PROBLEMS FOR MARCH
Please send your solutions to
Professor E. J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than April 30, 2001.
Notes. A realvalued function
$f$ defined on an
interval is concave iff
$f((1t)u+\mathrm{tv})\ge (1t)f(u)+\mathrm{tf}(v)$ whenever
$0<t<1$
and
$u$ and
$v$ are in the domain of definition of
$f(x)$. If
$f(x)$ is a oneone function defined on
a domain into a range, then the inverse function
$g(x)$ defined on the set of values assumed by
$f$ is determined by
$g(f(x))=x$ and
$f(g(y))=y$;
in other words,
$f(x)=y$ if and only if
$g(y)=x$.

67.

(a)
Consider the infinite integer lattice in
the plane (i.e., the set of points with integer
coordinates) as a graph, with the edges being the
lines of unit length connecting nearby points.
What is the minimum number of colours that can be
used to colour all the vertices and edges of this
graph, so that
(i) each pair of adjacent vertices gets two distinct
colours; AND
(ii) each pair of edges that meet at a vertex get
two distinct colours; AND
(iii) an edge is coloured differently that either
of the two vertices at the ends?
(b) Extend this result to lattices in real
$n$dimensional space.

68.

Let
$a,b,c>0$,
$a<\mathrm{bc}$
and
$1+{a}^{3}={b}^{3}+{c}^{3}$.
Prove that
$1+a<b+c$.

69.

Let
$n$,
${a}_{1}$,
${a}_{2}$,
$\dots $,
${a}_{k}$ be positive integers for which
$n\ge {a}_{1}>{a}_{2}>{a}_{3}>\dots >{a}_{k}$ and the
least common multiple of
${a}_{i}$ and
${a}_{j}$ does not exceed
$n$ for all
$i$ and
$j$. Prove that
${\mathrm{ia}}_{i}\le n$ for
$i=1,2,\dots ,k$.

70.

Let
$f(x)$ be a concave strictly increasing
function defined for
$0\le x\le 1$ such that
$f(0)=0$ and
$f(1)=1$. Suppose that
$g(x)$ is its
inverse. Prove that
$f(x)g(x)\le {x}^{2}$ for
$0\le x\le 1$.

71.

Suppose that lengths
$a$,
$b$ and
$i$ are given.
Construct a triangle
$\mathrm{ABC}$ for which
$\Vert \mathrm{AC}\Vert =b$.
$\Vert \mathrm{AB}\Vert =c$ and the length of the bisector
$\mathrm{AD}$ of angle
$A$ is
$i$ (
$D$ being the point where the
bisector meets the side
$\mathrm{BC}$).

72.

The centres of the circumscribed and the inscribed
spheres of a given tetrahedron coincide. Prove that the
four triangular faces of the tetrahedron are congruent.