PROBLEMS FOR JUNE
CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
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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 u denotes the greatest integer that does not exceed u.


13.
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 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 n0. For which values of r is it true that

x1 + x3 + x5 ++ x2m-1 = xm 2

for m=1,2,3,4,.

18.
Let a and b be integers. How many solutions in real pairs (x,y) does the system

x+2y=a


y+2x=b

have?

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