This department welcomes problems believed to be new and
at a level appropriate for the readers
of this journal. Old problems displaying novel and elegant methods ofsolution
are also invited.
Proposals should be accompanied by solutions if available and by any information
that will assist
the editor. An asterisk (*) preceding a problemnumber indicates that the
proposer did not submit a
All correspondence should be addressed to Harold Reiter,
Department of Mathematics , University
of North Carolina Charlotte, 9201 University City Boulevard, Charlotte, Electronic submissions using LATEX are
electronic submissions are also encouraged. Please submit each proposal and
typed or clearly written on a separate sheet ( one side only) properly identified
with name, affiliation,
and address. Solutions to problems in this issue should be mailed to arrive by
March 1, 2008.
Solutions identified as by students are given preference.
Problems for Solution.
1146. Proposed by Douglas Shafer, University of North
Carolina Charlotte This
problem first appeared here in fall 2006. It has not yet been solved
Because of a contribution from Professor Ali Amir-Moez, we are able to offer a
prize for the best undergraduate student solution to this problem.
Given six real constants a, b, c, d, e, and f, not all
zero, a conic section C :
ax2 + bxy + cy2 + dx + ey + f = 0 is determined. Since rescaling the six
by a non zero number does not change C, we may view (a, b, c, d, e, f) as lying
S5 ⊂ R6. If (a, b, c, d, e, f) is selected based on a uniform distribution on
S5, what is
the probability that C is an ellipse?
1159. Proposed by S.C. Locke, Florida Atlantic University,
Boca Raton, FL
for k = 2, 3, 4, 5.
1160. Proposed by Leo Schneider, John Carroll University,
Construct a proof that e is an ir rational number based on
the Alternating Series
1161. Proposed by Jos´e Luis D´ıaz-Barrero, Universidad
Polit´ecnica de Catalu˜na,
Find all continuous functions f : R → R such that
1162. Proposed by Paul S. Bruckman, Sointula, BC
prove the congruence
64M), for k ≥ 3.
Also, prove the ”near” corollaries:
1. 3M 1 + 20M(mod 32M), for k ≥ 2.
2. 3M 1 + 4M(mod 16M), for k ≥ 1.
3. 4M || (3M −1), for k ≥ 1; that is, the largest exponent t such that 2t
(3M − 1) is t = k + 2.
1163. Proposed by Stas Molchanov, University of North Carolina Charlotte
A two-pan balance and 16 coins of different weights are given. What is the
number of usages of the balance needed to determine the heaviest coin, the
heaviest coin, and the third heaviest coin?
where n is a positive integer . Your formula should demonstrate that each such
is a rational number. For example,
1165. Proposed by Marcin Kuczma, University of Warsaw, Warsaw, Poland
For positive integers n, k let F(n, k) be the number of mappings of an n-element
set into itself whose kth iterate is the identity map (e.g. F(3, 2) = 4 ) – and
number F(4, 2) + F(8, 2) + F(8, 3) be nice and lucky and happy for you!!
note: this puzzle was sent to friends of the poser in December of a certain year
gift. This is the seventh of several such problems we plan for this column.
1166. Proposed by Peter A. Lindstrom, Batavia, NY
Suppose that functions f, g, f\', and g\' are continuous over [0, 1] , g (x)
[0, 1] , f (0) = 0, g (0) =
π, f (1) = 1004, and g (1) = 1. Find the value of
1167. Proposed by Richard Armstrong, St. Louis Community College and Arthur
Holshouser, Charlotte, NC
Find necessary and sufficient conditions on positive integers a, b, c, and d
1168. Proposed by Sam Vandervelde, St. Lawrence University , Canton, NY
Define the Fibonacci numbers as usual by
for n ≥ 2. Determine the value of
1169. Proposed by James A. Sellers, Pennsylvania State University, University
A composition of the positive integer n is an ordered sequence of positive
which sum to n . So, for example, 3 + 4 + 2, 4 + 3 + 2, and 2 + 2 + 2 + 3 are
compositions of the number 9. Let co(n) be the number of compositions of n where
the last part is odd.
1. Find a Fibonacci- like recurrence satisfied by co(n).
2. Use the above recurrence to find a closed formula for co(n).
1170. Proposed by Andy Niedermaier, University of California San Diego
Consider a 10 × 10 grid of lights, each either on or off, which we denote using
, where, for each i = 1, 2, . . . , 10 and j = 1, 2, . . .
, 10, the entry in
row i and column j is
and its value is 0 or 1. We are allowed two types of
For each 1≤ u ≤ 8 and 1 ≤ v ≤ 8, we can change the status of all the lights
which both u ≤ i ≤ u + 2 and v ≤ j ≤ v + 2. This is called a small block move. The
other type move is, for each 1 ≤ u ≤ 6 and 1 ≤ v ≤ 6, we can change the status of all
for which both u
≤ i ≤ u + 4 and v ≤ j ≤ v + 4. This is called a large
block move. So essentially, we can change the status of all nine lights in each
3 × 3
subarray and of all the lights in each 5 × 5 subarray. Is it possible, beginning
the all on configuration, to achieve all possible on-off configurations of
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