 Proposed by Harry Tamvakis,
University of Pennsylvania, Philadelphia, PA.
(October 1998)
Let a_{1}, a_{2}, ... a_{n} be a sequence of nonzero
real numbers, exactly p of which are positive. Characterize the pairs (n, p)
such that exactly half of the possible products a_{i}a_{j}a_{k}
with i < j < k are positive.
(Reconstructed) Solution
(October 2000, "Splitting Triple Products by Sign")
 Proposed by Józef Przytycki,
The George Washington University, Washington, DC.
(January 2001)
A lattice simple closed curve in the plane is a simple
closed curve all of whose points have an integer coordinate. The area A(C) enclosed
by such a curve C is always a positive integer. A shrinking move on a
lattice simple closed curve C is one that replaces C with another such curve D
whose interior is inside C and with A(D)=A(C)1. Is is always possible to transform
a lattice simple closed curve to a 1by1 square via shrinking moves?
Solution
(January 2002, "Shrinking Lattice Curves")
 Proposed by Andrei Jorza,
"Moise Nicoara" High School, Arad, Romania.
(February 2001)
Find all bounded convex polyhedra such that no three faces have the same
number of edges.
Solution (with Generalization)
Mathematica Notebook with Pictures
(February 2006; "No Three Faces with the Same Number of Edges")
 Proposed by Gordon Rice, Davis, CA.
(November 2001)
It is clear from the law of
cosines that every angle that occurs in a triangle with integer sides has a rational
cosine. Is the converse true? Does every angle between 0 and π with a rational
cosine occur in some triangle with integer sides?
Solution (with Generalization)
(AugSept 2003, "Angles with Rational Cosines")
 Proposed by JeanMarie
De Koninck, Université Laval, Québec,
Canada.
(October 2002)
 Let φ denote the Euler φ function, and let
γ(n) be the product of the primes that divide into n, with γ(1)=1.
Prove that there are exactly six
positive integers such that φ(n)=(γ(n))^{2}.
 * Let σ(n) denote the sum of divisors of n. Prove or
disprove that the only solutions to σ(n)=(γ(n))^{2} are
n=1 and n=1782.
Solution (with Generalization)
(June/July 2004, "When the Totient is the Product of the Squared Prime Divisors")
 Proposed by Li Zhou, Polk Community College, Winter Haven, FL.
(December 2005)
Find a closed formula for the number of ways to tile a 4 by n rectangle with 1 by 2
dominoes.
Solution
(June/July 2007, "Tiling 4Rowed Rectangles with Dominoes")
 Proposed by Mohammad Hossein Mehrabi, Iran University of Science and Technology,
Tehran, Iran. (January 2006)
Let A and B be real n × n matrices.
Show that if AB−BA is invertible
and A^{2}+B^{2}=√3(AB−BA), then n is a multiple of 6.
Solution
(January 2008, "A Condition on the Cummutator")

Proposed by Li Zhou, Polk Community College, Winter Haven, FL.
(March 2006)
The stagen Menger Sponge M_{n} is generated recursively,
starting with the unit cube M_{0}. To drill a cube is to partition it into twentyseven
congruent subcubes and remove the closed central cube along with the six remaining subcubes sharing
a face with that cube, resulting in a solid that is the union of twenty congruent cubes. Given
M_{n1}, construct M_{n} by drilling each of the 20^{n1}
subcubes of edgelength 3^{1n} in M_{n1}. M_{4} is shown:
The chromatic number of a surface is the minimum number of colors that suffice to color any map
drawn on that surface (so that the chromatic number of M_{0} is 4 by the celebrated
"Four Color Theorem.") Find the chromatic number of the surface of M_{n}.
Solution (with Conjecture)
(November 2007, "Coloring Graphs on Sponges")
Update, March 2009: Li Zhou has informed me that my conjecture
(about the source of the problem) is false.

Proposed by Alexander Dubinov and Irina Dubinova, Russian Federal Nuclear Center, Sarov,
Russia. Consider the following system of n equations in n positive
unknowns x_{1}, ... x_{n} and n positive
given numbers a_{1}, ... a_{n}:
Find a formula in closed form for x_{1}, ... x_{n} in terms
of a_{1}, ... a_{n}.
Solution
(December 2007, "Can "Closed Form" Involve Lambert?")

Proposed by David Beckwith, Sag Harbor, NY. (March 2006)
Show that for an arbitrary positive integer n
Two Solutions
(April 2008, "A Vanishing Alternating Sum")

Proposed by Gary Gordon,
Lafayette College, Easton, PA. (April 2006)
Consider the following
algorithm, which takes as input a positive integer n and proceeds by rounds, listing
in each round certain positive integers between 1 and n inclusive, ultimately producing as
output a positive integer f(n), the last number to be listed. In the 0th round, list 1. In
the first round, list, in increasing order, all primes less than n. In the second round,
list in increasing order all numbers that have not yet been listed and are of the form 2p,
where p is prime. Continue in this fashion, listing numbers of the form 3p, 4p, and so on
until all numbers between 1 and n have been listed. Thus f(10)=8 because the list eventually
reaches the state {1,2,3,5,7,4,6,10,9,8}, while f(20)=16 and f(30)=27.
(a) Find f(2006).
(b) Describe the range of f.
(c) Find
Solution and Polytime Algorithm
(April 2008, "The Number Between 1 and n That is Least Prime")

Proposed by Roberto Tauraso,
Università di Roma "Tor Vergata", Rome, Italy. (AugustSeptember 2006)
Find a closed formula for
where F_{n} denotes the nth Fibonacci number (that is, F_{0}=0,
F_{1}=1, and F_{j}=F_{j}1+F_{j}2
when j ≥ 2) and S[k,n] is the set of all (k+1)tuples of
nonnegative integers that sum to nk.
Solution, Generalization, and Variations
(November 2008, "Fibonacci Numbers and Tiling a Board with Cuts")

Proposed by Manuel Kauers, Research Institute for
Symbolic Computation, Johannes Kepler University. (December 2006)
Let F_{n} denote
the nth Fibonacci number, and let i denote √(1). Prove that
Solution and Variation
(December 2008, "A Telescoping Fibonacci Sum")

Proposed by Ashay Burungale, Satara, Maharashtra, India. (December 2006)
In a certain town of population 2n+1, one knows those to whom one is known.
For any set A of n citizens, there is some person amongst the other
n + 1 who knows everyone in A. Show that some citizen of the town knows
all the others.
Solution
(November 2008, "Getting to Know You, Directly and Indirectly"; my solution published)

Proposed by Pablo Fernández Refolio, Universidad Autónoma de Madrid, Madrid, Spain. (December 2007)
Show that
Solution
(November 2009, "Baking a Pi with Factorials")

Proposed by David Beckwith, Sag Harbor, NY. (February 2008)
Show that when n is a positive
integer,
Solutions
(December 2009, "The Number of Balanced Senates from n States")

Proposed by Roger Cuculière, France. (February 2008)
Find all nondecreasing functions f from R to R such that
f(x+f(y)) = f(f(x)) + f(y)
for all real x and y.
Solution and Variation
(October 2009, "A Functional Equation")

Proposed by Bhavana Deshpande, Poona College of Arts, Science & Commerce Camp, Pune, India, and
M. N. Deshpande, Institute of Science, Nagpur, India. (March 2008)
Given a positive integer n and an integer k with 0 ≤ k ≤ n,
form a permutation a = (a_{1}, ..., a_{n}) of
(1, ..., n) by choosing the first k positions at random and filling the
remaining n  k positions in ascending order. Let E_{n,k} be the expected number of lefttoright
maxima. (For example,
E_{3,1} = 2, E_{3,2} = 11/6, and
E_{4,2} = 13/6.) Show that E_{n}+1,k  E_{n,k} = 1/(k+1).
(A lefttoright maximum occurs at
k when a_{j} < a_{k} for all j < k.)
Solution and Variation
(January 2010, "A Partially Random Permutation"; my variation published)

Proposed by David Callan, University of Wisconsin, Madison, WI. (May 2008)
A bit string arc diagram is an undirected graph in which the vertices are the positions
in a single string of bits and the edges are called arcs due to the visual representation
in which they are drawn joining positions in the string. To be a good diagram, arcs
must occur only between unequal bits, and each bit may be the left endpoint of at most one arc.
Thus, the first diagram is good but, for two reasons, the second is not.
There are six good diagrams on two bits, four with no arc and two with a single arc. How many
good diagrams are there on n bits?
Solution and Generalization
SOLUTION WAS NOT SUBMITTED ON TIME.
(February 2010, "Counting Good Bit String Arc Diagrams")

Proposed by Christopher Hillar, Texas A&M University, College Station, TX,
and Lionel Levine, Massachusetts Institute of Technology, Cambridge, MA. (Aug/Sep 2008)
Given a
monic polynomial p of degree n with complex coefficients, let A_{p}
be the (n + 1) × (n + 1)
matrix with p (–i + j) in position
(i, j), and let
D_{p} be the determinant of A_{p}. Show that
D_{p} depends only on n, and find its value in terms of n.
Solution
(October 2010, "A Determinant Generated by a Polynomial")

Proposed by K. S. Bhanu, Institute of Science, Nagpur, India, and M. N. Deshpande, Nagpur, India. (???)
A fair coin is tossed n times, with n ≥ 2. Let R be the resulting number of
runs of the same face, and X the number
of isolated heads. Show that the covariance of the random variables R and X is n / 8.
Solution
(December 2010, "Runs Versus Isolated Heads in Coin Tossing")

Proposed by A. A. Dzhumadil'daeva, Almaty, Republics Physics and Mathematics
School, Almaty, Kazakhstan. (January 2009)
Let n!! denote the product of all positive integers not
greater than n and congruent to n mod 2, and let 0!!=(1)!!=1.
For positive integer n, find
in closed form. Two solutions
(December 2010, "A Double Factorial Sum")

Proposed by Emeric Deutsch, Polytechnic University, Brooklyn, NY. (March 2009)
Find the number of bit strings of length n in which the number of 00 substrings is equal
to the number of 11 substrings. For example, when n = 4 we have 4 such bit strings: 0011,
0101, 1010, and 1100.
Solution and Generalization
(April 2011, "Equally Many Repetitions of Each Type"; my version published)

Proposed by M. L. Glasser, Clarkson University, Potsdam, NY. (April 2009)
Find
Γ(1/14) Γ(9/14) Γ(11/14)_{ } Γ(3/14) Γ(5/14) Γ(13/14) 
where Γ denotes the usual gamma function, given by
Γ(z) = ∫_{0}^{∞} t^{ z1} e^{–t} dt.
Solution and Variations
(November 2010, "Gamma Products")

Proposed by Estelle L. Basor, American Institute of Mathematics, Palo Alto,
CA, Steven N. Evans, University of California, Berkeley, CA, and Kent E. Morrison, California
Polytechnic State University, San Luis Obispo, CA. (JuneJuly 2010)
Find a closedform expression for
Solution

Proposed by Andrew McFarland, Płock, Poland ()
Given four concentric circles,
find a necessary and sufficient condition that there be a rectangle with one corner on each circle.
SOLUTION TO APPEAR AUGUST 1, 2011.
