Please send your solutions to
Professor E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than April 15, 2002.

133.

Prove that, if
$a$,
$b$,
$c$,
$d$ are real numbers,
$b\ne c$,
both sides of the equation are defined, and
$\frac{\mathrm{ac}{b}^{2}}{a2b+c}=\frac{\mathrm{bd}{c}^{2}}{b2c+d}\hspace{1em},$
then each side of the equation is equal to
$\frac{\mathrm{ad}\mathrm{bc}}{abc+d}\hspace{1em}.$
Give two essentially different examples of quadruples
$(a,b,c,d)$,
not in geometric progression, for which the conditions are satisfied.
What happens when
$b=c$?

134.

Suppose that
$a=\mathrm{zb}+\mathrm{yc}$
$b=\mathrm{xc}+\mathrm{za}$
$c=\mathrm{ya}+\mathrm{xb}\hspace{1em}.$
Prove that
$\frac{{a}^{2}}{1{x}^{2}}=\frac{{b}^{2}}{1{y}^{2}}=\frac{{c}^{2}}{1{z}^{2}}\hspace{1em}.$
Of course, if any of
${x}^{2}$,
${y}^{2}$,
${z}^{2}$ is equal to 1, then
the conclusion involves undefined quantities. Give the proper
conclusion in this situation. Provide two essentially different
numerical examples.

135.

For the positive integer
$n$, let
$p(n)=k$ if
$n$ is divisible by
${2}^{k}$ but not by
${2}^{k+1}$. Let
${x}_{0}=0$ and define
${x}_{n}$ for
$n\ge 1$ recursively by
$\frac{1}{{x}_{n}}=1+2p(n){x}_{n1}\hspace{1em}.$
Prove that every nonnegative rational number occurs exactly
once in the sequence
$\{{x}_{0},{x}_{1},{x}_{2},\dots ,{x}_{n},\dots \}$.

136.

Prove that, if in a semicircle of radius 1, five
points
$A$,
$B$,
$C$,
$D$,
$E$ are taken in consecutive order,
then
$\Vert \mathrm{AB}\Vert {}^{2}+\Vert \mathrm{BC}\Vert {}^{2}+\Vert \mathrm{CD}\Vert {}^{2}+\Vert \mathrm{DE}\Vert {}^{2}+\Vert \mathrm{AB}\Vert \Vert \mathrm{BC}\Vert \Vert \mathrm{CD}\Vert +\Vert \mathrm{BC}\Vert \Vert \mathrm{CD}\Vert \Vert \mathrm{DE}\Vert <4\hspace{1em}.$

137.

Can an arbitrary convex quadrilateral be decomposed
by a polygonal line into two parts, each of whose diameters is
less than the diameter of the given quadrilateral?

138.

(a) A room contains ten people. Among any three.
there are two (mutual) acquaintances. Prove that there are
four people, any two of whom are acquainted.


(b) Does the assertion hold if ``ten'' is replaced by
``nine''?