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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

${x}_{1}^{2}+{x}_{2}^{2}+\dots +{x}_{n}^{2}+{n}^{3}\le \left(2n-1\right)\left({x}_{1}+{x}_{2}+\dots +{x}_{n}\right)+{n}^{2} .$

Show that
(a) ${x}_{1},{x}_{2},\dots ,{x}_{n}$ are all nonnegative;
(b) ${x}_{1}+{x}_{2}+\dots +{x}_{n}+n+1$ is not a perfect square.

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

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

$f\left(f\left(1\right)\right)=f\left(f\left(2\right)\right)=f\left(f\left(3\right)\right) .$

59.
Let $\mathrm{ABCD}$ be a concyclic quadrilateral. Prove that

$‖\mathrm{AC}-\mathrm{BD}‖\le ‖\mathrm{AB}-\mathrm{CD}‖ .$

60.
Let $n\ge 2$ be an integer and $M=\left\{1,2,\dots ,n\right\}$. For every integer $k$ with $1\le k\le n-1$, let

${x}_{k}=\sum \left\{\mathrm{min} A+\mathrm{max} A:A\subseteq M,A \mathrm{has} k \mathrm{elements}\right\}$

where min $A$ is the smallest and max $A$ is the largest number in $A$. Determine $\sum _{k=1}^{n}\left(-1\right){}^{k-1}{x}_{k}$.

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