Training for the 2011 Putnam Competition
- 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!
