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 x1, x2, ¼,xn are nonnegative real numbers for which
x1 + x2 + ¼+ xn < 1/2.
Prove that
|
(1 - x1) (1 - x2) ¼(1 - xn) > |
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 xn recursively by
x0 = 0, x1 = 1, xn+2 = rxn+1 - xn
for n ³ 0. For which values of r is it true
that
|
x1 + x3 + x5 + ¼+ x2m-1 = xm2 |
|
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?
