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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{xy}{z} + \frac{yz}{x} + \frac{zx}{y} .

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

: (b) Prove that the midpioint of both diagonals of $MEKF$ is also the midpoint of the segment $OH$ .

33.: Prove the inequality $a^{2} + b^{2} + c^{2} + 2 abc < 2$ , 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.

: (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 .