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 x² + y² + z² = 1. Find the minimum value of

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 ÐCAB = 45^°.

(a) Prove, that the quadrilateral MEKF is a square.

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

(c) Find the length of EF, if the radius of Q has length 1 unit.

33.

Prove the inequality a² + b² + c² + 2abc < 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 Ö3?

36.

Prove that there are not three rational numbers x, y, z such that