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

213

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

221

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.

224 Counting Strategies

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

227

228 Counting Strategies

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.