Société mathématique du Canada
Société mathématique du Canada

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 u denotes the greatest integer that does not exceed u.

Suppose that x1, x2, ,xn are nonnegative real numbers for which x1 + x2 + + xn < 1/2. Prove that
(1 - x1) (1 - x2) (1 - xn) > 1

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?

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

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.

Suppose that r is a real number. Define the sequence xn recursively by x0 = 0, x1 = 1, xn+2 = rxn+1 - xn for n 0. For which values of r is it true that
x1 + x3 + x5 + + x2m-1 = xm2
for m = 1, 2, 3, 4, .

Let a and b be integers. How many solutions in real pairs (x, y) does the system
x + 2y = a
y + 2x = b

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