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# OLYMON

 Issue 3:8

 October, 2002

 PROBLEMS FOR OCTOBER
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 one-one and onto; this means that, if $f\left(u\right)=f\left(v\right)$, then $u=v$, and, if $w$ is some element of $B$, then $A$ contains an element $t$ for which $f\left(t\right)=w$. Such a function has an inverse ${f}^{-1}$ which is determined by the condition

${f}^{-1}\left(b\right)=a&lrArr;b=f\left(a\right) .$

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

$\sum _{i=1}^{n}\sqrt[n]{{\mathrm{ax}}_{i}+b}\le a+b+n-1$

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

$b={3}^{1}+{3}^{2}+{3}^{3}+\dots +{3}^{2002} .$

180.
Consider the function $f$ that takes the set of complex numbers into itself defined by $f\left(z\right)=3z+‖z‖$. 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

$a\sum _{i=1}^{n}\frac{1}{{a}_{i}}>2\pi .$

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

$\mathrm{MA}.\mathrm{MB}+\mathrm{MB}.\mathrm{MC}+\mathrm{MC}.\mathrm{MA}\ge 1 .$

183.
Simplify the expression

$\frac{\sqrt{1+\sqrt{1-{x}^{2}}}\left(\left(1+x\right)\sqrt{1+x}-\left(1-x\right)\sqrt{1-x}\right)}{x\left(2+\sqrt{1-{x}^{2}}\right)} ,$

where $0<‖x‖<1$.

184.
Using complex numbers, or otherwise, evaluate

$\mathrm{sin}{10}^{ˆ}\mathrm{sin}{50}^{ˆ}\mathrm{sin}{70}^{ˆ} .$

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