The Art and Craft of Problem Solving

Index

AAS condition, 260
absolute value, 51, 174
of a complex number, 121
of terms in a series, 345
of terms of a sequence, 319 , 328
abundant number, 355
add zero creatively tool, 149, 228
Affirmative Action problem, 20-22, 40,
62, 109
AGM inequality, see AM-GM inequality
AHSME, see American mathematical
contests
aikido, 23
AIME, see American mathematical con­
tests
algebraic closure, 145
algebraic methods
add zero creatively, 149, 228
extracting squares, 150
factoring, 6, 34, 42, 148-149, 185,
198,241, 243, 250, 322
simplification, 140, 150-155, 248
algorithm, 37
alternate interior angles, 261
altitude, 267
foot, 267
AM-GM inequality, 176- 1 81,243, 321
algebraic proof, 177, 186
algorithmic proof, 179
Cauchy's proof, 186
geometric proof, 178
American mathematical contests, 7
American Mathematical Monthly, 8
anagrams, 23
Andreescu, Titu, 183
Andretti, Mario, 339
angle bisector theorem, 258
angle chasing, see strategies, angle chas­
ing
angle(s)
alternate interior, 261
central, 264
complementary, 262
exterior, 259
inequalities in triangles, 260
inscribed, 264, 276
inscribed right, 266

360

interior, 259

measure, 259

of a parallelogram, 26 1

right, 259

straight, 259

subtending, 264

supplementary, 261

vertex, 262

vertical, 260

annulus, 128

antiderivative, 315

antidifferentiation, 335

Apostol, Tom, 357

approximation, 161,327,330,346, 353

arc, 264

area,270-274, 286-288

as proof tactic, 277

axioms, 270

of parallelogram, 27 1

of rectangle, 27 1

of rhombus, 279

of trapezoid, 279

of triangle, 27 1, 279, 280

ratios, 27 1, 286

Argand plane, 120

arithmetic

sequence, 9, 71,157, 229

series, 157

ARML, see American mathematical con­

tests

auxiliary object, see strategies, drawing

an auxiliary object

average principle

defined, 176

physical proof, 179

AWD, 40

axioms, 258

backburner problems, 15, 22, 37

backpacking, x

backward induction, 186

balls in urns formula, 204

base-2 representation, 51, 141

Bernoulli's inequality, 51, 330

bijection

defined, 145

used in combinatorics, 196-205

billiard problem, 72

binary representation, 51, 141

binomial coefficient, 190, 249

binomial theorem, 194

and generating functions, 133

and number theory, 234, 248

and Stirling numbers, 22 1

generalized, 353

bipartite graph, see graph theory

bisection method, 327

Boas, Ralph, 357

bowling, 23

box problem, 15-16

brain, 14

brain teasers, see recreational problems

brainstorming, 26, 37

breaking rules, 16, 20, 22, 23

Bugs problem, 65, 72

butt, sticking out, 61

cardinality of a set, 147

cards, 102

cards, playing, 210

Catalan numbers, 218

catalyst tool, 160

Cauchy property, 318

Cauchy-Schwarz inequality

applications, 184

defined, 182

generalizations, 342

proof, 183, 187

cautionary tales, 39

ceiling function, 146

Census-Taker problem, 2

center of symmetry, 300

centroid

as center of mass, 300

centroid theorem, 258

Ceva, Giovanni, 288

cevian, 286

changing point of view strategy, 58

chaos, creating order out of, 92

Chebyshev'S inequality, 185

checker problem, 104

chess, 23

China, 8

Chinese remainder theorem, 234