OLYMON
Please send your solution to
Dr. Mihai Rosu
135 Fenelon Drive, #205
Toronto, ON M3A 3K7
It is very important that the front page contain your
complete mailing address and your email address. The deadline
for this set is November 15, 2002.
Notes. A function
$f:A\to B$ is a bijection
iff it is oneone and onto; this means that, if
$f(u)=f(v)$, then
$u=v$, and, if
$w$ is some element of
$B$, then
$A$ contains
an element
$t$ for which
$f(t)=w$. Such a function has an
inverse
${f}^{1}$ which is determined by the condition
${f}^{1}(b)=a\&lrArr;b=f(a)\hspace{1em}.$

178.

Suppose that
$n$ is a positive integer and that
${x}_{1},{x}_{2},\dots ,{x}_{n}$ are positive real
numbers such that
${x}_{1}+{x}_{2}+\dots +{x}_{n}=n$. Prove that
$\sum _{i=1}^{n}\sqrt[n]{{\mathrm{ax}}_{i}+b}\le a+b+n1$
for every pair
$a,bor\; real\; numbers\; with\; all\mathrm{ax\_i\; +\; b}$ nonnegative. Describe the situation when equality occurs.

179.

Determine the units digit of the numbers
${a}^{2}$,
${b}^{2}$ and
$\mathrm{ab}$ (in base 10 numeration), where
$a={2}^{2002}+{3}^{2002}+{4}^{2002}+{5}^{2002}$
and
$b={3}^{1}+{3}^{2}+{3}^{3}+\dots +{3}^{2002}\hspace{1em}.$

180.

Consider the function
$f$ that takes the set of
complex numbers into itself defined by
$f(z)=3z+\Vert z\Vert $.
Prove that
$f$ is a bijection and find its inverse.

181.

Consider a regular polygon with
$n$ sides,
each of length
$a$, and an
interior point located at distances
${a}_{1}$,
${a}_{2}$,
$\dots $,
${a}_{n}$ from the sides. Prove that
$a\sum _{i=1}^{n}\frac{1}{{a}_{i}}>2\pi \hspace{1em}.$

182.

Let
$\mathrm{ABC}be\; an\; equilateral\; triangle\; with\; each\; side\; of\; unit\; length.\; LetMbe\; an\; interior\; point\; in\; the\; equilateral\; triangle\mathrm{ABC}$ with each side
of unit length.Prove that
$\mathrm{MA}.\mathrm{MB}+\mathrm{MB}.\mathrm{MC}+\mathrm{MC}.\mathrm{MA}\ge 1\hspace{1em}.$

183.

Simplify the expression
$\frac{\sqrt{1+\sqrt{1{x}^{2}}}((1+x)\sqrt{1+x}(1x)\sqrt{1x})}{x(2+\sqrt{1{x}^{2}})}\hspace{1em},$
where
$0<\Vert x\Vert <1$.

184.

Using complex numbers, or otherwise, evaluate
$\mathrm{sin}{10}^{\u02c6}\mathrm{sin}{50}^{\u02c6}\mathrm{sin}{70}^{\u02c6}\hspace{1em}.$