Bibliography. A. Books cited in text

Transcription

1 Bibliography A. Books cited in text [1] E. Beckenbach and R. Bellman, Inequalities, Springer-Verlag, Berlin Gottingen-Heidelberg, [2] A. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, New York-London, [3] P. S. Bullen, D. S. Mitrinovic, and P. M. Vasic, Means and Their Inequalities, Reidel, Dordrecht, [4] L. Childs, A Concrete Introduction to Higher Algebra, second edition, Springer-Verlag, New York, [5] P. Franklin, A Treatise on Advanced Calculus, Dover Publications, New York,1964. [6] H. W. Gould, Combinatorial Identities; a Standardized Set of Tables Listing 500 Binomial Coefficient Summations, revised edition, Morgantown, W.Va., [7] S. L. Greitzer, International Mathematical Olympiads , Mathematical Association of America, Washington, D.C., [8] S. Mac Lane and G. Birkhoff, Algebra, third edition, Chelsea Publishing Co., New York, [9] J. Riordan, Combinatorial Identities, Wiley, New York, [10] G. E. Shilov, Linear Algebra, Dover Publications, New York, [11] W. SierpiDski, Elementary Theory of Numbers, Panstwowe Wydawnictwo Naukowe, Warszawa, [12] I. M. Vinogradov, Elements of Number Theory, Dover Publications, New York,1954. [13] B. L. van der Waerden, Algebra, Vol. I, Springer-Verlag, New York, 1991.

2 338 Bibliography B. Additional English-language books [14] E. J. Barbeau, M. S. Klamkin, and W. O. J. Moser, Five Hundred Mathematical Challenges, Mathematical Association of America, Washington, DC, [15] E. Beckenbach and R. Bellman, An Introduction to Inequalities, Mathematical Association of America, Washington, DC, [16] G. Birkhoff and S. Mac Lane, A Survey of Modern Algebm, third edition, Collier-Macmillan, London, [17] M. Doob (ed.), The Canadian Mathematical Olympiad, , Canadian Mathematical Society, Ottawa, [18] H. Dorrie, 100 Great Problems of Elementary Mathematics, Dover Publications, Reprinted [19] M. J. Erickson and J. Flowers, Principles of Mathematical Problem Solving, Prentice-Hall, [20] A. Gardiner, The Mathematical Olympiad Handbook. An Introduction to Problem Solving Based on the First 92 British Mathematical Olympiads , Oxford University Press, [21] G. T. Gilbert, M. I. Krusemeyer, and L. C. Larson, The Wohascum County Problem Book, Mathematical Association of America, Washington, DC, [22] G. H. Hardy, J. E. Littlewood, and G. P61ya, Inequalities, second etition, Cambridge University Press, Cambridge, [23] K. Hardy and K. S. Williams, The Red Book of Mathematical Problems, Dover Publications, Mineola, NY, [24] K. Hardy and K. S. Williams, The Green Book of Mathematical Problems, Dover Publications, Mineola, NY, [25] R. Honsberger, From Erdos to Kiev. Problems of Olympiad Caliber, Mathematical Association of America, Washington, DC, [26] M. S. Klamkin, USA Mathematical Olympiads , Mathematical Association of America, Washington, DC, [27] P. P. Korovkin, Inequalities, Little Mathematics Library, Mir, Moscow; distributed by Imported Publications, Chicago, Ill., [28] J. Kiirschak, Hungarian Problem Book I, Based on the Eotvos Competitions, , Mathematical Association of America, Washington, DC, [29] J. Kiirschak, Hungarian Problem Book II, Based on the Eotvos Competitions, , Mathematical Association of America, Washington, DC, [30] L. C. Larson, Problem-Solving Through Problems, Springer-Verlag, New York,1983. [31] D. J. Newman, A Problem Seminar, Springer-Verlag, New York, [32] G. P6lya, Mathematical Discovery. On Understanding, Learning, and Teaching Problem Solving, reprint in one volume, John Wiley & Sons, Inc., New York,1981. [33] G. P6lya, How to Solve It. A New Aspect of Mathematical Method, Princeton University Press, Princeton, NJ, [34] J. Roberts, Elementary Number Theory. A Problem Oriented Approach, MIT Press, Cambridge, Mass.-London, 1977.

3 Bibliography 339 [35] D. O. Shklarsky, N. N. Chentzov, and I. M. Yaglom, The USSR Olympiad Problem Book, W. H. Freeman, San Francisco, Reprinted by Dover Publications, New York, [36] W. Sierpinski, 250 Problems in Elementary Number Theory, American Elsevier Publishing Co., Inc., New York; PWN Polish Scientific Publishers, Warsaw, [37] H. Steinhaus, One Hundred Problems in Elementary Mathematics, Basic Books, Inc., New York, Reprinted by Dover Publications, New York, [38] Ch. W. Trigg, Mathematical Quickies, Dover Publications, New York, [39] W. A. Wickelgren, How to Solve Mathematical Problems, Dover Publications, Mineola, NY, [40] P. Zeitz, The Art and Craft of Problem Solving, Wiley, New York, c. East European sources [41] I. L. Babinskaya, Zadachi matematicheskikh olimpiad, Nauka, Moskva, [42] V. I. Bernik, I. K. Zhuk, and O. V. Melnikov, Sbornik olimpiadnykh zadach po matematike, Narodnaya osveta, Minsk, [43] J. Fecenko, 0 nerovnostiach, ktore suvisia s aritmetickym a geometrickym priemerom, Rozhledy mat.-fyz. 66 (1987/88), no. 2, [44] M. Fiedler and J. Zemanek, Vybrane Ulohy matematicke olympiady (kategorie A + MMO), SPN, Praha, [45] G. A. GaIperin and A. K. Tolpygo, Moskovskiye matema~icheskiye olimpiady, Prosveshcheniye, Moskva, [46] V. M. Govorov, Sbornik konkursnykh zadach po matematike, Nauka, Moskva, [47] J. Herman and J. SimSa, 0 jednom uiiti Dirichletova principu v teorii Usel, Rozhledy mat.-fyz. 66 (1987/88), no. 9, [48] K. Horak et ai., Ulohy mezinarodnich matematickych olympiad, SPN, Praha, [49] S. V. Konyagin et ai., Zarubezhnyye matematicheskiye olimpiady, Nauka, Moskva, [50] I. Korec, Ulohy 0 velkych cislach, Mlada fronta, edice SMM, Praha, [51] V. A. Krechmar, Zadachnik po algebre, Nauka, Moskva, [52] G. A. Kudrevatov, Sbornik zadach po teorii chisel, Prosveshcheniye, Moskva, [53] V. B. Lidskiyet ai., Zadachi po elementarnoy matematike, Gos. izd. fiz.-mat. lit., Moskva, [54] P. S. Modenov, Sbornik zadach po spetsialnom kurse elementarnoy matematiky, Vysshaya shkola, Moskva, [55] N-ty rocnz'k matematicke olympiady, SPN, Praha, (from 1953 to 1993). [56] O. Odvarko et ai., Metody feseni matematickych uloh I, ucebnf text MFF UK, Praha, [57] I. Kh. Sivashinsldy, Neravenstva v zadachakh, Nauka, Moskva, [58] I. Kh. Sivashinskiy, Teoremy i zadachi po algebrei elementarnym funktsiyam, Nauka, Moskva, [59] S. Straszewicz, Zadania z olimpiad matematicznych, I, II, Warszawa, 1956, 1961.

4 340 Bibliography [60] S. Straszewicz and J. Browkin, Polskiye matematicheskiye olimipady, Mir, Moskva, [61] J. SedivY et al., Metody resent metematickych uloh II, ucebni text MFF UK, Praha, [62] V. S. Shevelyov, Tri formuly Ramanudzhana, Kvant, 1988, no. 6, [63] J. SimSa, Dolnt odhady rozdtlu prumeru, Rozhledy mat.-fyz. 65 (1986/87), no. 10 (June), [64] N. V. Vasilyevand A. A. Yegorov, Zadachi vsesoyuznykh matematicheskikh olimpiad, Nauka, Moskva, [65] N. V. Vasilyev et ai., Zaochnyye matematicheskiye olimpiady, Nauka, Moskva, [66] V. V. Vavilovet al., Zadachi po matematike: Algebra, Nauka, Moskva, [67] V. V. Vavilovet ai., Zadachi po matematike: Uravneniya i neravenstva, Nauka, Moskva, [68] A. Vrba and K. Horak, Vybrane ulohy MO kategorie A, SPN, Praha, [69] V. A. Vyshenskiy et al., Sbornik zadach kiyevskikh matematicheskikh olimpiad, Vishcha shkola, Kiyev, 1984.

5 Index Abel, Niels Henrik 25 AM-GM inequality 109, 144, 151, 155,160,162,163,167,232 for weighted means 162 algebra 23 algorithm division 28 Euclidean 178, 181 alternating sums 15 arithmetic mean 33, 109, 150 arithmetic progression 5 base of the natural logarithm 113 base representations 258 Bernoulli numbers 13 Bernoulli's formula 12 Bernoulli's inequality 109, 141, 166, 168, 233 general 165 binomial coefficients 2, 75 generalized 276 binomial theorem 2, 8, 175, 263 Bezout's equality 179 Bezout's theorem 28 Cauchy's inequality 127, 131, 133, 149, 166 Chebyshev's inequality 145, 148, 149, 150, 159 Chebyshev's theorem 187 Chinese remainder theorem 211 combinatorial identities 6 combinatorial numbers 1 common divisor 178, 180 common multiple 178, 180 completing the square 113 complex number 75 composite number 183 congruences 189 in one variable 201 linear 202 of higher degree 211 quadratic 216 conjugate 65 conjecture Goldbach 173 Shimura-Taniyama 173 twin prime 173 coprime 180 Cramer's rule 46 cubic polynomial 24

6 342 Index cubic (continued) de Moivre's theorem 75, 77, 78, 84 decimal expansion 261 degree of a polynomial 24, 38 of a power mean 167 digit patterns 265 digit sum 261, 267 digits 258 final 261 Diophantine equations 217 cardinality of solution set 245 linear 217 linear in some variable 220 Diophantus of Alexandria 217 Dirichlet's principle 268 Dirichlet's theorem 187 discriminant 124 divisibility of numbers 174 division algorithm 28 division theorem 176 Eisenstein's irreducibility criterion 280 elementary symmetric polynomials 32,39 in three variables 43 in two variables 39 elimination method 54 equivalent transformations 94 estimation method 101 Euclidean algorithm 178, 181 Euler's cp-function 194 Euler's criterion 216 Euler's theorem 198, 262, 264, 266 factorial 1 prime decomposition of 255 false roots 61 Fermat's last theorem 173 Fermat's theorem 197 finite induction 136 finite sums 5 fractional part 251 fundamental mean property 165 fundamental theorem of algebra 29 Gauss, Carl Friedrich 189 Gaussian elimination 46 generating functions 9 geometric mean 151, 167 geometric progression 6 geometric series 110 geometric-power mean inequalities 167 Goldbach conjecture 173 greatest common divisor 178, 180 harmonic mean 156 homogeneous inequality 105 homogeneous polynomial 38 Holder's inequality 166 implication method 61 incomplete quotient 176 induction principle 135, 136 finite 134 inequality 89 AM-GM 109, 144, 151, 155, 160, 162, 163, 167, 232 AM-GM, for weighted means 162 Bernoulli's 109, 141, 166, 168, 233 Bernoulli's, general 165 between power means 167 Cauchy's 127, 131, 133, 149, 166 Chebyshev's 145, 148, 149, 150, 159 geometric-power mean 167 homogeneous 105 Holder's 166 Jensen's 105, 169 Minkowski's 130, 166 symmetric 104 triangle 129 weak 91 Young's 163 infinite product 140 infinitude of primes in arithmetic progressions 187 of the form 3k of the form 4k integer part 251, 267 integer-valued polynomials 276 interpolation 23

7 Index 343 irrational equations 60 with a parameter 71 irrational numbers 169 irreversible transformations 96 Jensen's inequality 105, 169 least common multiple 178, 180 length of a vector 130 linear congruences 202 linear Diophantine equations 217 linear polynomial 24 lower bound 101 mathematical induction 5, 135 mean of degree zero 167 mean value 150 method of squares 113 substitution 66 symmetric polynomials 55 undetermined coefficients 19 Minkowski's inequality 130, 166 monomial 38 multiple 174 multiple root 28 multiple zero 28 multiplicity of a zero 28 nth roots 60 number theory 173 ordering of an n-tuple 146 pairwise relatively prime 180 partial fraction decomposition 21 partial summation 16 pigeonhole principle 268 polynomials 23, 84, 274 cubic 24 degree of 24, 38 elementary symmetric 32, 39, 43 homogeneous 38 in several variables 38 integer-valued 276 irreducible 280 linear 24 Taylor 23 with integer coefficients 274 polynomial division 27 power mean 151, 166, 167 prime number 183 Pythagorean equation 237 quadratic congruences 216 quadratic polynomial 24 quotient 28, 176 rational zeros 35 regular n-gon 80 relatively prime 180 remainder 28, 176 repunit 265 root of a polynomial equation 25 scalar product 128 Shimura-Taniyama conjecture 173 Sierpinski, W. 187 simple zero 28 strict inequalities 91 sum of digits 261, 267 sums of powers 11, 39 symmetric inequality 104 symmetric polynomial 38 systems of equations 46 equations of higher degree 54 irrational equations 73 linear congruences in one variable 206 linear equations 46 linear equations with parameters 48 Taylor polynomials 23 theorem binomial 2, 8, 175, 263 Bezout's 28 Chebyshev's 187 Chinese remainder 211 de Moivre's 75, 77, 78, 84 Dirichlet's 187 division 176 Euler's 198, 262, 264, 266

8 344 Index theorem (continued) Fermat's 197 Fermat's last 173 Wilson's 204 transitivity of inequalities 91 triangle inequality 129 trichotomy law 91 trigonometric functions 78 twin prime conjecture 173 unique factorization 185 upper bound 101 Vieta's relations 32, 38, 41, 56, 78, 80,82,83,125,132 Vinogradov, I. M. 216 weak inequalities 91 weighted arithmetic mean 162 weighted geometric mean 162 weighted means 151, 162 Wiles, Andrew 173 Wilson's theorem 204 Young's inequality 163 zero of a polynomial 23, 25, 275 multiple 28 simple 28 zero polynomial 24, 38