prasolov_zadachi_alg
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Olympiad
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Entries: 352
Прасолов В.В.
Орыс тілінде
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Entries
352
Preface
Other
Page 11
Chapter 1. Quadratic trinomial
Theory
Page 13
1.1. Minimum value of a quadratic trinomial
Theory
Page 13
1.2. Discriminant
Theory
Page 14
1.3. Various problems
Problem
Page 14
1.4. Intermediate value theorem
Theory
Page 15
1.5. Equation of a tangent to a conic
Theory
Page 16
1.6. Resultant
Theory
Page 16
Solutions
Answer
Page 16
Chapter 2. Equations
Theory
Page 23
2.1. Change of variables
Theory
Page 23
2.2. Guessing roots
Theory
Page 23
2.3. Equations with radicals
Theory
Page 24
2.4. Various equations
Problem
Page 24
Solutions
Answer
Page 24
Chapter 3. Systems of equations
Theory
Page 28
3.1. Finding all solutions
Theory
Page 28
3.2. Finding real solutions
Theory
Page 29
3.3. Positive solutions
Theory
Page 30
3.4. Number of solutions of a system of equations
Theory
Page 30
3.5. Systems of linear equations
Theory
Page 31
Solutions
Answer
Page 33
Chapter 4. Divisibility
Theory
Page 39
4.1. Even and odd
Theory
Page 39
4.2. Euclidean algorithm and the fundamental theorem of arithmetic
Theory
Page 40
4.3. Prime factorization
Theory
Page 41
4.4. Divisibility tests
Theory
Page 41
4.5. Greatest common divisor and least common multiple
Theory
Page 42
4.6. Exact divisibility
Theory
Page 42
4.7. Divisibility by a power of a prime
Theory
Page 43
4.8. Remainders of division
Theory
Page 44
4.9. Relatively prime numbers
Theory
Page 45
4.10. Prime numbers
Theory
Page 45
4.11. Arithmetic of remainders
Theory
Page 46
Solutions
Answer
Page 46
Chapter 5. Identities
Theory
Page 59
5.1. Factorization
Theory
Page 59
5.2. Proving identities
Theory
Page 59
5.3. Sums of squares
Theory
Page 60
5.4. Auxiliary identities
Theory
Page 60
5.5. Decompositions of rational functions
Theory
Page 61
5.6. Decompositions of quadratic functions
Theory
Page 61
5.7. Identities with integer parts
Theory
Page 61
Solutions
Answer
Page 62
Chapter 6. Rational and irrational numbers
Theory
Page 67
6.1. Comparing numbers
Theory
Page 67
6.2. Irrationality in denominators
Theory
Page 67
6.3. Identities with radicals
Theory
Page 68
6.4. Proofs of irrationality and rationality
Theory
Page 69
6.5. Conjugate numbers
Theory
Page 69
6.6. Farey sequence
Theory
Page 70
6.7. Problems with integer parts
Problem
Page 71
Solutions
Answer
Page 71
Chapter 7. Word problems
Problem
Page 79
7.1. Solutions without calculations
Theory
Page 79
7.2. Calculations
Theory
Page 79
7.3. Inequalities
Theory
Page 80
7.4. Integer approximations
Theory
Page 81
7.5. Correspondences
Theory
Page 81
Solutions
Answer
Page 82
Chapter 8. Inequalities
Theory
Page 86
8.1. x + 1/x ≥ 2 теңсіздігі
Theory
Page 86
8.2. Triangle inequality
Theory
Page 87
8.3. Cauchy's inequality
Theory
Page 87
8.4. Monotonicity
Theory
Page 88
8.5. Inequality between arithmetic and geometric means
Theory
Page 88
8.6. Inequalities with geometric interpretation
Theory
Page 89
8.7. Cyclic inequalities
Theory
Page 89
8.8. Various inequalities
Problem
Page 90
8.9. Convexity
Theory
Page 91
8.10. Hölder and Minkowski inequalities
Theory
Page 91
Solutions
Answer
Page 92
Chapter 9. Calculating sums and products
Theory
Page 104
9.1. Arithmetic and geometric progressions
Theory
Page 104
9.2. Changing the order of summation
Theory
Page 105
9.3. Sums S_k(n)=1^k+2^k+...+n^k
Theory
Page 105
9.4. Splitting into pairs
Theory
Page 106
9.5. Calculating one sum in two ways
Theory
Page 107
Solutions
Answer
Page 107
Chapter 10. Polynomials I
Theory
Page 112
10.1. Completing the square
Theory
Page 112
10.2. Roots of polynomials
Theory
Page 112
10.3. Coefficients of a polynomial
Theory
Page 113
10.4. Vieta's theorem
Theory
Page 113
10.5. Divisibility
Theory
Page 113
10.6. Inequalities for roots
Theory
Page 114
10.7. Number of real roots of a polynomial
Theory
Page 115
10.8. Various problems
Problem
Page 115
10.9. Interpolation polynomials
Theory
Page 116
10.10. Rational functions
Theory
Page 116
10.11. Integer-valued polynomials
Theory
Page 117
10.12. Polynomials in several variables
Theory
Page 117
Solutions
Answer
Page 117
Chapter 11. Trigonometry
Theory
Page 127
11.1. Inequalities and comparison of numbers
Theory
Page 127
11.2. Trigonometric identities
Theory
Page 128
11.3. Equations
Theory
Page 128
11.4. Sums of sines and cosines related to regular polygons
Theory
Page 129
11.5. Calculating sums and products
Theory
Page 129
11.6. Expressions for cos nφ and similar functions
Theory
Page 131
11.7. Auxiliary trigonometric functions
Theory
Page 131
11.8. Trigonometric polynomials
Theory
Page 132
Solutions
Answer
Page 132
Chapter 12. Equations in integers
Theory
Page 143
12.1. Pythagorean triples
Theory
Page 143
12.2. Finding all solutions
Theory
Page 144
12.3. Finding some solutions
Theory
Page 145
12.4. Proving finiteness of the number of solutions
Theory
Page 145
12.5. Pell equation
Theory
Page 145
12.6. Markov equation
Theory
Page 146
Solutions
Answer
Page 146
Chapter 13. Induction
Theory
Page 154
13.1. Calculating sums
Theory
Page 154
13.2. Inequalities
Theory
Page 154
13.3. Proving identities
Theory
Page 155
13.4. Various problems
Problem
Page 156
Solutions
Answer
Page 156
Chapter 14. Combinatorics
Theory
Page 160
14.1. Elements of combinatorics
Theory
Page 160
14.2. Identities for binomial coefficients
Theory
Page 161
14.3. Newton's binomial theorem in arithmetic
Theory
Page 162
14.4. Combinatorics in arithmetic
Theory
Page 162
14.5. Inequalities for binomial coefficients
Theory
Page 163
14.6. Arithmetic of binomial coefficients
Theory
Page 163
14.7. Inclusion-exclusion formula
Theory
Page 163
14.8. Analogues of binomial coefficients
Theory
Page 164
14.9. Catalan numbers
Theory
Page 164
14.10. Elements of probability theory
Theory
Page 166
Solutions
Answer
Page 167
Chapter 15. Recurrence sequences
Theory
Page 181
15.1. General properties
Theory
Page 181
15.2. Fibonacci numbers
Theory
Page 181
15.3. Fibonacci numbers and the Euclidean algorithm
Theory
Page 183
15.4. Fibonacci numbers in combinatorics
Theory
Page 183
15.5. Special recurrence sequences
Theory
Page 184
Solutions
Answer
Page 184
Chapter 16. Examples and constructions
Problem
Page 190
16.1. Sets of numbers
Theory
Page 190
16.2. Infinite sequences
Theory
Page 191
16.3. Sequences of operations
Theory
Page 191
16.4. Polynomials and rational functions
Theory
Page 191
16.5. Various examples and constructions
Problem
Page 192
Solutions
Answer
Page 193
Chapter 17. Dirichlet principle. Extremal principle
Theory
Page 197
17.1. Remainders of division
Theory
Page 197
17.2. Various problems
Problem
Page 198
17.3. Approximating irrational numbers by rational numbers
Theory
Page 198
17.4. Extremal principle
Theory
Page 199
Solutions
Answer
Page 200
Chapter 18. Invariants and semi-invariants
Theory
Page 206
18.1. Remainders of division
Theory
Page 206
18.2. Semi-invariants
Theory
Page 206
18.3. Parity of a permutation
Theory
Page 207
Solutions
Answer
Page 208
Chapter 19. Logic
Theory
Page 213
19.1. Logic problems
Problem
Page 213
19.2. Logical paradoxes
Theory
Page 214
19.3. Propositional logic
Theory
Page 215
Solutions
Answer
Page 216
Chapter 20. Strategies. Tournaments. Tables
Problem
Page 219
20.1. Choosing a strategy
Theory
Page 219
20.2. Pouring problems
Problem
Page 220
20.3. Tournaments
Theory
Page 220
20.4. Weighings
Problem
Page 221
20.5. Tables
Theory
Page 222
Solutions
Answer
Page 222
Chapter 21. Numeral systems
Theory
Page 231
21.1. Last digits
Theory
Page 231
21.2. First digits
Theory
Page 231
21.3. Other digits
Theory
Page 232
21.4. Sum of digits
Theory
Page 232
21.5. Various problems on decimal notation
Problem
Page 232
21.6. Periods of decimal fractions and repunits
Theory
Page 233
21.7. Determining the base-d representation of a number
Theory
Page 234
21.8. Binary system
Theory
Page 234
21.9. Other numeral systems
Theory
Page 235
21.10. Other representations of numbers
Theory
Page 235
Solutions
Answer
Page 236
Chapter 22. Graphs
Theory
Page 243
22.1. Traversals of graphs
Theory
Page 244
22.2. Directed graphs
Theory
Page 244
22.3. Matchings
Theory
Page 244
Solutions
Answer
Page 245
Chapter 23. Complex numbers
Theory
Page 249
23.1. Identities and inequalities for complex numbers
Theory
Page 250
23.2. De Moivre's formula
Theory
Page 250
23.3. Roots of unity
Theory
Page 251
23.4. Roots of polynomials
Theory
Page 253
Solutions
Answer
Page 253
Chapter 24. Equations solvable by radicals
Theory
Page 258
24.1. Solving cubic equations
Theory
Page 259
24.2. Discriminant of a cubic polynomial
Theory
Page 259
24.3. Solving quartic equations
Theory
Page 260
24.4. Other equations solvable by radicals
Theory
Page 260
Solutions
Answer
Page 260
Chapter 25. Limit of a sequence
Theory
Page 265
25.1. Properties of limits
Theory
Page 265
25.2. Weierstrass theorem
Theory
Page 266
25.3. Calculating limits
Theory
Page 267
25.4. The number e
Theory
Page 268
25.5. Conjugate numbers
Theory
Page 269
25.6. Least upper bound
Theory
Page 269
Solutions
Answer
Page 270
Chapter 26. Continuous and discontinuous functions
Theory
Page 281
26.1. Monotone functions
Theory
Page 281
26.2. Periodic functions
Theory
Page 281
26.3. Limit of a function
Theory
Page 281
26.4. Continuity
Theory
Page 282
26.5. Intermediate value theorem
Theory
Page 283
26.6. Properties of functions continuous on an interval
Theory
Page 283
26.7. Convex functions
Theory
Page 284
26.8. Uniform continuity
Theory
Page 285
26.9. Functions of bounded variation
Theory
Page 285
Solutions
Answer
Page 286
Chapter 27. Logarithm and exponential function
Theory
Page 292
27.1. Definition of the exponential function and logarithm
Theory
Page 292
27.2. Exponential function
Theory
Page 293
27.3. Identities for logarithms
Theory
Page 293
27.4. Inequalities and comparison of numbers
Theory
Page 294
27.5. Irrationality of logarithms
Theory
Page 294
27.6. Some remarkable limits
Theory
Page 294
27.7. Hyperbolic functions
Theory
Page 294
Solutions
Answer
Page 295
Chapter 28. Derivative
Theory
Page 301
28.1. Definition of the derivative
Theory
Page 301
28.2. Derivatives of elementary functions
Theory
Page 302
28.3. Multiple root of a polynomial
Theory
Page 303
28.4. Derivative of a polynomial
Theory
Page 303
28.5. Identities
Theory
Page 304
28.6. Tangent and normal
Theory
Page 304
28.7. Functions differentiable on an interval
Theory
Page 305
28.8. Inequalities
Theory
Page 306
28.9. L'Hôpital's rule
Theory
Page 307
28.10. Number of roots of an equation
Theory
Page 308
28.11. Periodic functions
Theory
Page 308
28.12. Normalized symmetric functions
Theory
Page 308
28.13. Algebraic and transcendental functions
Theory
Page 309
28.14. Taylor formula
Theory
Page 309
Solutions
Answer
Page 310
Chapter 29. Integral
Theory
Page 327
29.1. Indefinite integral
Theory
Page 327
29.2. Definite integral
Theory
Page 328
29.3. Calculating integrals
Theory
Page 330
29.4. Calculating areas
Theory
Page 330
29.5. Calculating volumes
Theory
Page 331
29.6. Length of a curve
Theory
Page 331
29.7. Surface area
Theory
Page 332
29.8. Inequalities
Theory
Page 333
29.9. Calculating limits
Theory
Page 334
29.10. Identity
Theory
Page 334
29.11. Examples and constructions
Problem
Page 334
29.12. Improper integrals
Theory
Page 335
Solutions
Answer
Page 335
Chapter 30. Series
Theory
Page 347
30.1. Calculating infinite sums
Theory
Page 347
30.2. Calculating infinite products
Theory
Page 347
30.3. Harmonic series
Theory
Page 348
30.4. Series for the logarithm
Theory
Page 349
30.5. Series for the number π
Theory
Page 350
30.6. Exponential in the complex domain
Theory
Page 350
30.7. Proofs of inequalities
Theory
Page 351
30.8. Convergent and divergent series
Theory
Page 351
30.9. Convergence of infinite products
Theory
Page 351
Solutions
Answer
Page 352
Chapter 31. Elements of number theory
Theory
Page 361
31.1. Fermat's little theorem
Theory
Page 361
31.2. Pseudoprimes
Theory
Page 361
31.3. Euler's function
Theory
Page 362
31.4. Wilson's theorem
Theory
Page 362
31.5. Problems on congruences
Problem
Page 363
31.6. Function σ_k(n). Divisors
Theory
Page 364
31.7. Quadratic residues
Theory
Page 365
31.8. Quadratic reciprocity law
Theory
Page 365
31.9. Gauss sums
Theory
Page 367
31.10. Sums of two squares
Theory
Page 368
31.11. Sums of four squares
Theory
Page 369
31.12. Primitive roots modulo a prime
Theory
Page 369
31.13. Primitive roots modulo a composite number
Theory
Page 370
31.14. Chebyshev's theorem on prime numbers
Theory
Page 371
Solutions
Answer
Page 372
Chapter 32. Polynomials II
Theory
Page 393
32.1. Separation of roots
Theory
Page 393
32.2. Irreducible polynomials
Theory
Page 395
32.3. Symmetric polynomials
Theory
Page 398
32.4. Chebyshev polynomials
Theory
Page 401
32.5. Algebraic and transcendental numbers
Theory
Page 402
32.6. Adjoining a root of a polynomial
Theory
Page 404
Solutions
Answer
Page 404
Chapter 33. Algorithms and computations
Theory
Page 416
33.1. Calculating some numbers
Theory
Page 416
33.2. Arithmetic operations. Polynomials
Theory
Page 417
33.3. Sorting
Theory
Page 417
33.4. Public-key cryptography
Theory
Page 419
Solutions
Answer
Page 420
Chapter 34. Functional equations
Theory
Page 425
34.1. Substitution method
Theory
Page 425
34.2. Functional equations for arbitrary functions
Theory
Page 426
34.3. Functional equations for continuous functions
Theory
Page 426
34.4. Functional equations for differentiable functions
Theory
Page 427
34.5. Functional equations for polynomials
Theory
Page 427
Solutions
Answer
Page 429
Chapter 35. Continued fractions
Theory
Page 438
35.1. Definition and basic properties
Theory
Page 438
35.2. Best approximations
Theory
Page 440
35.3. Continued fractions and Pell equation
Theory
Page 440
Solutions
Answer
Page 441
Chapter 36. Formal series and generating functions
Theory
Page 446
36.1. Formal series
Theory
Page 446
36.2. Formal derivative
Theory
Page 447
36.3. Root of a formal series
Theory
Page 447
36.4. Exponential and logarithm
Theory
Page 447
36.5. Identities for formal series
Theory
Page 448
36.6. Generating functions
Theory
Page 449
36.7. Bernoulli numbers and polynomials
Theory
Page 450
36.8. Number of partitions
Theory
Page 450
36.9. Waring's formulas
Theory
Page 451
Solutions
Answer
Page 452
Chapter 37. Finite difference calculus
Theory
Page 462
37.1. Properties of finite differences
Theory
Page 462
37.2. Generalized power
Theory
Page 463
37.3. Euler summation formula
Theory
Page 464
Solutions
Answer
Page 464
Chapter 38. Curves in the plane
Theory
Page 467
38.1. Polar coordinates
Theory
Page 468
38.2. Envelope of a family of curves
Theory
Page 468
38.3. Curvature
Theory
Page 471
38.4. Osculating circle
Theory
Page 472
38.5. Focal points. Evolute
Theory
Page 473
Solutions
Answer
Page 474
Chapter 39. Set theory
Theory
Page 482
39.1. Finite sets
Theory
Page 482
39.2. Operations on sets
Theory
Page 482
39.3. Equipotent sets
Theory
Page 483
39.4. Countable sets
Theory
Page 484
39.5. Cardinality of the continuum
Theory
Page 484
39.6. Properties of cardinality
Theory
Page 485
39.7. Paradoxes of set theory
Theory
Page 485
Solutions
Answer
Page 486
Supplement
Other
Page 490
1. Rational parametrization of a circle
Theory
Page 490
2. Sums of squares of polynomials
Theory
Page 493
3. Representation of numbers as a sum of two squares
Theory
Page 496
4. Construction of a regular 17-gon
Theory
Page 498
5. Constructions with compass and straightedge
Theory
Page 501
6. Chromatic polynomial of a graph
Theory
Page 507
7. Transcendence of the numbers e and π
Theory
Page 510
8. Solvability of equations by radicals
Theory
Page 514
9. Diophantine equations for polynomials
Theory
Page 525
10. Van der Waerden's theorem on arithmetic progressions
Theory
Page 529
11. Origin of mathematical terms
Other
Page 533
Name index
Other
Page 536
Subject index
Other
Page 538
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