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| Àâòîðèçàöèÿ |
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| Ïîèñê ïî óêàçàòåëÿì |
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| Tao T. — Solving Mathematical Problems: A Personal Perspective |
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| Ïðåäìåòíûé óêàçàòåëü |
(mod n) notation 10
18, multiples of 12—13
2, powers of 14—18
9, multiples of 9 11—13
Algebra 35
Algebra, examination marks problem 86—89
Algebra, polynomials 41—47
Ami-symmetry 25 26 30
Analysis of functions 36—40
Analytic geometry, line segments 77—79
Analytic geometry, partitioning of rectangles 74—77
Analytic geometry, square swimming pool problem 79—82
Analytic geometry, vector arithmetic 69—74
Angles in circles 50—51
Angles of triangles 50—54
Angles, notation 51
Angles, proof of equality 58 66—68
Areas of triangles 58
Arithmetic progression, lengths of triangle 1 2—7
Bernoulli polynomials 24
Chains, partitioning of rectangle 76—77
Chameleon colour combinations 83—85
Chessboard problem 84
Chocolate-breaking game 90—93
Chords, subtended angles 51 67
Circle theorems 49—50 57 67
Circles, angles in 50—51
Colour combinations, chameleons 83—85
Concurrence, perpendicular bisectors of triangle ix
Conjectures 4
Consequences, proof of 4
Constant polynomials 42
Constructions 58—61
Contradiction, proof by 65—66 76—77
Coordinate geometry 55 58
Coordinate geometry, use in constructions 59
Coprime numbers 10
Cosine rule 3 53
Cubes, sum of 35
Cubic polynomials 42
Cyclic manoeuvres 85
Cyclic quadrilaterals 58
Data, omitting it from problem 5
Data, recording it 3
Data, understanding it 2
Degree of a polynomial (n) 41 42
Degrees of freedom, polynomials 46
Diagonals, lengths of 69—74
Diagrams 3 4
Diameter, angle subtended by (Thales’ theorem) 49—50 57 67
Digits, rearrangement 14—19
Digits, summing 10—14
Digits, summing, powers of 2 16—18
Diophantmc equations 19—22
Direct (forward) approach 54—55
Divisibility, sums of powers 23—26
Divisibility, sums of reciprocals 27—33
Division system, price of penknife 95—97
Elegance of solutions ix
Elimination of variables 61
Equations, use of 3—4
Equilateral triangles, construction 58—60
Euclidean geometry 49—50
Euclidean geometry, angles in circles 50—51
Euclidean geometry, angles of triangles 51—54
Euclidean geometry, constructions 58—61
Euclidean geometry, equating angles 66—68
Euclidean geometry, ratios 55—58
Euclidean geometry, squares and rectangles 62—66
Examination marks problem 86—89
Exponent variables 20—22
Factorization techniques, Diophantine equations 21
Factors 87
Factors of polynomials 42 44 45—47
Factors of polynomials, roots of 43
Fermat’s Last Theorem 20
Finite problems, simplification 13
Formulae, use of 3—4
Forward (direct) approach 54—55
Functions, analysis 36—40
Functions, polynomials 41—47
Generalization 4 15
Geometry see “Analytic geometry” “Euclidean
guessing 14 80—82
Heron’s formula 3 4 6—7
Homogeneous polynomials 42
Induction approach 92
Induction, proof by 37—38 40
Inequalities 88
Inequalities in analysis of functions 36—40
Inequalities in Euclidean geometry 65—66
Infinite products 28
Integer lengths, rectangles 74—77
Irreducible polynomials 42
Isosceles triangles 52
Lagrange’s Theorem 9
Levels of difficulty viii
Line segments, analytic geometry 77—79
Matrix algebra 35
Modification of problems 4—5
Modular arithmetic 9 10
| Modular arithmetic, Diophantine equations 21—22
Modular arithmetic, powers of 2 17—18
Modular arithmetic, squares 96
Modular arithmetic, sums of powers 23 24—26
Modular arithmetic, sums of reciprocals 28—33
Modular arithmetic, vectors 85
Multiples of 9 9 11—13
Natural numbers 10
NIM 91
Notation 3
Notation, vectors 84 90
Number theory 9—10
Number theory, digit rearrangement 14—19
Number theory, digit summing 10—14 16
Number theory, Diophantine equations 19—22
Number theory, sums of powers 23—26
Number theory, sums of reciprocals 27—33
Numerators, reduced 27—28
Objectives of problems 2
p-adics 9
Pairwise cancelling 25 26 30 32
Parallel lines 58—60
Parameterization 44
Partitioning of rectangle 74—77
periodicity 23—24
physical constraints 3
Polygons, lengths of line segments 69—74
Polynomials 41—43
Polynomials and reciprocals 43—44
Polynomials, factorization 44 45—47
Powers of 2, digit rearrangement 14—19
Powers of 2, digit sums 16—18
Powers, sums of 23—26
Prime numbers 10
Problem types 1—2
Proof by contradiction 65—66 76—77
Proof by induction 37—38
Proof by induction, strong induction 40
Proving results 6
Pseudo-coordinate geometry 58
Pythagoras’ theorem 57
Quadnlaterals, midpoints of sides 50
Quadratic formula 20 42
Quadratic polynomials 42
Ratios, Euclidean geometry 55—58
Rearrangement of digits 14—19
Reciprocals and polynomials 43—41
Reciprocals, sums of 27—33
Rectangles, chocolate-breaking game 90—93
Rectangles, in a square 62—66
Rectangles, partitioning 74—77
Reduced numerators 27—28
Reformulation of problems 4
Representation of data and objectives 3
Reversal of problems 5
Roots of factors of polynomials 43
Roots of polynomials 42 44 46
Rotations 60
Similar triangles 56 57
Simplification of problems 4 5 6 13 19 91—92
Sine rule 3 53 54
Skill games 93
Special cases 4—5
Special cases in geometry problems 58
Square roots 20
Square swimming pool problem 79—82
Squares (Euclidean geometry) 62—66
Squares, modular arithmetic 96
Steiner’s theorem of parallel axes 74
Steps in problem-solving 1
Strong induction 40
Sums of cubes 35
Sums of digits 10—14
Sums of lengths of line segments 79
Sums of powers 23—26
Sums of powers, powers of 2 16—18
Sums of reciprocals 27—33
Symmetry 30 32 73—74
Tables 89
Tangents to circle 57
Thales’ Theorem 49—50 57 67
Triangle inequality 3 79
Triangles, angles of 50—54
Triangles, areas of 58
Triangles, concurrence of perpendicular bisectors ix
Triangles, lengths in arithmetic progression 1 2—7
Triangles, similar 56 57
Trigonometry 53—54
Trivial polynomials 42
Variables 3
Variables, elimination of 61
Variables, exponent 20—22
Vector geometry 55 69 73—74
Vectors, chameleon colour combinations 84—85
Vectors, chocolate-breaking game 90
Wilson’s Theorem 9
“Evaluate...” problems 1—2
“Find a...”/“Find all...” problems 1 2
“Is there a...” questions 1 2
“Pocket mathematics” 36
“Show thar...” problems 1—2
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