# Portal through Mathematics: Journey to Advanced Thinking

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*O. A. Ivanov*

Translated by Robert G. Burns

MAA Press: An Imprint of the American Mathematical Society

Reprinted edition available: NML/52

Portal through Mathematics is a collection of
puzzles and problems mostly on topics relating to secondary
mathematics. The problems and topics are fresh and interesting and
frequently surprising. One example: the puzzle that asks how much
length must be added to a belt around the Earth's equator to raise it
one foot has probably achieved old chestnut status. Ivanov, after
explaining the surprising answer to this question, goes a step further
and asks, if you grabbed that too long belt at some point and raised
it as high as possible, how high would that be? The answer to that is
more surprising than the classic puzzle's answer. The book is
organized into 29 themes, each a topic from algebra, geometry or
calculus and each launched from an opening puzzle or problem. There
are excursions into number theory, solid geometry, physics and
combinatorics. Always there is an emphasis on surprise and
delight. And every theme begins at a level approachable with minimal
background requirements. With well over 250 puzzles and problems,
there is something here sure to appeal to everyone.

Portal through Mathematics will be useful for prospective
secondary teachers of mathematics and may be used (as a supplementary
resource) in university courses in algebra, geometry, calculus, and
discrete mathematics. It can also be used for professional development
for teachers looking for inspiration. However, the intended audience
is much broader. Every fan of mathematics will find enjoyment in
it.

# Table of Contents

## Portal through Mathematics: Journey to Advanced Thinking

- Cover cov11
- Half Title Page i2
- Copyright ii3
- Title Page iii4
- Contributors iv5
- Anneli Lax New Mathematical Library v6
- Contents vii8
- Foreword ix10
- Preface for anAmerican Readership xi12
- Author's Preface xiii14
- Part I Surprising and Easy 116
- Part II Algebra, Calculus, and Geometry: problems 2742
- 6 Five problems and a function 2944
- 7 Five solutions of a routine problem 3348
- 8 Equations of the form f(x, y) = g(x, y) and their generalizations 3954
- 9 The generalized version of Viete's formula 4964
- 10 Multiple roots of polynomials 5570
- 11 Non-routine applications of the derivative 6378
- 12 Complex numbers, polynomials, and trigonometry 7186
- 13 Complex numbers and geometry 7994
- 14 Areas of triangles and quadrilaterals 85100
- 15 Constructions in solid geometry 93108
- 16 Inequalities 101116
- 17 Diophantine equations 111126
- 18 Combinatorial tales 119134
- 19 Integrals 129144

- Part III Algebra, Calculus, and Geometry: theory (a little way beyond high school mathematics) 139154
- 20 Functional equations of elementary functions 143158
- 21 Sequences given by recurrence relations 151166
- 22 The "golden ratio" or solving equations of the form f( x) = x 161176
- 23 Convex functions: inequalities and approximations 167182
- 24 Taylor's formula, Euler's formula, and a combinatorial problem 177192
- 25 Derivatives of vector-functions 187202
- 26 Polynomials and trigonometric relations 199214
- 27 Areas and volumes as functions of co-ordinates 207222
- 28 Values of trigonometric functions and sequences satisfying certain recurrence relation 217232
- 29 Do there exist further "numbers" beyond complex numbers? 223238

- Solutions of the supplementary problems 231246
- Index 303318