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.
Let x, y, z be positive real numbers
for which x2 + y2 + z2 = 1. Find the minimum
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
(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.
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.
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.
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
Prove that there are not three rational
numbers x, y, z such that
x2 + y2 + z2 + 3(x + y + z) + 5 = 0 .|