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
Wednesday,
4:10 – 5:00 pm,
in Math 306
 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.
Sep 2001,
Nov 2001,
Feb 2002,
Oct 2002,
Mar 2003,
Oct 2003,
Jan 2004,
Oct 2004,
Jan 2005,
Mar 2005,
Sep 2005,
Jan 2006,
Mar 2006,
Apr 2006,
Sep 2006,
Nov 2006,
Jan 2007,
Feb 2007,
Sep 2007,
Oct 2007,
Nov 2007,
Jan 2008,
Feb 2008,
Mar 2008,
Sep 2008,
Sep 2010,
Oct 2010,
Nov 2010,
Feb 2011,
Feb 2012,
Sep 2012
 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 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
and
A · B ≠ −1.
Prove that
(A + B) / (1 + A · B)
is a real number.

Find a positive number such that onefifth of it multiplied by
oneseventh 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 longrun
success if this game will be repeated many times?

Find all integers n
such than
1 + n + n^{2} + n^{3} + n^{4}
is the square of an integer.
ASCM
Problem Solving Competition, September 2012
Last modified: 7 November 2012, Wednesday 10:42