probability that their greatest common divisor is 1? Slightly more precisely, consider all natural numbers less than N and assume a uniform distribution on all pairs (a,b) with a,b< N. Consider the probability that the gcd(a,b) = 1 and determine the limit as N tends to infinity.
- Second problem: Suppose that f:R -> Rwith the property that for every x > 0, the limit as n tends to infinity of f(nx) = 0. Does this imply that f(x) tends to 0 as x tends to infinity? Either prove this or produce a counter-example. Bonus: Does your answer change if we assume f is continuous?
- Friday, Oct. 7: Since so many people have been asking what the Putnam exam looks like, for this week, let's look at last years test and see the order of difficulty of the problems. One can find last year's test at http://amc.maa.org/a-activities/a7-problems/putnamindex.shtml The solutions are also there as well but try to problems first (at least try A1) before looking at the answers.
- Friday, Oct. 14: Here is a short summary of the first two sessions and some additional problems to look at this week.
- Friday, Oct. 21: I will post some comments about last week's session later. For this week, we will look at some geometry problems of the Putnam variety. Have a look at B1 from 2008, A2 from 2007 and B3 from 2006. Also consider the following warm-up problem suggested by Matt: Suppose you have an isoceles triangle (imagine it with the unequal side on the bottom) and you inscribe a circle (the circle touches all three sides). In the space above the circle, inscribe another circle (it touches two sides and the first circle). Repeat this, constructing an infinite stack of circles inside the triangle. Question: what is the total sum of all the circumferences of all the circles?
- Friday, Oct. 28: Here is a short commentary on the last two sessions. Matt has produced a nice explanation and picture regarding the hyperbola question from last week. For this week, have a look at some combinatorial problems from the 1996 exam: A3, A4 and B1.
- Friday, Nov. 4: We will look at B1 from 1996, A4 from 2002, B2 from 1997 and B3 from 2005. Volunteers to present solutions are welcome.
- Friday, Nov. 18: Let's look at A2 and B3 from the 2000 exam, A2 and B1 from 2001. Again, volunteers are welcome to present.
- Friday, Nov. 25: Back to 1998; let's look at A1, A2, A3, B1, B2 and B5.
- Friday, Dec. 2: Sorry about the late posting; this will be like a dry run for Saturday. Let's look at A1, A2, A3, B1 and B2 from 2004; I'll also say something about square roots.
- For
Saturday, let's meet at 10 in HH 104. The exam runs from 10
- 1 and from 3 - 6; lunch is on the department!