Problems and Contests (Math 394) --- Fall 2010
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 2010 semester, we meet
Thursday,
4:10 – 5:00 pm,
in Math 211
- 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.
- Faculty problem posers & coaches
- UM students can participate in several mathematical
contests (including UM's
Lennes Competition).
Here is our third set of problems for the Fall 2010 semester.
- Given a circle, is it possible to circumscribe two
noncongruent triangles of equal areas about the circle?
- Is there a value of b such that b > 1 and
bx = x has a unique solution?
- Let N be a positive integer and consider the N-by-N array
whose element in row I and column J is the smaller of I and J.
For example, if N = 4, the array would be
| 1 | 1 | 1 | 1 |
| 1 | 2 | 2 | 2 |
| 1 | 2 | 3 | 3 |
| 1 | 2 | 3 | 4 |
Show that the sum of all entries in this array is
12 + 22 + 32 + ... + N2
- Suppose a cubic polynomial with leading coefficient of one
and with inflection point at the origin passes through (c,0) and (a,b)
where a > c > 0.
A translated copy of the cubic has its inflection point at (a,b) and
passes through the origin.
Prove that twice the area between the two cubic polynomials equals
a4.
ASCM
Problem Solving Competition, November 2010
- If
sin(α) · cos(β) = −1/2 ,
what are the possible values of
cos(α) · sin(β) ?
- Consider a set S and a binary operation *
[i.e., if a and b are in S, then a*b is also in S].
- Assume
(a * b) * a = b
for all a and b in S.
- Prove that
a * (b * a) = b
for all a and b in S.
This was problem A1 of the
Putnam Competition in 2001.
Last modified: 18 November 2010, Thursday 10:51