PROBLEMS FOR SEPTEMBER
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# 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

$S=\frac{\mathrm{xy}}{z}+\frac{\mathrm{yz}}{x}+\frac{\mathrm{zx}}{y} .$

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 $\angle \mathrm{CAB}={45}^{ˆ}$.
(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

${x}^{2}+{y}^{2}+{z}^{2}+3\left(x+y+z\right)+5=0 .$

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