PROBLEMS FOR OCTOBER
Solutions should be submitted to
Prof. E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than November 30, 2000.

37.

Let
$\mathrm{ABC}$ be a triangle with sides
$a$,
$b$,
$c$,
inradius
$r$ and circumradius
$R$ (using the conventional
notation). Prove that
$\frac{r}{2R}\le \frac{\mathrm{abc}}{\sqrt{2({a}^{2}+{b}^{2})({b}^{2}+{c}^{2})({c}^{2}+{a}^{2})}}\hspace{1em}.$
When does equality hold?

38.

Let us say that a set
$S$ of nonnegative real
numbers if hunkydory if and only if, for all
$x$ and
$y$ in
$S$, either
$x+y$ or
$\Vert xy\Vert $ is in
$S$. For instance, if
$r$ is positive and
$n$ is a natural
number, then
$S(n,r)=\{0,r,2r,\dots ,\mathrm{nr}\}$
is hunkydory. Show that every hunkydory set
with finitely many elements is
$\{0\}$, is of the form
$S(n,r)$ or has exactly four
elements.

39.

(a)
$\mathrm{ABCDEF}$ is a convex hexagon, each of
whose diagonals
$\mathrm{AD}$,
$\mathrm{BE}$ and
$\mathrm{CF}$ pass through a common
point. Must each of these diagonals bisect the area?
(b)
$\mathrm{ABCDEF}$ is a convex hexagon, each of whose diagonals
$\mathrm{AD}$,
$\mathrm{BE}$ and
$\mathrm{CF}$ bisects the area (so that half the area of
the hexagon lies on either side of the diagonal). Must the
three diagonals pass through a common point?

40.

Determine all solutions in integer pairs
$(x,y)$
to the diophantine equation
${x}^{2}=1+4{y}^{3}(y+2)$.

41.

Determine the least positive number
$p$ for which
there exists a positive number
$q$ such that
$\sqrt{1+x}+\sqrt{1x}\le 2\frac{{x}^{p}}{q}$
for
$0\le x\le 1$. For this least value of
$p$, what
is the smallest value of
$q$ for which the inequality is
satisfied for
$0\le x\le 1$?

42.

$G$ is a connected graph; that is, it consists of
a number of vertices, some pairs of which are joined by edges,
and, for any two vertices, one can travel from one to another
along a chain of edges. We call two vertices adjacent
if and only if they are endpoints of the same edge. Suppose
there is associated with each vertex
$v$ a nonnegative integer
$f(v)$ such that all of the following hold:
(1) If
$v$ and
$w$ are adjacent, then
$\Vert f(v)f(w)\Vert \le 1$.
(2) If
$f(v)>0$, then
$v$ is adjacent to at least one vertex
$w$
such that
$f(w)<f(v)$.
(3) There is exactly one vertex
$u$ such that
$f(u)=0$.
Prove that
$f(v)$ is the number of edges in the chain with the
fewest edges connecting
$u$ and
$v$.