Problems and Contests (Math 394) --- Fall 2008
Do you enjoy solving problems?
Do you like contests?
The undergraduate mathematics seminar on Problems and Contests
may be for you. Come and participate when you can.
For the Fall 2008 semester, we meet
Thursday,
2:10 – 3:00 pm,
in Math 108.
- Our current set of problems is posted on the internet at the URL
www.math.umt.edu/394
- Some previous sets of problems are available.
Sep 2001,
Nov 2001,
Feb 2002,
Oct 2002,
Mar 2003,
Oct 2003,
Jan 2004,
Oct 2004,
Jan 2005,
Mar 2005,
Sep 2005,
Jan 2006,
Mar 2006,
Apr 2006,
Sep 2006,
Nov 2006,
Jan 2007,
Feb 2007,
Sep 2007,
Oct 2007,
Nov 2007,
Jan 2008,
Feb 2008,
Mar 2008.
- Faculty problem posers & coaches
- UM students can participate in several mathematical
contests.
Here is our first set of problems for the Fall 2008 semester.
- Sally selects 2008 points in the plane so that no three are collinear.
Cathy colors them so that half of the points are colored Blue
while the other half are colored Green.
Penelope then pairs each Blue point with a Green point
and joins each pair with a line segment.
Show that Penelope can do her pairing
so that no two line segments intersect.
This was problem 1 of our
Lennes Competition in 2008.
- Note: 16 has prime factorization 24,
17 is prime, 18 = 2 * 32,
19 is prime, 20 = 22 * 5, and 21 = 3 * 7.
- A is the sum of 1 and three odd positive integers.
What can be said about the prime factorization of A?
- B is the sum of 1 and three
consecutive odd positive integers.
What can be said about the prime factorization of B?
- C is the sum of 1 and the squares of three
consecutive odd positive integers.
What can be said about the prime factorization of C?
- Suppose g is a positive-valued function which is
differentiable on the interval [a,b].
Prove there is some t in (a,b) such that
g'(t) 1 1
------- = ------- + -------
g(t) a - t b - t
- Given three points in the plane, construct a line
such that the sum of their distances to the line
is a minimum.
- Farmer Jones has 65 hens.
If he had one more solid colored hen,
then exactly one-third of his hens would be speckled.
From years of experience, Farmer Jones knows that one-half
of the speckled hens will lay speckled eggs and that each
hen-and-a-half will lay an egg-and-a-half in a day-and-a-half.
- After how many full days will Farmer Jones have
five dozen speckled eggs to sell?
- After how many full days will Farmer Jones have
four dozen speckled eggs to sell?
- Let P be a nonconstant polynomial with
positive integer coefficients.
Prove that if N is a positive integer, then
P(N) divides P(1+P(N)) if and only if N = 1.
This was problem B-1 of the
Putnam Competition in 2007.
Last modified: 11 September 2008, Thursday 15:53