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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 (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) .

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]{{ax}_{i} + b} \leq a + b + n - 1

for every pair

a, b or
real numbers with all ax_i + b

nonnegative. Describe the situation when equality occurs.

179.: Determine the units digit of the numbers $a^{2}$ , $b^{2}$ and $ab$ (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 $f (z) = 3 z + ‖ z ‖$ . 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 π .

182.: Let $ABC be an equilateral triangle with each side
of unit length. Let M be an interior point in the equilateral triangle ABC$ with each side of unit length.Prove that

MA . MB + MB . MC + MC . MA \geq 1 .

183.: Simplify the expression

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

where

0 < ‖ x ‖ < 1

184.: Using complex numbers, or otherwise, evaluate

\sin 10^{ˆ} \sin 50^{ˆ} \sin 70^{ˆ} .

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