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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) > 12 ,

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
 ëx û+ 2y = a
 ëy û+ 2x = b
have?

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