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**.

- 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

- 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

- 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

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

have?