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# OLYMON

 Issue 3:8

 October, 2002

 PROBLEMS FOR OCTOBER
Dr. Mihai Rosu
135 Fenelon Drive, #205
Toronto, ON M3A 3K7
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

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