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2000 | Buch

Geometry and Trigonometry Kapitel lesen Erstes Kapitel lesen

Mathematical Olympiad Challenges

verfasst von: Titu Andreescu, Răzvan Gelca

Verlag: Birkhäuser Boston

Enthalten in: Professional Book Archive

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SUCHEN

Über dieses Buch

Mathematical Olympiad Challenges is a rich collection of problems put together by two experienced and well-known professors and coaches of the U.S. International Mathematical Olympiad Team. Hundreds of beautiful, challenging, and instructive problems from algebra, geometry, trigonometry, combinatorics, and number theory were selected from numerous mathematical competitions and journals. An important feature of the work is the comprehensive background material provided with each grouping of problems.

The problems are clustered by topic into self-contained sections with solutions provided separately. All sections start with an essay discussing basic facts and one or two representative examples. A list of carefully chosen problems follows and the reader is invited to take them on. Additionally, historical insights and asides are presented to stimulate further inquiry. The emphasis throughout is on encouraging readers to move away from routine exercises and memorized algorithms toward creative solutions to open-ended problems.

Aimed at motivated high school and beginning college students and instructors, this work can be used as a text for advanced problem- solving courses, for self-study, or as a resource for teachers and students training for mathematical competitions and for teacher professional development, seminars, and workshops.

Inhaltsverzeichnis

Frontmatter

Problems

Frontmatter
Chapter 1. Geometry and Trigonometry
Abstract
Given two points A and B, if one rotates B around A through 60° to a point B’, then the triangle ABB’ is equilateral. A consequence of this result is the following property of the equilateral triangles, which was noticed by the Romanian mathematician D. Pompeiu in 1936. Pompeiu’s theorem is a simple fact, part of classical plane geometry. Surprisingly, it was discovered neither by Euler in the eighteenth century nor by Steinitz in the nineteenth.
Titu Andreescu, Răzvan Gelca
Chapter 2. Algebra and Analysis
Abstract
In this section we will consider some applications of the simplest inequality in algebra:
$$ {{X}^{2}} \geqslant 0 $$
, where equality holds if and only if x = 0.
Titu Andreescu, Răzvan Gelca
Chapter 3. Number Theory and Combinatorics
Abstract
This section is about a problem-solving technique that although simple, can be very powerful. As the title says, the idea is to arrange the elements in order. Here is an example.
Titu Andreescu, Răzvan Gelca

Solutions

Frontmatter
Chapter 1. Geometry and Trigonometry
Abstract
1. The idea is to look at what happen in a neighborhood of a vertex. Let ABC be a triangle that is not equilateral and suppose that AB < BC. Let a = BC − AB. Choose P inside ABC such that PB < a/4, PA − AB < a/4 and BC − PC < a/4. This is possible because of the continuity of the distance function.
Titu Andreescu, Răzvan Gelca
Chapter 2. Algebra and Analysis
Abstract
1. If the inequalities
$$ a - {{b}^{2}} > \frac{1}{4},{\text{ }}b - {{c}^{2}} > \frac{1}{4},{\text{ }}c - {{d}^{2}} > \frac{1}{4},{\text{ }}d - {{a}^{2}} > \frac{1}{4} $$
hold simultaneously, then by adding them we obtain a+b+c+d−(a2+b2+c2+) > 1.
Titu Andreescu, Răzvan Gelca
Chapter 3. Number Theory and Combinatorics
Abstract
1. Let a,b,c,d,e,f be the six digits of the number, arranged in increasing order of their valaue. Choose for the first three digits f,c,aand for the last three digits e,d,b. Then f+c+a−e−d−b = (f−e)+(c−d)+(a−b) ≤ f−e≤9, since c−d and a−b are negative or zero. Also, e+d+b−c−a = (e−f)+(b−c)+(d−a)≤d−a≤9. Hence the number fcaedb satisfies the desired property.
Titu Andreescu, Răzvan Gelca
Backmatter
Metadaten
Titel
Mathematical Olympiad Challenges
verfasst von
Titu Andreescu
Răzvan Gelca
Copyright-Jahr
2000
Verlag
Birkhäuser Boston
Electronic ISBN
978-1-4612-2138-8
Print ISBN
978-0-8176-4155-9
DOI
https://doi.org/10.1007/978-1-4612-2138-8