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
$\lfloor u\rfloor $ denotes the greatest
integer that does not exceed
$u$.

13.

Suppose that
${x}_{1},{x}_{2},\dots ,{x}_{n}$ are nonnegative real numbers for which
${x}_{1}+{x}_{2}+\dots +{x}_{n}<\frac{1}{2}$.
Prove that
$(1{x}_{1})(1{x}_{2})\dots (1{x}_{n})>\frac{1}{2}\hspace{1em},$

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
$x(y+1)=y(z+1)=z(x+1)\hspace{1em}.$

16.

Suppose that
$\mathrm{ABCDEZ}$ 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
$\mathrm{AB}$,
$\mathrm{AC}$,
$\mathrm{AD}$ respectively such that
$\mathrm{AF}=\mathrm{AG}=\mathrm{AH}$.


(a) Show that
$\mathrm{EF}$ and
$\mathrm{DG}$ must intersect
in a point
$K$, and that
$\mathrm{BG}$ and
$\mathrm{EH}$ must intersect
in a point
$L$.


(b) Let
$\mathrm{EG}$ meet the plane of
$\mathrm{AKL}$ in
$M$.
Show that
$\mathrm{AKML}$ is a square.

17.

Suppose that
$r$ is a real number.
Define the sequence
${x}_{n}$ recursively by
${x}_{0}=0$,
${x}_{1}=1$,
${x}_{n+2}={\mathrm{rx}}_{n+1}{x}_{n}$
for
$n\ge 0$. For which values of
$r$ is it true
that
${x}_{1}+{x}_{3}+{x}_{5}+\dots +{x}_{2m1}={x}_{m}^{2}$
for
$m=1,2,3,4,\dots $.

18.

Let
$a$ and
$b$ be integers. How many solutions
in real pairs
$(x,y)$ does the system
$\lfloor x\rfloor +2y=a$
$\lfloor y\rfloor +2x=b$
have?