364 INDEXperipheral vision, 18, 19, 22, 24, 54, 58,
63
permutation, 191
permutations, see combinations and per
mutations
perpendicular, 259
phantom point, see strategies, phantom
point
I/I-function, see functions, number theo-
retic
piano, 12
Pick's theorem, 52, 54
picture, draw a, 53, 75, 25 1, 340
PIE, see combinatorial strategies and tac-
tics, inclusion-exclusion
pigeonhole principle, 84-92, 204, 250
Platonism, 17
Pleiades contstellation, 22
Poe, Edgar Allan, 23
Poincare, Henri, 257
point at infinity, 308
polar form of complex number, 121
P61ya, 14
P6lya, George, 3, 6
polyhedra, 25, 37, 93
polynomials, 164-173, see coefficients of
a polynomial
division algorithm, 165
factor theorem, 166
fundamental theorem of algebra, 166
monic, 164, 171
operations, 164
primitive, 171
relationship between zeros and coeffi
cients, 168-170
remainder, 165
remainder theorem, 166
postulates, 258
power of a point (quantity), 257, 295
power of a point theorem, 93, see theo-
rem, power of a point
prime numbers, see fundamental theorem
of arithmetic
importance of, 23 1
infinitude of
classical proof, 51, 223
Euler's proof, 348
Prime Power Factorization (PPF), 223,
224,229, 243
primitive
polynomial, 171, 229
root of unity, 254
solution, 23 1
principle of inclusion-exclusion, see
combinatorial strategies and tactics,
inclusion-exclusion
problems
contest, 7
open-ended, 9
recreational, 6
to find, 6
to prove, 6
problems to find, 26
problems to prove, 26
problemsolvingology, II
product
and Gaussian pairing, 68
and parity, 95, 97
Cartesian, 145
catalyst tool, 160
complex numbers, 122, 130, 131
consecutive integers, 249
generating functions, 136, 138
indicator functions, 146, 212
notation, 156
optimize, 177, 179, 183, 185, 348
polynomial, 136
roots, 151, 169
telescope tool, 160
progression
arithmetic, 157
geometric, 158
proof by contradiction, 22, 36,41,43, 74
proportions, 288
Propp, Jim, 21, 73, 130, 205
psychological strategies, see strategies,
psychological
Ptolemy'S theorem, 66, see theo
rem,Ptolemy's
Putnam Exam, see American mathemati
cal contests
Pythagorean theorem, see theorem,
Pythagorean
Pythagorean triples, 150, 242QED, 40
quadratic formula, 165, 187, 219
quadratic residue, 242
quadrilateral
cyclic, 266radical axis, 292, 295
radius
and chords, 264
Ramanujan, S., 347
rate of change, 316 , 328
ratio of similitude, 278
rational numbers (1Ql), see sets
rationalize the denominator, 181
ray, 259real numbers (lR), see sets
recasting problems, 54, 116, 195, see re-
formulating a problem
receptiveness to new ideas, 17, 18, 23
recreational problems, 6, 23
reflection, see transformations, rigid mo
tions, reflection
reflection tool, 64, 66
reformulating a problem, 26, 53-55, 58,
109, 136, 179, 236, 315, see geo
metric interpretation; crossover
relationship between zeros and coeffi-
cients, 168-170,329, 349
relatively prime, 51, 224
remainder
and Taylor series, 315, 345
integral, 44, 83, 85, 94, 100, 225, 228,
230
polynomial, I64
theorem, 166
residue
modulo m, 230
quadratic, 242
restating a problem, 23, 26, 30, 109, 179
retina, 18
rhombus, 279
right angle, 259
right triangle, 266
inscribed in circle, 266
right triangles
and similarity, 275
rigor, 39, 346, 347
rooted tree, 112
roots of unity, 126
and cyclotomic polynomials, 254
as invariant, 130
filter, 141
rotation, see transformations, rigid mo
tions, rotation
routines, 23
rules, breaking, 16, 20, 23SAS condition, 260
Schwarz inequality, 342
secant line, 330, 332
semi-perimeter, 279
sequence
of functions, 343
sequences
and continuity, 323
and monotonizing, 81, 185
arithmetic, 157
Catalan, 218
Cauchy property, 318
convergence of, 317 -322