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| Cofman J. — What to Solve? Problems and Suggestions for Young Mathematicians |
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| Ïðåäìåòíûé óêàçàòåëü |
 , the number, Archimedes' algorithm for calculating 137
, the number, Leibniz' series for 138 164
, the number, problems 137—140
, the number, solutions 160—166
Abel, Niels Henrik 234
Algebra, fundamental theorem of 140
Algebraic structures 144 214—218
Archimedes of Syracuse 137 146 234
Argand diagrams 220—221
Arithmetic progressions 12—13 41—43 133—134
Axioms 145
Axonometry 141—142 166—167
Binomial coefficients 38 230
Block designs 183 192—194 231 232
Block designs, balanced incomplete 231—232
Bolyai, Janos 145 234
Buffon's Needle Problem 139—140 165—166
Buffon, Comte de, Georges Louis Leclerc 139 234
Catalan's problem 149—150 172—173
Cayley, Arthur 145 235 240
Centres of gravity 131
Chameleons problem 97 115
Chessboards, generalized 22—23 74—77
Chessboards, king moves on 11 12 39—40
Chessboards, knight moves on 10—11 35—37
Chessboards, natural numbers on 91—92
Chessboards, number patterns from 11—12 37—40
Chessboards, rook moves on 11 22—23 37—38 74—77
Circles and lattice points 186 199
Circles, carrying at least three points 182—183 191—192
Circles, circumferences of 137
Circles, circumscribed about polygons 221
Circles, construction of 96 112
Circles, inscribed in polygons 221
Circles, inscribed in triangles 25—26 85—87
Circles, inversion with respect to 225—226
Circles, properties of points on 21 69—70
Circles, rolling 97 116—117
Coefficients of polynomials 213
Coefficients, binomial 38 230
Coefficients, multinomial 230
Coefficients, trinomial 40 76 230
Combinations 229
Combinatorics 149 183 229
complex numbers 208 220
Complex numbers, applications of problems 140—145
Complex numbers, applications of solutions 166—168
Composite numbers 208
Congruences 208
Contradiction, proof by 94
Contradiction, proof by, problems 99
Contradiction, proof by, solutions 127
Converses 4—5
Converses, problems 23—27
Converses, solutions 77—88
Coordinate systems 141 219—221
Coordinate systems, barycentric 107—110 220
Coordinate systems, Cartesian 70 98 103 120—121 186 219—220
Copernicus, Nicholas 97 235
Counting, the art of 149
Counting, the art of, problems 149—153
Counting, the art of, solutions 172—179
Cube numbers 41 210
Cube numbers, sum of 45
Cubes, division of 98 118—120
Cycles 105 231
de Bruijn, N.G. 183 234
de Fermat, Pierre 187 188 236
Desargues, Girard 235
Desargues, theorem of, in plane 97 113—115
Desargues, theorem of, in space 25 85
Descartes, Rene 235
Diophantine equations 203 235
Diophantus of Alexandria 235
Dirichlet's principle 184
Dirichlet, Peter Gustav Lejeune 134 235
Dodecahedra 112
Dominoes 95 99—101
Duality, principle of 171
Edges 230—231
Erdoes, Paul 181 182 183 185 235
Euclid of Alexandria 133 145 153 236
Euclid's parallel postulate 145
Euler's problem on polygon division 150—151 174
Euler, Leonhard 150 188 236
Eulerian paths 100
Exceptions and special cases 3—4
Exceptions and special cases, problems 17—19
Exceptions and special cases, solutions 55—66
Extending the field of investigation 90—91
Extending the field of investigation, problems 96—97
Extending the field of investigation, solutions 110—115
Extremal elements, use of 91—92
Extremal elements, use of, problems 97—98
Extremal elements, use of, solutions 117—118
Factors 208
Faltings, Gerd 188 236
Fermat's last theorem 187—188 201—204
Fermat's little theorem 24 81 155 157
Fibonacci numbers 15 38 48 210 236
Fibonacci numbers, properties of 8 32
Fibonacci, son of Bonaccio 236
Figurate numbers 208—210
Fox, Captain 140
Fractions 205
Fractions in simplest terms 16 53
Fractions, decimal 205—206
Fractions, successors of 16—17 53—54
Gabriel-Marie, Fr 127
Gallai, problem of Sylvester — Gallai 180—183 188—194
Gallai, Tibor 181 236
Gauss' fundamental theorem of axonometry 140—142 166—167
Gauss, Carl Friedrich 140 236
Generalizing given problems 4
Generalizing given problems, problems 19—23
Generalizing given problems, solutions 66—77
Geometry, Euclidean problems 145—146 180—183
Geometry, Euclidean solutions 168—170 188—192
Geometry, non-Euclidean 145
Geometry, non-Euclidean, problems 146—149
Geometry, non-Euclidean, solutions 170—172
graphs 230—231
Graphs, connected 104 231
Graphs, oriented 99—101 231
Gregory's arc tangent series 138—139 161—164
Gregory, James 138 237
groups 214—216
Hamilton, William Rowen 143 144 237
Hanani, Haim 183 194 232 237
Heron of Alexandria 126 237
Hexagons, regular 121
Hilbert, David 237—238 240 244
Hurwitz, Adolf 134 238
Hyperbolic geometry 145
imaginary numbers 140 207
incidence 148 170—172
Incident pairs 123 124
Induction, mathematical 94
Induction, mathematical, problems 98—99
Induction, mathematical, solutions 121—124
Infinite descent, method of 92—94
Infinite descent, method of, problems 98
Infinite descent, method of, solutions 118—121
Integers, positive 205
Integers, positive, as solutions to equations 96 97 102—104 117
Invariants of transformations, use of 91
Invariants of transformations, use of, problems 97
Invariants of transformations, use of, solutions 115—117
inversion 191 225—227
iterating 1—2
| Iterating, problems 5—10
Iterating, solutions 27—37
Ladder, sliding 91
Lagrange's identity 142—144
Lagrange, Joseph Louis 134 142 238
Language, different, expressing problem in 89—90
Language, different, expressing problem, problems 95—96
Language, different, expressing problem, solutions 99—110
Lattice points 98 120—121 186—187 199—200
Leibniz' series for 138 164
Leibniz' theorem 135 157—158
Leibniz, Gottfried Wilhelm 8 135 138 238
Light rays 126
Limits 232—233
Line segment, division of 96 107—110 122—123
Line, ideal 147 148
Lobachevsky, Nicolai Ivanovitch 145 238
Loops 231
Markoff numbers 85
Markoff, A.A. 238
Matijasevic, Jurii Vladimirovic 136 238
Matrices 21—22 73—74 212—213
Minkowski, Hermann 180 186 238
Moebius 107
Motzkin, T.S. 181 239
Mouseholes problem 18 58—59
n-gons see "Polygons"
Natural numbers 205
Natural numbers as solutions of equations 19 63—64 187—188 201—204
Natural numbers as sums of square numbers 142—145 167—168
Natural numbers, number expressed as sum of 19 64—65
Natural numbers, odd divisors of 17 54—55
Natural numbers, prime divisors of 154—155
Natural numbers, square numbers between 110—112
Natural numbers, sum of powers of 98 121—122
Necessary conditions 4—5
News transmission problem 96 104—106
Newton, Isaac 239
Norms 144
Number sequences 232—233
Number series 7—8
Number theory, analytic 134
Operations, algebraic 214
Pappus Of Alexandria 19 146 239
parallelograms 18 19—20 48 57—58 66—67
Pascal's triangle 37 38 211 239
Pascal, Blaise 239
Paths 104—106 231
Patterns, search for 2—3
Patterns, search for, problems 11—17
Patterns, search for, solutions 37—55
Peano, Giuseppe 239
Pentagonal numbers 209
Pentagons 24 79—80
Pentagons, regular 8 30 31 98 120—121
Pentagrams 8 30 31
permutations 229 230
Permutations, zigzag 151—153 175—177
Physics, employment of 95
Physics, employment of, problems 99
Physics, employment of, solutions 125—132
Pigeon-hole principle 184 194—197
Plane, division of 21 71—72
Planes in three-dimensional space 181—183 189—191
Planes, projective 146—149
Plutarch 24
Point sets, convex 221—223
Points, ideal 147 148
Polygons (n-gons) 24 80—81 99 109—110 125 221
Polygons (n-gons), convex 18 56—57 150—151 174 221
Polygons (n-gons), regular 15 47—48 98 120—121 137 160—161
Polyhedra 19 65—66
Polynomial equations 140
Polynomials 98 121—122 134—135 155—157 213—214
Polynomials with prime number values 135—136 157—158
Positive numbers, products of 20—21 68—69
Prime numbers 24 208
Prime numbers, problems on 10 13 24 96 133—136
Prime numbers, problems on, solutions to 34 43 110 153—159
Prime numbers, relatively 15 53 208
Prime numbers, relatively, properties of 25 82 89—90 188 203—204
Probability theory 139
Projection, stereographic 112 227—228
Pyramids 20 67—68
Pythagoras of Samos 239
Pythagoras of Samos, theorem of 239
Quadrilaterals, areas of 19 61—62
Quadrilaterals, cyclic 62 129 223
Quaternions 143—144 167—168 208 225
Ramsey numbers 184—186 197—199
Ramsey, Frank Plumpton 185 239—240
Rational numbers 205 207
Real numbers 207
Rectangles 18 24 58 79
Reflection 228—229
Regions, division of plane into 21 71—72
Regions, division of space into 21 72—73
Rhombuses 15 47—48
Rotation 143 144 223—225
Scalars 216
Schoenberg, Isaac 186—187 240
Schwartz 129
sec x 151 153 178—179
Sequences, number 110—112 232—233
Series, number 7—8
Series, number, harmonic 212
Shoemaker's knife 146 168—170
Sierpiriski, Waclaw 71 154 186 240 244 246
Space, Division of 21 72—73
Sphere, cut by planes 26—27 87—88
Sphere, inversion with respect to 226—227
Sphere, properties of points on 21 70
Square numbers 19 24 42 43 62—63 209
Square numbers between terms of sequences 96 110—112
Square numbers, natural numbers as sums of 142—145 167—168
Square numbers, prime numbers as sums/differences of 136 158—159
Square, Latin 21
squares 15 18 23 48—50 58 78—79
Steinhaus, Hugo 186 240 242
Structures, algebraic 144 214—218
Sufficient conditions 4—5
Sundaram's sieve 13 43
Sylvester, James Joseph 181 240
Sylvester, problem of Sylvester — Gallai 180—183 188—194
Tan x 151 153 177—179
Tetrahedra 98—99 109 110 124 132
Tetrahedral numbers 209
transformations, geometric 223—229
Translation 228
tree diagrams 17 25 53 82—83 85
Trees 105—106
Triangle, harmonic 8 211—212
Triangle, Pascal's 37 38 211 239
Triangles, areas of 97—98 117—118
Triangles, circles inscribed in 25—26 85—87
Triangles, congruent 77 78
Triangles, construction of, problems 6—7 96 97—98 99
Triangles, construction of, solutions 29 107—108 118 125—130
Triangles, division of sides of 99 131—132
Triangles, equilateral 121
Triangles, isosceles 14 18 23 45—47 77—78
Triangles, right-angled 17—18 55—56 103—104
Triangular numbers 24 41 47 209
Tribonacci numbers 40
Trinomial coefficients 40 76 230
Triples, Pythagorean 21 210—211
tripods 141 142
Vector spaces, real 216—218
Vectors 35 69—70 216—218
Vertices 230
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