Problems and Contests (Math 394) --- Fall 2014

Do you enjoy solving problems?
Do you like contests?

The undergraduate mathematics seminar on

Tuesday,
8:10 – 9:00 am,
in Math 305

- 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, Feb 2012, Sep 2012, Nov 2012
- 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 Fall 2014 semester.

- Find all positive rational numbers Q such that R = Q + (1/Q) is an integer.
- This problem ask you to consider some
small sets of positive integers.
- Is there a set of three integers such the sum of each pair is prime?
- Is there a set of four integers such the sum of each triple is prime?
- Is there a set of five integers such the sum of each triple is prime?

- Suppose five points in the plane are arranged so that they do not fall on a line and no four are on a circle. Show there must be a circle passing through three of the points such that one of the remaining points is inside the circle and the other is outside the circle.
- Suppose 50 is written as the sum of some positive integers (which do not need to be distinct) such that the product of those summands is divisible by 100. Find the largest possible value for that product.
- Find the sum of this expression involving factorials.
1 · (1!) + 2 · (2!) + 3 · (3!) + … + n · (n!)
- Create a sequence {p
_{i}} as follows. Choose p_{0}to be a particular prime. If p_{0}, p_{1}, …, p_{n−1}have already been chosen, then let p_{n}be the**largest**prime factor of 1 + p_{0}p_{1}··· p_{n−1}.- Is there a choice for p
_{0}such that {p_{i}} is not strictly increasing? - If p
_{0}= 2, then p_{1}= 3, p_{2}= 7, and p_{3}= 43. Will 5 appear in this sequence? - What changes if "
*largest prime factor*" is replaced with "*smallest prime factor*"?

- Is there a choice for p

Last modified: 9 September 2014, Tuesday 07:03