chord, 264
andradii, 264
Chvatal, Vaclav, 55
circles
and arcs, 265
arc, 264
chord, 264
general, 264-266
generalized, 308
inscribed angle, 264
inscribed in triangle, 266
relationship between chords and radii,
264
tangent, 264
circumcenter, 266
circumcircle, 266
construction, 267
existence and uniqueness, 267
circumradius, 266, 279
circumscribed circle, see circumcircle
Cis 8, 121
climber, 4, 13, 61
coefficients of a polynomial, 36, 69, 133
and zeros, 168, 170, 171, 226
collinear points, 289-29 1
and Menalaus's theorem, 295
coloring
of graphs, 21, 49
use of, 55, 101
coloring problems, 21, 49, 55, 84, 101
combination, 191
combinations and permutations, 188--191
combinatorial arguments, 191
combinatorial strategies and tactics
count the complement, 200, 207, 211,
236
encoding, 197, 199, 203, 218
inclusion-exclusion, 207-2 14
partitioning, 196, 199, 206
complete graph, see graph theory
complete theory, 115, 242, 250
completing the square tool, 149
complex numbers, 120--13 2
absolute value, 121
and vectors, 121, 122, 128
as transformations, 123
Cis8, 121
conjugation, 121
Euler's formula, 123
geometric interpretation, 120-- 126
magnitude, 120
Mobius transformations, 124
multiplication, 122
polar form, 121
roots of unity, 126, 130
Conan Doyle, 23
concentration, 14, 23
mental calculation, 23
concurrent lines
altitudes, 267, 294
angle bisectors, 267, 294
chords, 292
conditions for, 288
medians, 294
perpendicular bisectors, 267
concyclic points, 266, 282 -286
confidence, x, 14, 15,27, 154
congruence
definitions and properties, 44
multiplicative inverse, 44
congruence (geometric)
AAS condition, 260
conditions for, 260
SAS condition, 260
SSS condition, 260
congruence (number theoretic)
definitions and properties, 230
multiplicative inverse, 23 1
congruence theorems
Wilson's, 68
congruence theorems (number theoretic)
Chinese remainder, 234
Euler's extension of Fermat's little,
239
Fermat's little, 232 -233
combinatorial proof, 249
induction proof, 234
Wilson's, 252
congruent triangles, 259
conjecture, 2, 5, 6, 10, 28, 37, 195, 23 1
conjugation, 121
connected component of graph, 111 , 118
constructions (compass and ruler), 269,
280
contest problems
American, 7
other nations, 8
continued fractions, 246, 326
continuity, 317 -325
and fundamental theorem of calculus,
325
defined, 323
uniform, 324
contradiction, 22, 36, 41, 43, 74
contrapositive, 41
convergence
of sequences, 317 -322
of sums, 162, 344
uniform, 343
converse, 41convex polygon, 294
Conway, JohnINDEX 361checker problem of, 104
creativity, 17
crossover, see reformulating a problem
defined, 54
examples of, 247
tactics, 109- 142
crossword puzzles, 23
crux move, 4, 6, 21, 46, 51, 70
culture
problem solving, xi
cycles, see graph theory
cyclic
permutation, 70, 249
quadrilateral, 66
sum, 71
symmetry, 70, 101, 15 4
cyclic quadrilateral, 266
cyclic quadrilaterals, 283
cyclotomic polynomial, 255d-function, see fu nctions, number theo-
retic
de Bruijn, 98, 120
decimal representation, 91
deck of cards, 102
deductive argument, 40, 41
definite integral
as area under a curve, 317
as sum, 336
degree of vertex, see graph theory
dense sets, 326
derangement, 214, 220
derivative
algebraic interpretation, 330
dynamic interpretation, 328
geometric interpretation, 328
determinant, 34, 89
dice, 8, 142
differentiation of series, 344
digraph, 115
diophantine equations, 240
Fermat's Last Theorem, 230
linear, 228
Pell's, 246
strategies and tactics, 240
sum of two squares, 250-- 253
directed length, 306
Dirichlet, 84
disjoint sets, 196
dissection, 263
distance from a point to a line, 279
distance-time graph, 53
divisibility