location:
 PROBLEMS FOR MARCH
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

$\frac{\mathrm{ac}-{b}^{2}}{a-2b+c}=\frac{\mathrm{bd}-{c}^{2}}{b-2c+d} ,$

then each side of the equation is equal to

$\frac{\mathrm{ad}-\mathrm{bc}}{a-b-c+d} .$

Give two essentially different examples of quadruples $\left(a,b,c,d\right)$, not in geometric progression, for which the conditions are satisfied. What happens when $b=c$?

134.
Suppose that

$a=\mathrm{zb}+\mathrm{yc}$

$b=\mathrm{xc}+\mathrm{za}$

$c=\mathrm{ya}+\mathrm{xb} .$

Prove that

$\frac{{a}^{2}}{1-{x}^{2}}=\frac{{b}^{2}}{1-{y}^{2}}=\frac{{c}^{2}}{1-{z}^{2}} .$

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 $p\left(n\right)=k$ 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

$\frac{1}{{x}_{n}}=1+2p\left(n\right)-{x}_{n-1} .$

Prove that every nonnegative rational number occurs exactly once in the sequence $\left\{{x}_{0},{x}_{1},{x}_{2},\dots ,{x}_{n},\dots \right\}$.

136.
Prove that, if in a semicircle of radius 1, five points $A$, $B$, $C$, $D$, $E$ are taken in consecutive order, then

$‖\mathrm{AB}‖{}^{2}+‖\mathrm{BC}‖{}^{2}+‖\mathrm{CD}‖{}^{2}+‖\mathrm{DE}‖{}^{2}+‖\mathrm{AB}‖‖\mathrm{BC}‖‖\mathrm{CD}‖+‖\mathrm{BC}‖‖\mathrm{CD}‖‖\mathrm{DE}‖<4 .$

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''?