Problems and Contests (Math 394) —
Spring 2015

Do you enjoy solving problems?
Do you like contests?

The undergraduate mathematics seminar on

(day) ,
(time) ,
in Math (room)

- 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 online.
- Faculty problem posers & coaches
- UM students can participate in several mathematical contests (including UM's Lennes Competition).

Here is our second set of problems for the Spring 2015 semester.

- 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?
- 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).] - 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?
- Suppose a, b, c are positive real numbers such that
A = a
^{2}, B = b^{2}, and C = c^{2}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. - 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.
- Consider all of those numbers that can be gotten by removing ten digits from D(10). What is the largest such number?
- Find the largest of all those numbers obtainable by removing fifteen digits from D(15). Show how you found your answer.
- Find the maximum among all numbers gotten by deleting twenty digits from D(20). Explain your reasoning.
- Find the largest number obtainable by deleting fifty digits from D(50). Prove your answer is correct.

This problem was part of UM's
Lennes Competition
in 2010.

Last modified: 26 February 2015, Thursday 15:17