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TRANSCRIPT
Glossary
Convexity A function f(x) is concave up (down) on [a, b] ~ lR if f(x) lies under (over) the line connecting (a1 , f (a1 )) and (b1 , f(b1))
for all
a :"::: a1 < x < b1 :"::: b.
A function g(x) is concave up (down) on the Euclidean plane if it is concave up (down) on each line in the plane, where we identify the line naturally with R
Concave up and down functions are also called convex and concave, respectively.
If f is concave up on an interval [a, b] and >.1, >.2, ... , An are nonnegative numbers with sum equal to 1, then
for any x 1 , x 2 , . •. , Xn in the interval [a, b] . If the function is concave down, the inequality is reversed. This is Jensen's Inequality.
Lagrange's Interpolation Formula Let xo, x1 , ... , Xn be distinct real numbers, and let yo, Yl, ... , Yn be arbitrary real numbers. Then there exists a unique polynomial P(x) of degree at most n such that P(xi ) = Yi, i = 0, 1, ... , n. This polynomial is given by
Ln (x- xo) · · · (x- Xi-l)(x- Xi+!)·· · (x- Xn) P(x)= Yi~---7--~----~--~-7~7-~~
i= O (xi- xo ) · ··(xi - Xi-l)(xi- Xi+!)··· (xi - Xn) ·
Maclaurin Series Given a function f(x), the power series
00 J(k) (0) k L-k!-x k=O
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214 Counting Strategies
is the Maclaurin Series of f(x), where J Ckl (x ) denotes t he kth
derivative of f ( x).
Pigeonhole Principle If n objects are distributed among k < n boxes, some box contains at least two objects.
Power Mean Inequality Let a1o a2, ... , an be any positive numbers for which ai + a 2 + · · · + an = 1. For positive numbers XI, x2, ... , Xn we define
where tis a non-zero real number. Then
for s ::; t.
Root Mean Square-Arithmetic Mean Inequality For posit ive numbers XI , x2, ... , Xn ,
JXI + X~ + · · · +X~ XI + X 2 + · · · + Xk ~--~~------~ > .
n - n
The inequality is a special case of the Power Mean Inequality.
Triangle Inequality In a non-degenerated triangle, the sum of t he lengths of any two sides of the t riangle is bigger than the length of t he t hird side.
Vandermonde Matrix A Vandermonde Matrix M is a matrix of t he form
1 1 1
l I x, X2 Xn
M = -~~ X~ x2 n
n -I n -I x~- I x l x2
Its determinant is
II (Xj - X;) , l :S:i<j :S: n
Glossary 215
which is nonzero if and only if x 1 , x 2 , •.• , Xn are distinct. The Vandermonde matrix is closely related to the Lagrange's Interpolation Formula. Indeed, it arises in the problem of finding a polynomial
p(x) = an-1Xn-l + an-2Xn-2 + · · · + a1X + ao
such that p(xi) = Yi for all i with 1 ::::; i ::::; n. Because
n-1 + n-2 + + + an-1x1 an-2x1 · · · a1x1 ao = Yb
it follows that
[ -~~- j = MT . [ ;~. j Yn an- 1
= [ . ~1. . :~ Xn X~
where MT is the transpose of M. (Note that a matrix and its transpose have the same determinant.)
Index
addition principle
base p representation Becheanu's formula Bernoulli-Euler Formula bijection, one-to-one correspondence binomial coefficient, binomial numbers Bonferonni's inequalities
Catalan numbers Catalan path circular permutation combinations congruency of polynomials modulo p
convexity concave down concave up
Deutsch's covering problem derangement direct product
Euler ( totient) function
2
59 137 128 15
45, 47
130
82 111 19 25 59
213 213 213
189 128 144
124
217
218
Fibonacci number Fibonacci sequence fixed point Fubini's principle function
generating functions of the first type of the second type
image Inclusion-Exclusion Principle, Boole-Sylvester formula
Jensen's Inequality
Kummer's Theorem
Lagrange's Interpolation Formula Lucas's Theorem
Maclaurin Series main diagonal map, mapping
injective, one-to-one surjective, onto
multiplication principle
partition height increasing length parts
Pascal's triangle permutation Pigeonhole Principle Power Mean inequality prime decomposition probability Probleme des menages
Counting Strategies
53 53 128 144 15
47 165 173
14 12, 120
213
60
213 59
169, 172, 213 205 15 15 15 10
86 86 86 86 86 46 15 124, 214 214 11 10 134
Index
recursive relation, recursion Root Mean Square-Arithmetic Mean inequality
Szego & P6lya's formula Sperner's Theorem
Triangle inequality
Vandermonde identity Vandermonde matrix Young's diagram
91 58, 160, 214
126 149
5, 214
55 100, 214 87
219
Further Reading
1. Andreescu, T.; Feng, Z., USA and International Mathematical Olympiads 2002 , Mathematical Association of America, 2003.
2. Andreescu, T.; Feng, Z., USA and International Mathematical Olympiads 2001 , Mathematical Association of America, 2002.
3. Andreescu, T.; Feng, Z. , USA and International Mathematical Olympiads 2000 , Mathematical Association of America, 2001.
4. Andreescu, T.; Feng, Z.; Lee, G.; Loh, P., Mathematical Olympiads: Problems and Solutions from around the World, 2001-2002, Mathematical Association of America, 2004.
5. Andreescu, T.; Feng, Z.; Lee, G., Mathematical Olympiads: Problems and Solutions from around the World, 2000-2001, Mathematical Association of America, 2003.
6. Andreescu, T.; Feng, Z. , Mathematical Olympiads: Problems and Solutions from around the World, 1999- 2000 , Mathematical Association of America, 2002.
7. Andreescu, T.; Feng, Z., Mathematical Olympiads: Problems and Solutions from around the World, 1998- 1999, Mathematical Association of America, 2000.
8. Andreescu, T.; Kedlaya, K. , Mathematical Contests 1997-1998: Olympiad Problems from around the World, with Solutions, American Mathematics Competitions, 1999.
9. Andreescu, T.; Kedlaya, K., Mathematical Contests 1996-1997: Olympiad Problems from around the World, with Solutions , American Mathematics Competitions, 1998.
10. Andreescu, T.; Kedlaya, K.; Zeitz, P., Mathematical Contests
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222 Counting Strategies
1995-1996: Olympiad Problems from around the World, with Solutions, American Mathematics Competitions, 1997.
11. Andreescu, T.; Feng, Z., 101 Problems in Algebra from the Training of the USA IMO Team, Australian Mathematics Trust, 2001.
12. Andreescu, T.; Feng, Z., 102 Combinatorial Problems from the Training of the USA IMO Team, Birkhauser, 2002.
13. Andreescu, T.; Enescu, B., Mathematical Olympiad Treasures, Birkhauser, 2003.
14. Andreescu, T.; Gelca, R., Mathematical Olympiad Challenges, Birkhauser, 2000.
15. Andreescu, T.; Andrica, D., 360 Problems for Mathematical Contests, GIL, 2002.
16. Andreescu, T .; Andrica, D., An Introduction to Diophantine Equations, GIL, 2002.
17. Barbeau, E., Polynomials, Springer-Verlag, 1989.
18. Beckenbach, E. F.; Bellman, R., An Introduction to Inequalities, New Mathematical Library, Vol. 3, Mathematical Association of America, 1961.
19. Bollobas, B., Graph Theory, An Introductory Course, SpringerVerlag, 1979.
20. Chinn, W. G.; Steenrod, N. E., First Concepts of Topology, New Mathematical Library, Vol. 27, Random House, 1966.
21. Cofman, J., What to Solve?, Oxford Science Publications, 1990.
22. Coxeter, H. S. M.; Greitzer, S. L., Geometry Revisited, New Mathematical Library, Vol. 19, Mathematical Association of America, 1967.
23. Coxeter, H. S. M., Non-Euclidean Geometry, The Mathematical Association of American, 1998.
24. Doob, M., The Canadian Mathematical Olympiad 1969-1993, University of Toronto Press, 1993.
25. Engel, A., Problem-Solving Strategies, Problem Books in Mathematics, Springer, 1998.
26. Fomin, D.; Kirichenko, A., Leningrad Mathematical Olympiads 1987- 1991, MathPro Press, 1994.
Further Reading 223
27. Fomin, D.; Genkin, S.; Itenberg, I., Mathematical Circles , American Mathematical Society, 1996.
28. Graham, R. L.; Knuth, D. E .; Patashnik, 0., Concrete Mathematics, Addison-Wesley, 1989.
29. Gillman, R., A Friendly Mathematics Competition, The Mathematical Association of American, 2003.
30. Greitzer, S. L., International Mathematical Olympiads, 1959-1977, New Mathematical Library, Vol. 27, Mathematical Association of America, 1978.
31. Grossman, I.; Magnus, W., Groups and Their Graphs, New Mathematical Library, Vol. 14, Mathematical Association of America, 1964.
32. Holton, D., Let's Solve Some Math Problems , A Canadian Mathematics Competition Publication, 1993.
33. Ireland, K.; Rosen, M., A Classical Introduction to Modern Number Theory, Springer-Verlag, 1982.
34. Kazarinoff, N. D., Geometric Inequalities , New Mathematical Library, Vol. 4, Random House, 1961.
35. Kedlaya , K; Poonen, B .; Vakil, R., The William Lowell Putnam Mathematical Competition 1985-2000, The Mathematical Association of American, 2002.
36. Klamkin, M., International Mathematical Olympiads, 1978- 1985, New Mathematical Library, Vol. 31, Mathematical Association of America, 1986.
37. Klamkin, M., USA Mathematical Olympiads, 1972-1986, New Mathematical Library, Vol. 33, Mathematical Association of America, 1988.
38. Klee, V.; Wagon, S, Old and N ew Unsolved Problems in Plane Geometry and Number Theory , The Mathematical Association of American, 1991.
39. Kiirschak, J., Hungarian Problem Book, volumes I f3 II , New Mathematical Library, Vols. 11 & 12, Mathematical Association of America, 1967.
40. Kuczma, M., 144 Problems of the Austrian-Polish Mathematics Competition 1978-1993, The Academic Distribution Center, 1994.
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41. Landau, E., Elementary Number Theory, Chelsea Publishing Company, New York, 1966.
42. Larson, L. C., Problem-Solving Through Problems, SpringerVerlag, 1983.
43. Lausch, H. The Asian Pacific Mathematics Olympiad 1989-1993, Australian Mathematics Trust, 1994.
44. Leveque, W. J., Topics in Number Theory, Volume 1, Addison Wesley, New York, 1956.
45. Liu, A., Chinese Mathematics Competitions and Olympiads 1981-1993, Australian Mathematics Trust, 1998.
46. Liu, A., Hungarian Problem Book III, New Mathematical Library, Vol. 42, Mathematical Association of America, 2001.
47. Lozansky, E .; Rousseau, C. Winning Solutions, Springer, 1996.
48. Mordell, L. J., Diophantine Equations, Academic Press, London and New York, 1969.
49. Ore, 0., Graphs and Their Use, Random House, 1963.
50. Ore, 0., Invitation to Number Theory, Random House, 1967.
51. Savchev, S.; Andreescu, T. Mathematical Miniatures, Anneli Lax New Mathematical Library, Vol. 43, Mathematical Associat ion of American, 2002.
52. Sharygin, I. F., Problems in Plane Geometry, Mir, Moscow, 1988.
53. Sharygin, I. F., Problems in Solid Geometry, Mir, Moscow, 1986.
54. Shklarsky, D. 0; Chentzov, N. N; Yaglom, I. M., The USSR Olympiad Problem Book, Freeman, 1962.
55. Slinko, A., USSR Mathematical Olympiads 1989- 1992, Australian Mathematics Trust, 1997.
56. Sierpinski, W., Elementary Theory of Numbers, Hafner, New York, 1964.
57. Soifer, A., Colorado Mathematical Olympiad: The first ten years, Center for excellence in mathematics education, 1994.
58. Szekely, G. J ., Contests in Higher Mathematics, Springer-Verlag, 1996.
59. Stanley, R. P., Enumerative Combinatorics, Cambridge University Press, 1997.
Further Reading 225
60. Tabachnikov, S. Kavant Selecta: Algebra and Analysis I, American Mathematics Society, 1991.
61. Tabachnikov, S. Kavant Selecta: Algebra and Analysis II , American Mathematics Society, 1991.
62. Tabachnikov, S. Kavant Selecta: Combinatorics I, American Mathematics Society, 2000.
63. Taylor, P. J., Tournament of Towns 1980-1984, Australian Mathematics Trust, 1993.
64. Taylor, P. J., Tournament of Towns 1984-1989, Australian Mathematics Trust, 1992.
65. Taylor, P. J., Tournament of Towns 1989-1993, Australian Mathematics Trust, 1994.
66. Taylor, P. J.; Storozhev, A., Tournament of Towns 1993-1997, Australian Mathematics Trust, 1998.
67. Tomescu, I., Problems in Combinatorics and Graph Theory , Wiley, 1985.
68. Vanden Eynden, C., Elementary Number Theory, McGraw-Hill, 1987.
69. Vaderlind, P.; Guy, R.; Larson, L., The Inquisitive Problem Solver, The Mathematical Association of American, 2002.
70. Wilf, H. S., Generatingfunctionology, Academic Press, 1994.
71. Wilson, R., Introduction to graph theory, Academic Press, 1972.
72. Yaglom, I. M., Geometric Transformations, New Mathematical Library, Vol. 8, Random House, 1962.
73. Yaglom, I. M., Geometric Transformations II , New Mathematical Library, Vol. 21, Random House, 1968.
74. Yaglom, I. M., Geometric Transformations III, New Mathematical Library, Vol. 24, Random House, 1973.
75. Zeitz, P. , The Art and Craft of Problem Solving, John Wiley & Sons, 1999.
Afterword
This book is the product of many years of work and is based on the authors' extensive experience in mathemat ics and mathematical education. Both authors have extensive experience in the development and composition of original mathematics problems and in the applications of advanced methodologies in mathematical science teaching and learning.
This book is aimed at three major types of audiences: (a) Students ranging from high school juniors to college seniors. This
book will prove useful to those wanting to tie up many loose ends in their study of combinatorics and to develop mathematically, in general. Students with interest in mathematics competitions should have this book in their personal libraries.
(b) Numerous teachers who are implementing problem solving across the nation. This book is a perfect match for teachers wanting to teach advanced problem-solving classes and to organize mathematical clubs and circles.
(c) Amateur mathematicians longing for new mathematical gems and brain teasers. This book presents sophisticated applications of genuine mathematical ideas in real-life examples. It will help them to recall the experience of reading the wonderful stories by Martin Gardner in his monthly column in Scientific American.
By studying this book, readers will be well-equipped to further their knowledge in more abstract combinatorics and its related fields in mathematical and computer science. This book serves as a solid stepping stone for advanced mathematical reading, such as Combinatorial
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Theory by M. Aigner, Concrete Mathematics - A Foundation for Computer Science by R. L. Graham; D. E. Knuth; and 0. Patashnik, and Enumerative Combinatorics by R. P. Stanley.