The Art and Craft of Problem Solving

inequalities
AM-OM, see AM-OM inequality
Bernoulli's, 51, 330
Cauchy-Schwan, see Cauchy-
Schwan inequality
Chebyshev, 185
Euler's, see Euler's inequality
Ptolemy's, 132
Schwan, 342
triangle, 51, see triangle inequality
inexperienced problem solver
attitude, 4, 13,61,200, 331
lack of confidence, 14
poor concentration, 14
infinitude of primes, see prime numbers,
infinitude of
information
free, 63, 99, 202
inhibitions, reducing, 256
inradius, 266, 279
inscribed angle, 264
inscribed circle, see incircle
integers (2:), see sets
integral
definite, 317, 336
interior angle, 259
International Mathematical Olympaid, xi,
8
invariant, 265
invariants, 92-10 6
inverse
of a function, 145
inversion, see transformations,inversion
invertible matrix, 89
investigation, x, 4, 13, 18, 25, 27,39,40,
43,75, 347
irrational numbers, 9, 50, 144, 146, 252,
326
proving irrationality, 50, 171
irreducibility of polynomials, 83
isosceles triangle, 92
ISTS, 40
iteration, 32, 37
IVT (Intermediate value theorem), 323

jazz, 12
Josephus problem, 39
judo, 102
Jumble puzzle, 23
Jungreis, Doug, 339, 350

Kao, John, 35 1
karate, 22
Kedlaya, Kiran, 253
Klee, Victor, 55

Klein, Felix, 257

Lansing, Alfred, 23

lateral thinking, 23

lattice point, 38, 52, 53, 90, 104, 107, 132

law of cosines, 280

law of sines, 280

LCM, see least common multiple

least common multiple, 77, 79, 83, 224,

229

Leningrad, 8

L'Hopital's rule, 340

lightbulb problem, 7

limit, 330, 343

of a sequence, 318 , 322

of a sum, 336, 342

line segment, 259

linear approximation, 331, 346

linear combination, 224, 225 , 247

Liu, Andy, 264

locker problem, 29, 54, 68, 71, 195

logarithmic differentiation, 334

magazines

problems in, 8

magnitude

of complex number, 120, 228

of error, 345

massage, 161, 163, 175, 181, 227

Math Horizons, 8

Mathematical Association of America, 8

mathematical induction, see induction

matrix, 34, 52, 89, 108

mean

arithmetic, 158

geometric, 177

medial triangle, 75, 269

median, 24, 258

mental calculation, 23

mental toughness, 15

midline, 278

midpoint, 269

Mississippi formula, 189, 190

Mobius

function, 238

inversion formula, 239

transformation, 124

modular arithmetic

as invariant, 100

modulo m filter, 24 1, 243

monic polynomial, 164 , 171, 226

monk problem, 7, 17, 53

monotonic function, 176, 327, 337

monotonic sequence, 318

monotonize tactic, 75, 79, 81, 202

INDEX 363

monovariant, 102- 106

Motel Room Paradox, 92

mountaineering, 3, 43, 61

moving curtain, 315, 325

,u-function, see functions, number theo-

retic

multigraph, 109

multinomial theorem, 196, 253

multiplication

of complex numbers, 122, 132

of polynomials, 133, 164

multiplicative function, 235

multiplicative inverse, 44, 23 1

natural numbers (N), see sets

Needham, T., 120, 131, 132

Newman, Donald, 20

non-Euclidean geometry, 26 1

number line, 144, 252

olympiads

other olympiads, 8

one-to-one correspondence, 145

opportunistic strategy, 43, 346

optimistic strategy, 15, 17

optimization, 115, 179, 348

order, created from chaos, 92

orthic triangle, 294

orthocenter, 267

overcounting, 190, 200, 207

packing, 11

palindrome, 83

pantograph, 312

parallel lines, 260-26 1

alternate interior angles, 26 1

and similar triangles, 275

parallelogram, 54, 73, 26 1

angles, 26 1

diagonals, 26 1

edges, 26 1

parity, 94-99

partition, 48

partitioning, 196

Pascal's Triangle

binomial theorem and, 192, 210

combinatorial properties of, 193, 20 1

defined,lO

Fibonacci numbers and, 10, 24, 220

parity and, 10, 39, 253

patterns,look for, 5, 10, 19,26,61, 147

PelI's equation, 228, 246

penultimate step strategy, 95

perfect number, 254, 355