PROBLEMS FOR SEPTEMBER
Solutions should be submitted to
Dr. Valeria Pendelieva
708  195 Clearview Avenue
Ottawa, ON K1Z 6S1
Solution to these problems should be
postmarked no later than October 31, 2000.

31.

Let
$x$,
$y$,
$z$ be positive real numbers
for which
${x}^{2}+{y}^{2}+{z}^{2}=1$. Find the minimum
value of
$S=\frac{\mathrm{xy}}{z}+\frac{\mathrm{yz}}{x}+\frac{\mathrm{zx}}{y}\hspace{1em}.$

32.

The segments
$\mathrm{BE}$ and
$\mathrm{CF}$ are altitudes of the
acute triangle
$\mathrm{ABC}$, where
$E$ and
$F$ are points on the
segments
$\mathrm{AC}$ anbd
$\mathrm{AB}$, resp[ectively.
$\mathrm{ABC}$ is
inscribed in the circle Q with centre
$O$. Denote the
orthocentre of
$\mathrm{ABC}$ be
$H$, and the midpoints of
$\mathrm{BC}$ and
$\mathrm{AH}$ be
$M$ and
$K$, respectively. Let
$\angle \mathrm{CAB}={45}^{\u02c6}$.


(a) Prove, that the quadrilateral
$\mathrm{MEKF}$ is a
square.


(b) Prove that the midpioint of both diagonals
of
$\mathrm{MEKF}$ is also the midpoint of the segment
$\mathrm{OH}$.


(c) Find the length of
$\mathrm{EF}$, if the radius of
Q has length 1 unit.

33.

Prove the inequality
${a}^{2}+{b}^{2}+{c}^{2}+2\mathrm{abc}<2$,
if the numbers
$a$,
$b$,
$c$ are the lengths of the sides
of a triangle with perimeter 2.

34.

Each of the edges of a cube is 1 unit in length,
and is divided by two points into three equal parts.
Denote by K the solid with vertices at these points.


(a) Find the volume of K.


(b) Every pair of vertices of K is
connected by a segment. Some of the segments are
coloured. Prove that it is always possible to find
two vertices which are endpoints of the same number
of coloured segments.

35.

There are
$n$ points on a circle whose
radius is 1 unit. What is the greatest number of
segments between two of them, whose length exceeds
$\sqrt{3}$?

36.

Prove that there are not three rational
numbers
$x$,
$y$,
$z$ such that
${x}^{2}+{y}^{2}+{z}^{2}+3(x+y+z)+5=0\hspace{1em}.$