Problems and Contests (Math 394) --- Fall 2012
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 2012 semester, we meet
4:10 – 5:00 pm,
in Math 306
- Our current set of problems is posted on the internet at the URL
- Some previous sets of problems are available.
- Faculty problem posers & coaches
- UM students can participate in several mathematical
contests (including UM's
Here is our second set of problems for the Fall 2012 semester.
Prove: If the vertices of a trapezoid lie on a circle,
then its diagonals have the same length.
Click on a figure to display an interactive webpage
which lets you explore by moving points A, B, C.
This was problem 5 of our
Lennes Competition in 2010.
Suppose A and B are complex numbers such that
|A| = 1 = |B|
A · B ≠ −1.
(A + B) / (1 + A · B)
is a real number.
Find a positive number such that one-fifth of it multiplied by
one-seventh of it equals the number.
Is there a negative number with the same property?
A standard Bridge deck of 52 playing cards is shuffled
and placed facedown on a table;
then the cards are shown one after another (dealing from the top).
Suppose you are allowed to bet (in advance) when the first black Ace
will appear. What number should you pick to maximize your long-run
success if this game will be repeated many times?
Find all integers n
1 + n + n2 + n3 + n4
is the square of an integer.
Problem Solving Competition
, September 2012
Last modified: 7 November 2012, Wednesday 10:42