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
$f(\mathrm{ab})$ 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
$\theta $
so that
$f(x)\equiv \mathrm{cos}{}^{2}x+\mathrm{cos}{}^{2}(x+\theta )\mathrm{cos}x\mathrm{cos}(x+\theta )$
is a constant function of
$x$.

104.

Prove that there exists exactly one sequence
$\{{x}_{n}\}$ of positive integers for which
${x}_{1}=1\hspace{1em},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_{2}>1\hspace{1em},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_{n+1}^{3}+1={x}_{n}{x}_{n+2}$
for
$n\ge 1$.

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
$f(x,y)=\frac{{x}^{4}}{{y}^{4}}+\frac{{y}^{4}}{{x}^{4}}\frac{{x}^{2}}{{y}^{2}}\frac{{y}^{2}}{{x}^{2}}+\frac{x}{y}+\frac{y}{x}$
is attained. Determine that minimum value.

107.

Given positive numbers
${a}_{i}$ with
${a}_{1}<{a}_{2}<\dots <{a}_{n}$, for which permutation
$({b}_{1},{b}_{2},\dots ,{b}_{n})$ of these numbers is the
product
$\underset{i=1}{\overset{n}{\Pi}}({a}_{i}+\frac{1}{{b}_{i}})$
maximized?

108.

Determine all realvalued functions
$f(x)$ of a real variable
$x$ for which
$f(\mathrm{xy})=\frac{f(x)+f(y)}{x+y}$
for all real
$x$ and
$y$ for which
$x+y\ne 0$.