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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

(1 - x_{1}) (1 - x_{2}) \dots (1 - x_{n}) > \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 $(x, y, z)$ of real numbers for which

x (y + 1) = y (z + 1) = z (x + 1) .

16.: Suppose that $ABCDEZ$ 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 $AB$ , $AC$ , $AD$ respectively such that $AF = AG = AH$ .

: (a) Show that $EF$ and $DG$ must intersect in a point $K$ , and that $BG$ and $EH$ must intersect in a point $L$ .

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

17.: Suppose that $r$ is a real number. Define the sequence $x_{n}$ recursively by $x_{0} = 0$ , $x_{1} = 1$ , $x_{n + 2} = {rx}_{n + 1} - x_{n}$ for $n \geq 0$ . For which values of $r$ is it true that

x_{1} + x_{3} + x_{5} + \dots + x_{2 m - 1} = 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

⌊ x ⌋ + 2 y = a

⌊ y ⌋ + 2 x = b

have?

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