PROBLEMS FOR MARCH
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.

133.

Prove that, if a, b, c, d are real numbers,
b ¹ c,
both sides of the equation are defined, and

ac  b^{2} a  2b + c

= 
bd  c^{2} b  2c + d

, 

then each side of the equation is equal to
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?

134.

Suppose that
Prove that

a^{2} 1  x^{2}

= 
b^{2} 1  y^{2}

= 
c^{2} 1  z^{2}

. 

Of course, if any of x^{2}, y^{2}, z^{2} is equal to 1, then
the conclusion involves undefined quantities. Give the proper
conclusion in this situation. Provide two essentially different
numerical examples.

135.

For the positive integer n, let p(n) = k if
n is divisible by 2^{k} but not by 2^{k+1}. Let
x_{0} = 0 and define x_{n} for n ³ 1 recursively by

1 x_{n}

= 1 + 2p(n)  x_{n1} . 

Prove that every nonnegative rational number occurs exactly
once in the sequence { x_{0}, x_{1}, x_{2}, ¼, x_{n}, ¼}.

136.

Prove that, if in a semicircle of radius 1, five
points A, B, C, D, E are taken in consecutive order,
then
AB ^{2} + BC ^{2} + CD ^{2} +DE ^{2} + AB BC CD + BC CD DE  < 4 . 


137.

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?

138.

(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''?