PROBLEMS FOR MAY
Please send your solutions to
Professor E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than June 30, 2001
Notes. A set in any space is convex if and only
if, given any two points in the set, the line segment
joining them is also contained in the set. A closed
set is one that contains its boundary. A real sequence
$\{{x}_{n}\}$
converges if and only if there is a number
$c$,
called its limit, such that, as
$n$ increases,
the number
${x}_{n}$ gets closer and closer to
$c$. If the
sequences is increasing (i.e.,
${x}_{n+1}\ge {x}_{n}$ for each index
$n$) and bounded above
(i.e., there is a number
$M$ for which
${x}_{n}\le M$
for each
$n$, then it must converge. [Do you see why this is
so?] Similarly, a decreasing sequence that is bounded
below converges. [Supply the definitions and justify the
statement.] An infinite series is an expression of the
form
$\sum _{k=a}^{\infty}{x}_{k}={x}_{a}+{x}_{a+1}+{x}_{a+2}+\dots +{x}_{k}+\dots $, where
$a$ is an integer, usually
0 or 1. The
$n$th partial sum of the series is
${s}_{n}\equiv \sum _{k=a}^{n}{x}_{k}$. The series has sum
$s$ if
and only if its sequence
$\{{s}_{n}\}$ of partial sums
converges and has limit
$s$; when this happens, the
series converges. If the sequence of partial sums
fails to converge, the series diverges. If every
term in the series is nonnegative and the sequence of
partial sums is bounded above, then the series converges.
If a series of nonnegative terms converges, then it is
possible to rearrange the order of the terms without changing
the value of the sum.

79.

Let
${x}_{0}$,
${x}_{1}$,
${x}_{2}$ be three positive real numbers.
A sequence
$\{{x}_{n}\}$ is defined, for
$n\ge 0$ by
${x}_{n+3}=\frac{{x}_{n+2}+{x}_{n+1}+1}{{x}_{n}}\hspace{1em}.$
Determine all such sequences whose entries consist solely
of positive integers.

80.

Prove that, for each positive integer
$n$, the
series
$\sum _{k=1}^{\infty}\frac{{k}^{n}}{{2}^{k}}$
converges to twice an odd integer not less than
$(n+1)!$.

81.

Suppose that
$x\ge 1$ and that
$x=\lfloor x\rfloor +\{x\}$, where
$\lfloor x\rfloor $
is the greatest integer not exceeding
$x$ and the
fractional part
$\{x\}$ satisfies
$0\le x<1$.
Define
$f(x)=\frac{\sqrt{\lfloor x\rfloor}+\sqrt{\{x\}}}{\sqrt{x}}\hspace{1em}\hspace{1em}.$


(a) Determine the small number
$z$
such that
$f(x)\le z$ for each
$x\ge 1$.


(b) Let
${x}_{0}\ge 1$ be given, and for
$n\ge 1$, define
${x}_{n}=f({x}_{n1})$. Prove that
$\underset{n\to \infty}{lim}{x}_{n}$ exists.

82.

(a) A regular pentagon has side length
$a$ and diagonal length
$b$. Prove that
$\frac{{b}^{2}}{{a}^{2}}+\frac{{a}^{2}}{{b}^{2}}=3\hspace{1em}.$


(b) A regular heptagon (polygon with seven equal
sides and seven equal angles) has diagonals of two
different lengths. Let
$a$ be the length of a side,
$b$ be the length of a shorter diagonal and
$c$ be the
length of a longer diagonal of a regular heptagon
(so that
$a<b<c$). Prove that:
$\frac{{a}^{2}}{{b}^{2}}+\frac{{b}^{2}}{{c}^{2}}+\frac{{c}^{2}}{{a}^{2}}=6$
and
$\frac{{b}^{2}}{{a}^{2}}+\frac{{c}^{2}}{{b}^{2}}+\frac{{a}^{2}}{{c}^{2}}=5\hspace{1em}.$

83.

Let
$C$ be a circle with centre
$O$ and radius 1, and let
$F$ be a closed
convex region inside
$C$. Suppose from each point
$C$, we can draw two rays tangent to
$F$
meeting at an angle of
${60}^{\u02c6}$. Describe
$F$.

84.

Let
$\mathrm{ABC}$ be an acuteangled triangle,
with a point
$H$ inside. Let
$U$,
$V$,
$W$ be
respectively the reflected image of
$H$ with respect
to axes
$\mathrm{BC}$,
$\mathrm{AC}$,
$\mathrm{AB}$. Prove that
$H$ is the
orthocentre of
$\Delta \mathrm{ABC}$ if and only if
$U$,
$V$,
$W$ lie on the circumcircle of
$\Delta \mathrm{ABC}$,