1995 Konhauser Problemfest: Results and Problems
The third annual Konhauser Problemfest,
a problem-solving contest for teams from Macalester, St.
Olaf, and Carleton Colleges was held on Saturday, February 25, 1995.
Information about this competition has been provided
by Stan Wagon (wagon@macalstr.edu)
and appears here with his permission.
A noteworthy feature of this
year's contest was the presentation of a Helaman Ferguson
sculpture which will serve as a trophy to travel to the
winning school each year. The sculpture uses 16 pieces in
two colors of polished granite to illustrate a dissection
proof of the pizza theorem (see Mathematics Magazine. 67(1994)267).
Results:
1. Carleton #1 70 pts Daniel Martin and Neal Weiner
2. Carleton #2 68 pts John Costello, Curtis Mitchell, Liz Stanhope
3. Macalester #1 58 pts Michael Dekker, Sim Simeonov, Michael Wolfe
4. Carleton #3 44 pts
5. St. Olaf #1 34 pts
6. Carleton 4 32 pts
7. Macalester 2 25 pts
8. Macalester 2A 25 pts
9. St Olaf 2 16 pts
10. St Olaf 3 13 pts
Here is the complete text of the 1995 Konhauser Problemfest problems (which
were selected by Stan Wagon from various places, mostly the old
Konhauser files). The hardest ones from the point of
view of the contest were #5, #6, #7, and #8. Wagon would have
predicted that #8 was hardest, but two teams solved it.
Only one team solved #6.
Problem 1. Circular Surprises
In the diagram, the large circle has radius one, the
inscribed figures are a diameter (a), an equilateral
triangle (b), and a square (c), the shaded circle is
tangent to the top of the circle, and the other
tangencies are as they appear. Determine the radii of the
shaded and unshaded circles.
Problem 2. A Vintage Year
The year 1979 was unusual
in that it results from stringing together distinct 2-digit primes,
namely 19, 97, and 79. The next time this
happens is in 2311, which comes from 23, 31, and 11. When
will this happen for the last time? Remember: the primes
must be distinct (and the examples are not meant to
suggest that there are only three primes).
Problem 3. A Circular Committee
A committee with 1,995
members sits around a circular table. Every hour there is
a vote and each member must vote either YES or NO.
Everyone votes his or her conscience on the first round,
but after that the following rule is obeyed: On the nth
vote, if a person's vote is the same as at least one of
the two votes of adjacent committee members, the member
votes the same way on the (n+1)st round as on the nth.
Otherwise the person's (n+1)st vote is the opposite of
his or her nth. Prove that, regardless of the first-round
votes, there will come a time after which no one's votes
will change.
Problem 4. Complex Relationships
Let a, b, c, d denote
complex numbers. True or False:
- If a + b = 0 and |a| = |b|, then a^2 = b^2.
- If a + b + c = 0 and |a| = |b| = |c|, then
a^3 = b^3 = c^3.
- If a + b + c + d = 0 and |a| = |b| = |c| = |d|,
then a^4 = b^4 = c^4 = d^4.
Problem 5. The Middle of a Moving Line
Suppose that L and
M are two nonintersecting lines in 3-space that are
perpendicular to each other. A line segment PQ of fixed
length moves so that P is on L and and Q is on M. What is
the locus of the midpoint of PQ?
Note: A line L in 3-space is perpendicular to another
line M if M is contained in a plane perpendicular to L.
Problem 6. Visible People
Suppose n people, all having
distinct heights, are standing in a single-file line.
Call a person "visible" if he or she is taller than
anyone in front of him or her (and so is visible to a
person looking at the line from the front). Assuming a
random distribution of the people into the lines, how
large must n be in order that the expected number of
visible people is 10?
Problem 7. Find the Pattern
The squares of an infinite chessboard are numbered as
illustrated. The number 0 is placed in the lower left-
hand corner; each remaining square is numbered with the
smallest nonnegative integer that does not already appear
to the left of it in the same row or below it in the same
column. If the first row (column) is called the zeroth
row (column), which number will appear in the 666th row
and 401st column?
| | | | | | |
----------------------------------------
| 5 | 4 | 7 | 6 | 1 | 0 |
----------------------------------------
| 4 | 5 | 6 | 7 | 0 | 1 |
----------------------------------------
| 3 | 2 | 1 | 0 | 7 | 6 |
----------------------------------------
| 2 | 3 | 0 | 1 | 6 | 7 |
----------------------------------------
| 1 | 0 | 3 | 2 | 5 | 4 |
----------------------------------------
| 0 | 1 | 2 | 3 | 4 | 5 |
----------------------------------------
Problem 8. Count the Tilings
The diagram shows a tiling
of a 2x7 rectangle with 1x1 and 1x2 tiles
(singletons and dominoes; dominoes may be placed
horizontally or vertically). How many such tilings of a
2x7 grid are there?
+--+-----+--+--+-----+
| | | | | |
+--+--+--+ | +--+--+
| | | | | | |
+-----+--+--+--+--+--+
Problem 9. 11111111111111...
Prove that there are
infinitely many integers n for which the base-10 number
obtained by stringing together n 1's is divisible by n.
For example, 111 is divisible by 3.
Problem10. Three Rising Vectors
A rising vector in the
plane is one whose vertical component is nonnegative. The
sum of two rising vectors of length one can be very
short. Prove that the sum of three rising vectors, all
having length one, cannot have length less than one.
Notes:
- Problem 1 is a 1955 Leon Bankoff problem -- the result is quite surprising!
- Problem 3 is a Chinese Olympiad practice problem; from a recent issue
of Crux Mathematicorum.
- Problem 8 was suggested by Phyllis Chinn of Humboldt State Univ.,
Arcata, Calif.
Here are some references:
- Robert C. Brigham, Richard M. Caron, Phyllis Z. Chinn, and Ralph Grimaldi,
A tiling scheme for the Fibonacci numbers,
Journal of Recreational Mathematics (to appear).
- Robert C. Brigham, Phyllis Z. Chinn, Linda Holt, and Steve Wilson,
Finding the recurrence relation for tiling 2 x n rectangles,
Congressus Numerantium (to appear)
- Problem 10 is an old U.S. Olympiad problem. The result is true if three
is replaced by an odd integer.
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