PROBLEMS FOR OCTOBER
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
$v$.