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**.

- 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

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

and

- 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

- 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

- 183.
- Simplify the expression

where $0<\Vert x\Vert <1$.

- 184.
- Using complex numbers, or otherwise, evaluate