Prof. E.J. Barbeau

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
${x}_{1}^{2}+{x}_{2}^{2}+\dots +{x}_{n}^{2}+{n}^{3}\le (2n1)({x}_{1}+{x}_{2}+\dots +{x}_{n})+{n}^{2}\hspace{1em}.$
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
$\angle \mathrm{DAE}={15}^{\u02c6}$. 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
$f(x)={\mathrm{ax}}^{2}+\mathrm{bx}+c$ satisfies
$f(f(1))=f(f(2))=f(f(3))\hspace{1em}.$

59.

Let
$\mathrm{ABCD}$ be a concyclic quadrilateral.
Prove that
$\Vert \mathrm{AC}\mathrm{BD}\Vert \le \Vert \mathrm{AB}\mathrm{CD}\Vert \hspace{1em}.$

60.

Let
$n\ge 2$ be an integer and
$M=\{1,2,\dots ,n\}$. For every integer
$k$
with
$1\le k\le n1$, let
${x}_{k}=\sum \{\mathrm{min}\hspace{1em}A+\mathrm{max}\hspace{1em}A:A\subseteq M,A\hspace{1em}\mathrm{has}\hspace{1em}k\hspace{1em}\mathrm{elements}\}$
where min
$A$ is the smallest and max
$A$ is the largest
number in
$A$.
Determine
$\sum _{k=1}^{n}(1){}^{k1}{x}_{k}$.