PROBLEMS FOR SEPTEMBER
Please send your solutions to
Professor E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than October 31, 2001.
Notes. A unit cube (tetrahedron) is a cube
(tetrahedron) all of whose side lengths are
1.

90.

Let m be a positive integer, and
let f(m) be the smallest value of n for which
the following statement is true:


given any set of n integers, it
is always possible to find a subset of m integers
whose sum is divisible by m
Determine f(m).
[Comment. This problem is being reposed, as no
one submitted a complete solution to this problem
the first time around. Can you conjecture what
f(m) is? It is not hard to give a lower bound for
this function. One approach is to try to relate
f(a) and f(b) to f(ab) and reduce the problem
to considering the case that m is prime; this give
access to some structure that might help.]

103.

Determine a value of the parameter q
so that
f(x) º cos^{2} x + cos^{2} (x + q) cosx cos(x + q) 

is a constant function of x.

104.

Prove that there exists exactly one sequence
{ x_{n} } of positive integers for which
x_{1} = 1 , x_{2} > 1 , x_{n+1}^{3} + 1 = x_{n} x_{n+2} 

for n ³ 1.

105.

Prove that within a unit cube, one can place two
regular unit tetrahedra that have no common point.

106.

Find all pairs (x, y) of positive real numbers
for which the least value of the function
f(x, y) = 
x^{4} y^{4}

+ 
y^{4} x^{4}

 
x^{2} y^{2}

 
y^{2} x^{2}

+ 
x y

+ 
y x



is attained. Determine that minimum value.

107.

Given positive numbers a_{i} with
a_{1} < a_{2} < ¼ < a_{n}, for which permutation
(b_{1}, b_{2}, ¼, b_{n}) of these numbers is the
product

n Õ
i=1


æ ç
è

a_{i} + 
1 b_{i}


ö ÷
ø



maximized?

108.

Determine all realvalued functions
f(x) of a real variable x for which
for all real x and y for which x + y ¹ 0.