PROBLEMS FOR OCTOBER
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# 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

$\frac{r}{2R}\le \frac{\mathrm{abc}}{\sqrt{2\left({a}^{2}+{b}^{2}\right)\left({b}^{2}+{c}^{2}\right)\left({c}^{2}+{a}^{2}\right)}} .$

When does equality hold?

38.
Let us say that a set $S$ of nonnegative real numbers if hunky-dory if and only if, for all $x$ and $y$ in $S$, either $x+y$ or $‖x-y‖$ is in $S$. For instance, if $r$ is positive and $n$ is a natural number, then $S\left(n,r\right)=\left\{0,r,2r,\dots ,\mathrm{nr}\right\}$ is hunky-dory. Show that every hunky-dory set with finitely many elements is $\left\{0\right\}$, is of the form $S\left(n,r\right)$ 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 $\left(x,y\right)$ to the diophantine equation ${x}^{2}=1+4{y}^{3}\left(y+2\right)$.

41.
Determine the least positive number $p$ for which there exists a positive number $q$ such that

$\sqrt{1+x}+\sqrt{1-x}\le 2-\frac{{x}^{p}}{q}$

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 $f\left(v\right)$ such that all of the following hold:
(1) If $v$ and $w$ are adjacent, then $‖f\left(v\right)-f\left(w\right)‖\le 1$.
(2) If $f\left(v\right)>0$, then $v$ is adjacent to at least one vertex $w$ such that $f\left(w\right).
(3) There is exactly one vertex $u$ such that $f\left(u\right)=0$.
Prove that $f\left(v\right)$ is the number of edges in the chain with the fewest edges connecting $u$ and $v$.