Challenges in Geometry for Mathematical Olympians Past and Present

by Bradley, Christopher J.

• ISBN13: 9780198566915
• ISBN10: 0198566913
• Format: Hardcover
• Copyright: 2005-04-28
• Publisher: Oxford University Press

Summary

The International Mathematical Olympiad (IMO) is the World Championship Mathematics Competition for High School students and is held annually in a different country. More than eighty countries are involved. Containing numerous exercises, illustrations, hints and solutions, presented in alucid and thought-provoking style, this text provides a wide range of skills required in competitions such as the Mathematical Olympiad.More than fifty problems in Euclidean geometry involving integers and rational numbers are presented. Early chapters cover elementary problems while later sections break new ground in certain areas and are a greater challenge for the more adventurous reader. The text is ideal for MathematicalOlympiad training and also serves as a supplementary text for students in pure mathematics, particularly number theory and geometry.Dr. Christopher Bradley was formerly a Fellow and Tutor in Mathematics at Jesus College, Oxford, Deputy Leader of the British Mathematical Olympiad Team and for several years Secretary of the British Mathematical Olympiad Committee.

Author Biography

Dr. Christopher Bradley was formerly a Fellow and Tutor in Mathematics at Jesus College, Oxford, Deputy Leader of the British Mathematical Olympiad Team and for several years Secretary of the British Mathematical Olympiad Committee.

Table of Contents

Glossary of symbols	xi
1 Integer-sided triangles	1	(18)
1.1 Integer-sided right-angled triangles	2	(2)
1.2 Integer-sided triangles with angles of 60° and 120°	4	(3)
1.3 Heron triangles	7	(4)
1.4 The rectangular box	11	(4)
1.5 Integer-related triangles	15	(1)
1.6 Other integer-related figures	16	(3)
2 Circles and triangles	19	(24)
2.1 The circumradius R and the inradius r	20	(2)
2.2 Intersecting chords and tangents	22	(2)
2.3 Cyclic quadrilaterals and inscribable quadrilaterals	24	(5)
2.4 The medians of a triangle	29	(5)
2.5 The incircle and the excircles	34	(1)
2.6 The number of integer-sided triangles of given perimeter	35	(3)
2.7 Triangles with angles u, 2u, and 180° - 3u	38	(1)
2.8 Integer r and integer internal bisectors	39	(2)
2.9 Triangles with angles 'a, nu, and 180° - (n + 1)u	41	(2)
3 Lattices	43	(10)
3.1 Lattices and the square lattice	43	(3)
3.2 Pick's theorem	46	(4)
3.3 Integer points on straight lines	50	(3)
4 Rational points on curves	53	(18)
4.1 Integer points on a planar curve of degree two	53	(5)
4.2 Rational points on cubic curves with a singular point	58	(2)
4.3 Elliptic curves	60	(5)
4.4 Elliptic curves of the form y² = x³ - ax - b	65	(6)
5 Shapes and numbers	71	(18)
5.1 Triangular numbers	71	(4)
5.2 More on triangular numbers	75	(3)
5.3 Pentagonal and N-gonal numbers	78	(5)
5.4 Polyhedral numbers	83	(3)
5.5 Catalan numbers	86	(3)
6 Quadrilaterals and triangles	89	(18)
6.1 Integer parallelograms	89	(3)
6.2 Area of a cyclic quadrilateral	92	(4)
6.3 Equal sums of squares on the sides of a triangle	96	(2)
6.4 The integer-sided equilateral triangle	98	(9)
7 Touching circles and spheres	107	(16)
7.1 Three circles touching each other and all touching a line	107	(2)
7.2 Four circles touching one another externally	109	(3)
7.3 Five spheres touching each other externally	112	(3)
7.4 Six touching hyperspheres in four-dimensional space	115	(2)
7.5 Heron triangles revisited	117	(6)
8 More on triangles	123	(22)
8.1 Transversals of integer-sided triangles	123	(3)
8.2 The pedal triangle of three Cevians	126	(5)
8.3 The pedal triangle of a point	131	(3)
8.4 The pivot theorem	134	(2)
8.5 The symmedians and other Cevians	136	(1)
8.6 The Euler line and ratios 2 : 1 in a triangle	137	(4)
8.7 The triangle of excentres	141	(1)
8.8 The lengths of O I and O H	142	(2)
8.9 Feuerbach's theorem	144	(1)
9 Solids	145	(12)
9.1 Tetrahedrons with integer edges and integer volume	145	(4)
9.2 The circumradius of a tetrahedron	149	(4)
9.3 The five regular solids and six regular hypersolids	153	(4)
10 Circles and conics	157	(6)
10.1 Sequences of intersecting circles of unit radius	157	(2)
10.2 Simson lines and Simson conics	159	(2)
10.3 The nine-point conic	161	(2)
11 Finite geometries	163	(4)
11.1 Finite projective and affine geometries	163	(4)
Appendix
A Areal co-ordinates	167	(18)
A.1 Preliminaries	167	(1)
A.2 The co-ordinates of a line	168	(1)
A.3 The vector treatment of a triangle	169	(2)
A.4 Why the co-ordinates (l, m, n) are called areal co-ordinates	171	(2)
A.5 The area of a triangle PQR and the equation of the line PQ	173	(1)
A.6 The areal co-ordinates of key points in the triangle	174	(1)
A.7 Some examples	175	(2)
A.8 The areal metric	177	(2)
A.9 The condition for perpendicular displacements	179	(1)
A.10 The equation of a circle	180	(5)
Answers to exercises	185	(16)
References	201	(2)
Index	203