PROBLEMS FOR JUNE
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# 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 $⌊u⌋$ 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

$\left(1-{x}_{1}\right)\left(1-{x}_{2}\right)\dots \left(1-{x}_{n}\right)>\frac{1}{2} ,$

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 $\left(x,y,z\right)$ of real numbers for which

$x\left(y+1\right)=y\left(z+1\right)=z\left(x+1\right) .$

16.
Suppose that $\mathrm{ABCDEZ}$ is a regular octahedron whose pairs of opposite vertices are $\left(A,Z\right)$, $\left(B,D\right)$ and $\left(C,E\right)$. 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

${x}_{1}+{x}_{3}+{x}_{5}+\dots +{x}_{2m-1}={x}_{m}^{2}$

for $m=1,2,3,4,\dots$.

18.
Let $a$ and $b$ be integers. How many solutions in real pairs $\left(x,y\right)$ does the system

$⌊x⌋+2y=a$

$⌊y⌋+2x=b$

have?