Problems and Contests (Math 394) — Spring 2015
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 2015 semester, we meet
(day) ,   (time) ,   in Math (room)

Here is our second set of problems for the Spring 2015 semester.
  1. In a 100–yard race, Alex beat Bob by 10 yards and Bob beat Charlie by 20 yards. Assuming each runner ran with a constant speed, by how much did Alex beat Charlie?
  2. This problem was part of UM's Lennes Competition in 2010.
  3. A set W of real numbers has property P provided   (x in W)   implies   (x + |x| is also in W).  
    Find all finite sets of real numbers with property P. [If there are infinitely many such finite sets, then identify their crucial characteristic(s).]
  4. Three vertices are chosen randomly from the vertices of a cube. What is the probability those three vertices are also vertices of an equilateral triangle?
  5. Suppose a, b, c are positive real numbers such that A = a2, B = b2, and C = c2 are consecutive terms in an arithmetic progression, i.e., there is some D such that B = A + D and C = B + D.   Show that 1/(b+c), 1/(c+a), 1/(a+b) are also in an arithmetic progression.
  6. For each positive integer n, let D(n) be the number whose decimal digits are the concatenation of the decimal digits for integers 1 through n. For example, D(5) = 12345 and D(13) = 12345678910111213.
    1. Consider all of those numbers that can be gotten by removing ten digits from D(10). What is the largest such number?
    2. Find the largest of all those numbers obtainable by removing fifteen digits from D(15). Show how you found your answer.
    3. Find the maximum among all numbers gotten by deleting twenty digits from D(20). Explain your reasoning.
    4. Find the largest number obtainable by deleting fifty digits from D(50). Prove your answer is correct.

Last modified: 26 February 2015, Thursday 15:17