Problems for March

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

\frac{ac - b^{2}}{a - 2 b + c} = \frac{bd - c^{2}}{b - 2 c + d},

then each side of the equation is equal to

\frac{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

134.: Suppose that

a = zb + yc

b = xc + za

c = ya + xb .

Prove that

\frac{a^{2}}{1 - x^{2}} = \frac{b^{2}}{1 - y^{2}} = \frac{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 $n \geq 1$ recursively by

\frac{1}{x_{n}} = 1 + 2 p (n) - x_{n - 1} .

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

‖ 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''?

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