Problems from the Book

1 Some Useful Substitutions 1
1.1 Theory and examples ........................ 3
1.2 Practice Problems .......................... 20
2 Always Cauchy-Schwarz... 25
2.1 Theory and examples ........................ 27
2.2 Practice problems .......................... 43
3 Look at the Exponent 51
3.1 Theory and examples ........................ 53
3.2 Practice problems .......................... 71
4 Primes and Squares 77
4.1 Theory and examples ........................ 79
4.2 Practice problems .......................... 93
5 T2's Lemma 97
5.1 Theory and examples ........................ 99
5.2 Practice problems .......................... 115
6 Some Classical Problems in Extremal Graph Theory 119
6.1 Theory and examples ........................ 121
6.2 Practice problems .......................... 132
7 Complex Combinatorics 137
7.1 Theory and examples ........................ 139
7.2 Practice Problems .......................... 154
8 Formal Series Revisited 159
8.1 Theory and examples ........................ 161
8.2 Practice problems .......................... ' 179
9 A Brief Introduction to Algebraic Number Theory 185
9.1 Theory and examples ........................ 187
9.2 Practice problems .......................... 206
10 Arithmetic Properties of Polynomials 213
10.1 Theory and examples ........................ 215
10.2 Practice problems .......................... 235
11 Lagrange Interpolation Formula 241
11.1 Theory and examples ........................ 243
11.2 Practice problems .......................... 267
12 Higher Algebra in Combinatorics 271
12.1 Theory and examples ........................ 273
12.2 Practice problems .......................... 290
13 Geometry and Numbers 299
13.1 Theory and examples ........................ 301
13.2 Practice problems .......................... 319
14 The Smaller, the Better 325
14.1 Theory and examples ........................ 327
14.2 Practice problems .......................... 339
15 Density and Regular Distribution 345
15.1 Theory and examples ........ ' ................ 347
15.2 Practice problems .......................... 362
16 The Digit Sum of a Positive Integer 367
16.1 Theory and examples ........................ 369
16.2 Practice problems .......................... 383
17 At the Border of Analysis and Number Theory 387
17.1 Theory and examples .- ....................... 389
17.2 Practice problems .......................... 406
18 Quadratic Reciprocity 413
18.1 Theory and examples ........................ 415
18.2 Practice problems .......................... 433
19 Solving Elementary Inequalities Using Integrals 437
19.1 Theory and examples ........................ 439
19.2 Practice problems .......................... 457
20 Pigeonhole Principle Revisited 463
20.1 Theory and examples ........................ 465
20.2 Practice problems .......................... 485
21 Some Useful lrreducibility Criteria 491
21.1 Theory and examples ........................ 493
21.2 Practice problems .......................... 513
22 Cycles. Paths, and Other Ways 519
22.1 Theory and examples ........................ 521
22.2 Practice problems .......................... 533
23 Some Special Applications of Polynomials 537
23.1 Theory and examples ........................ 539
23.2 Practice problems .......................... 557
Bibliography 563
Index 570