Please send your solutions to
Professor E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than April 15, 2002.

Prove that, if a, b, c, d are real numbers, bc, both sides of the equation are defined, and

ac- b2 a-2b+c = bd- c2 b-2c+d ,

then each side of the equation is equal to

ad-bc a-b-c+d .

Give two essentially different examples of quadruples (a,b,c,d), not in geometric progression, for which the conditions are satisfied. What happens when b=c?

Suppose that




Prove that

a2 1- x2 = b2 1- y2 = c2 1- z2 .

Of course, if any of x2 , y2 , z2 is equal to 1, then the conclusion involves undefined quantities. Give the proper conclusion in this situation. Provide two essentially different numerical examples.

For the positive integer n, let p(n)=k if n is divisible by 2k but not by 2k+1 . Let x0 =0 and define xn for n1 recursively by

1 xn =1+2p(n)- xn-1 .

Prove that every nonnegative rational number occurs exactly once in the sequence { x0 , x1 , x2 ,, xn ,}.

Prove that, if in a semicircle of radius 1, five points A, B, C, D, E are taken in consecutive order, then


Can an arbitrary convex quadrilateral be decomposed by a polygonal line into two parts, each of whose diameters is less than the diameter of the given quadrilateral?

(a) A room contains ten people. Among any three. there are two (mutual) acquaintances. Prove that there are four people, any two of whom are acquainted.
(b) Does the assertion hold if ``ten'' is replaced by ``nine''?