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, $b\ne c$, both sides of the equation are defined, and

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

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 $n\ge 1$ recursively by

Prove that every nonnegative rational number occurs exactly once in the sequence $\{{x}_{0},{x}_{1},{x}_{2},\dots ,{x}_{n},\dots \}$.

- 136.
- Prove that, if in a semicircle of radius 1, five points $A$, $B$, $C$, $D$, $E$ are taken in consecutive order, then

- 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''?