PROBLEMS FOR JUNE
Solutions should be submitted to
Prof. E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than July 31, 2000.
Notes: The word unique means
exactly one. A regular octahedron
is a solid figure with eight faces, each of
which is an equilateral triangle. You can think
of gluing two square pyramids together along the
square bases. The symbol
ëu û denotes the greatest
integer that does not exceed u.
 13.

Suppose that x_{1}, x_{2}, ¼,x_{n} are nonnegative real numbers for which
x_{1} + x_{2} + ¼+ x_{n} < ^{1}/_{2}.
Prove that
(1  x_{1}) (1  x_{2}) ¼(1  x_{n}) > 
1 2

, 

 14.

Given a convex quadrilateral, is it
always possible to determine a point in its interior
such that the four line segments joining the point
to the midpoints of the sides divide the
quadrilateral into four regions of equal area?
If such a point exists, is it unique?
 15.

Determine all triples (x, y, z)
of real numbers for which
x(y + 1) = y (z + 1) = z(x + 1) . 

 16.

Suppose that ABCDEZ is a
regular octahedron whose pairs of opposite
vertices are (A, Z), (B, D) and (C, E).
The points F, G, H are chosen on the segments
AB, AC, AD respectively such that
AF = AG = AH.


(a) Show that EF and DG must intersect
in a point K, and that BG and EH must intersect
in a point L.


(b) Let EG meet the plane of AKL in M.
Show that AKML is a square.
 17.

Suppose that r is a real number.
Define the sequence x_{n} recursively by
x_{0} = 0, x_{1} = 1, x_{n+2} = rx_{n+1}  x_{n}
for n ³ 0. For which values of r is it true
that
x_{1} + x_{3} + x_{5} + ¼+ x_{2m1} = x_{m}^{2} 

for m = 1, 2, 3, 4,
¼.
 18.

Let a and b be integers. How many solutions
in real pairs (x, y) does the system
have?