Challenges in Geometry for Mathematical

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.

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.

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

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Challenges in Geometry for Mathematical Olympians Past and Present

Summary

Author Biography

Table of Contents

Supplemental Materials

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

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Challenges in Geometry for Mathematical Olympians Past and Present

Summary

Author Biography

Table of Contents

Supplemental Materials

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