Problems and Contests (Math 394) --- Spring 2011
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 2011 semester, we meet
Thursday,
2:10 – 3:00 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
- Faculty problem posers & coaches
- UM students can participate in several mathematical
contests (including UM's
Lennes Competition).
Here is our first set of problems for the Spring 2011 semester.
- Set S is partitioned into subsets A and B
if and only if S = A ∪ B and A ∩ B = ∅
- Partition {1,2,3,4,5} into two disjoint subsets, A and B.
Verify that there is at least one choice of integers x and y such that
x, y, x − y are in the same subset.
For example, if A={1,4,5}, then we could choose x=5 and y=4;
if A={1,3,5}, then x=4 and y=2 are in B.
- Partition {1,2,3,4,5,6,7,8,9,10} into subsets A and B.
Show there must be
a pair (x,y) such that x,y,x−y are in the same subset and
another pair (s,t) such that s,t,s−t are in the same subset.
Note: (x,y) could be in A and (s,t) in B.
- How can those results be generalized?
- How many times a day do the hands of clock overlap?
- Big Red Triangle has vertices A, B, C;
real number t is between 0 and 1.
Point D is obtained by moving A along segment AB so that
distance(A,D) = t ∗ distance(A,B);
points E and F are obtained similarly by moving B toward C and C toward
A using the same factor.
(This operation is called a dilation.)
Smaller Green Triangle has vertices D, E, F.
Are there any non-trivial choices of t for which the Red and Green
triangles are similar?
Click on either figure to display an interactive webpage
(in a new window)
which lets you explore by moving points A,B,C and change value of t.
- If n is a positive integer, then
n! ≤ ([1 + n] / 2)n
- Suppose A and B are consecutive odd prime integers;
let S = A + B. Suppose S =
p1 · p2
··· pn
where each pj is a prime and
there are n (not necessarily distinct) terms in that factorization.
- Show n > 1 (i.e., S is not prime).
- Find the minimal value for n (together with explicit values for
A and B to show that minimal value is actually attained).
- Find several choices of (A,B) for which n is larger than the minimal
value identified in the preceding part.
Last modified: 3 February 2011, Thursday 11:56
