Problems and Contests (Math 394) —
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
in Math (room)
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 = 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.
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.
Last modified: 26 February 2015, Thursday 15:17