Problems and Contests (Math 394) --- Spring 2012
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 Spring 2012 semester, we meet
Thursday,
2:10 – 3:00 pm,
in Math 306
- 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,
Sep 2008,
Sep 2010,
Oct 2010,
Nov 2010,
Feb 2011
- Faculty problem posers & coaches
- UM students can participate in several mathematical
contests (including UM's
Lennes Competition).
Here is our first set of problems for the Spring 2012 semester.
- In a 100-yard race, Alex beat Bob by 10 yards
and Bob beat Charlie by 20 yards.
Assuming that each runner ran with a constant speed,
by how much did Alex beat Charlie?
- Suppose x0 = 3, x1 = 4, and
the remainder of this sequence satisfies the recursion
xn+1 =
(xn−1)2 − n · xn
Find an explicit expression for xn
which is true for any integer n ≥ 1.
- Lines L1 and L2 are parallel while a
different pair of lines L3 and L4 are also
parallel. Point P moves so that the sum of its distances from the
four lines is constant. Describe the path of P's motion.
- Can the following pattern be generalized?
- 2 + 1 = 3 and 22 + 1 = 5 are relatively prime.
- 2 + 1 = 3 and 24 + 1 = 17 are relatively prime.
- 22 + 1 = 5 and 28 + 1 = 257
are relatively prime.
Are any two members of the sequence
2 + 1,
22 + 1,
24 + 1,
28 + 1,
216 + 1,
etc.
relatively prime?
- An envelope of a set of lines is a curve
tangent to all of them. For positive number c, let Lc
be the line with equation y = (2/c) + (1 − 1/c2) x.
Find the envelope for this family of lines?
Last modified: 15 February 2012, Wednesday 13:07