Send your solutions to Prof. E.J. Barbeau, Department of Mathematics,
University of Toronto, Toronto, ON M5S 3G3 no later than
October 15, 2002. Please make sure that the front page of
your solution contains your complete mailing address and your email
address.

171.

Let
$n$ be a positive integer. In a roundrobin
match,
$n$ teams compete and each pair of teams plays exactly
one game. At the end of the match, the
$i$th team has
${x}_{i}$ wins and
${y}_{i}$ losses. There are no ties. Prove that
${x}_{1}^{2}+{x}_{2}^{2}+\dots +{x}_{n}^{2}={y}_{1}^{2}+{y}_{2}^{2}+\dots +{y}_{n}^{2}\hspace{1em}.$

172.

Let
$a$,
$b$,
$c$,
$d$.
$e$,
$f$ be different
integers. Prove that
$(ab){}^{2}+(bc){}^{2}+(cd){}^{2}+(de){}^{2}+(ef){}^{2}+(fa){}^{2}\ge 18\hspace{1em}.$

173.

Suppose that
$a$ and
$b$ are positive real numbers
for which
$a+b=1$. Prove that
$(a+\frac{1}{a}{)}^{2}+(b+\frac{1}{b}{)}^{2}\ge \frac{25}{2}\hspace{1em}.$
Determine when equality holds.

174.

For which real value of
$x$ is the function
$(1x){}^{5}(1+x)(1+2x){}^{2}$
maximum? Determine its maximum value.

175.

$\mathrm{ABC}$ is a triangle such that
$\mathrm{AB}<\mathrm{AC}$. The point
$D$
is the midpoint of the arc with endpoints
$B$ and
$C$ of that arc of
the circumcircle of
$\Delta \mathrm{ABC}$ that contains
$A$. The foot of the
perpendicular from
$D$ to
$\mathrm{AC}$ is
$E$. Prove that
$\mathrm{AB}+\mathrm{AE}=\mathrm{EC}$.

176.

Three noncollinear points
$A$,
$M$ and
$N$ are given in
the plane. Construct the square such that one of its vertices is
the point
$A$, and the two sides which do not contain this vertex are
on the lines through
$M$ and
$N$ respectively. [Note: In such
a problem, your solution should consist of a description of the
construction (with straightedge and compasses) and a proof in
correct logical order proceeding from what is given to what is
desired that the construction is valid. You should deal with
the feasibility of the construction.]

177.

Let
${a}_{1}$,
${a}_{2}$,
$\dots $,
${a}_{n}$ be nonnegative
integers such that, whenever
$1\le i$,
$1\le j$,
$i+j\le n$, then
${a}_{i}+{a}_{j}\le {a}_{i+j}\le {a}_{i}+{a}_{j}+1\hspace{1em}.$


(a) Give an example of such a sequence which is not
an arithmetic progression.


(b) Prove that there exists a real number
$x$ such that
${a}_{k}=\lfloor \mathrm{kx}\rfloor $ for
$1\le k\le n$.
Since solutions are still being marked for the June set of problems,
their solutions will not be published until October.