Problems and Contests (Math 394) --- Fall 2014
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 2014 semester, we meet
Tuesday,   8:10 – 9:00 am,   in Math 305

Here is our first set of problems for the Fall 2014 semester.
  1. Find all positive rational numbers Q such that R = Q + (1/Q) is an integer.
  2. This problem ask you to consider some small sets of positive integers.
    1. Is there a set of three integers such the sum of each pair is prime?
    2. Is there a set of four integers such the sum of each triple is prime?
    3. Is there a set of five integers such the sum of each triple is prime?
  3. 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.
  4. 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.
  5. Find the sum of this expression involving factorials.
    1 · (1!)   +   2 · (2!)   +   3 · (3!)   +   …   +   n · (n!)
  6. Create a sequence {pi} as follows. Choose p0 to be a particular prime. If p0, p1, …, pn−1 have already been chosen, then let pn be the largest prime factor of 1 + p0 p1 ··· pn−1.
    1. Is there a choice for p0 such that {pi} is not strictly increasing?
    2. If p0 = 2, then p1 = 3, p2 = 7, and p3 = 43. Will 5 appear in this sequence?
    3. What changes if "largest prime factor" is replaced with "smallest prime factor"?

Last modified: 9 September 2014, Tuesday 07:03