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PROBLEMS FOR JANUARY

Solutions should be submitted to

Prof. E.J. Barbeau

Department of Mathematics

University of Toronto

Toronto, ON M5S 3G3

no later than February 28, 2001

55.: A textbook problem has the following form: A man is standing in a line in front of a movie theatre. The fraction $x$ of the line is in front of him, and the fraction $y$ of the line is behind him, where $x$ and $y$ are rational numbers written in lowest terms. How many people are there in the line? Prove that, if the problem has an answer, then that answer must be the least common multiple of the denominators of $x$ and $y$ .

56.: Let $n$ be a positive integer and let $x_{1}, x_{2}, \dots, x_{n}$ be integers for which

x_{1}^{2} + x_{2}^{2} + \dots + x_{n}^{2} + n^{3} \leq (2 n - 1) (x_{1} + x_{2} + \dots + x_{n}) + n^{2} .

Show that

57.: Let $ABCD$ be a rectangle and let $E$ be a point in the diagonal $BD$ with $∠ DAE = 15^{ˆ}$ . Let $F$ be a point in $AB$ with $EF ⊥ AB$ . It is known that $EF = \frac{1}{2} AB$ and $AD = a$ . Find the measure of the angle $∠ EAC$ and the length of the segment $EC$ .

58.: Find integers $a$ , $b$ , $c$ such that $a \neq 0$ and the quadratic function $f (x) = {ax}^{2} + bx + c$ satisfies

f (f (1)) = f (f (2)) = f (f (3)) .

‖ AC - BD ‖ \leq ‖ AB - CD ‖ .

60.: Let $n \geq 2$ be an integer and $M = {1, 2, \dots, n}$ . For every integer $k$ with $1 \leq k \leq n - 1$ , let

x_{k} = \sum {\min A + \max A : A \subseteq M, A has k elements}

where min

A

is the smallest and max

A

is the largest number in

A

. Determine

\sum_{k = 1}^{n} (- 1)^{k - 1} x_{k}