## Problems From the Book

Tôi giới thiệu với các bạn chuẩn bị tham dự kì thi chọn đội tuyển Toán Việt Nam tham dự IMO (Vietnam TST) hai cuốn sách sau đây:

1) “Problems From the Book” của Titu Andreescu và Gabriel Dospinescu.

Đây là đoạn mô tả trên trang của nhà xuất bản XYZ: “The authors provide a combination of enthusiasm and experience which will delight any reader. In this volume they present innumerable beautiful results, intriguing problems, and ingenious solutions. The problems range from elementary gems to deep truths. A trully delightful and highly instructive book, this will prepare the engaged reader not only for any mathematics competition they may enter but also for a life time of mathematical enjoyment. A must for the bookshelves of both aspiring and seasoned mathematicians.”

Bạn mua từ nhà xuất bản hoặc tìm E-book.

2) “Straight from the Book” của Titu Andreescu và Gabriel Dospinescu.

Cuốn 1) có rất nhiều bài tập về nhà, và nhiều bài rất khó. Cuốn 2) sẽ có lời giải của hầu hết các bài tập về nhà trong 1). Đây là đoạn mô tả trên trang của nhà xuất bản XYZ: “This book is a compilation of many suggestions, much advice, and even more hard work. Its main objective is to provide solutions to the problems which were originally proposed in the first 12 chapters of “Problems from the Book”. The volume is far more than a collection of solutions. The solutions are used as motivation for the introduction of some very clear expositions of mathematics. And this is modern, current, up-to-the-minute mathematics. This is absolutely state-of-the-art material. Everyone who loves mathematics and mathematical thinking should acquire this book.”

Editorial Reviews

(Đoạn này được lấy từ amazon. )

This is an exceptionally well-written book. The material is arranged in small chapters, with brief theory in the beginning of each chapter followed by a set of exceptionally difficult problems with solutions. These solutions are elegant, innovative and beautiful. You learn a lot from the solutions. In every page, you will discover one or more clever steps/tricks that will make you wonder “How come I could not think of that?”. If you are preparing for Mathematics Olympiads, working through this book will boost your confidence 100 fold. If you are a math enthusiast, you will enjoy the material – most of it is “Mathematical poetry”. Grab it before it gets sold out! –Dr S Muralidharan

Problems from the Book is rife with elegant mathematical pursuits that are well worth the effort of exploring and solving. For high schoolers up through University students, the book’s problems will illustrate important concepts and provide hours of fun at every sitting. –David Cordeiro

This book is a treasure of the mathematical gems: many many very nice problems and results, historic notes and useful comments. Readers will also find many very interesting original problems from the authors of the book and from others. If you want to develop your mathematical skills in problem solving and your knowledge in diverse mathematical branches, you will definitely find many instructive topics throughout this book. Many thanks to Prof. Andreescu and his colleagues for their invaluable books and problems. I do highly recommend this book and all other books by Prof. Andreescu to all mathematics lovers: from the pupils preparing to participate in mathematical contests to people searching excitement in mathematics. The book contains the following 23 chapters, in addition to preface, bibliography and index: 1. Some Useful Substitutions 2. Always Cauchy-Schwarz … 3. Look at the Exponent 4. Primes and Squares 5. T2’s Lemma 6. Some Classical Problems in Extremal Graph theory 7. Complex Combinatorics 8. Formal Series Revisited 9. A Brief Introduction to Algebraic Number Theory 10. Arithmetic Properties of Polynomials 11. Lagrange Interpolation Formula 12. Higher Algebra in Combinatorics 13. Geometry and Numbers 14. The Smaller, The Better 15. Density and Regular Distribution 16. The Digit Sum of Positive Integer 17. At the Border of Analysis and Number Theory 18. Quadratic Reciprocity 19. Solving Elementary Inequalities Using Integrals 20. Pigeonhole Principle Revisited 21. Some Useful Irreducibility Criteria 22. Cycles, Paths and Other Ways 23. Some Special Applications of Polynomials –H. A. Shah Ali.

Bạn mua từ nhà xuất bản hoặc tìm E-book. Continue reading “Problems From the Book”

## IMC training 2016 (4)

Problem 1. The number 16 is placed in the top left corner square of a  table. The remaining 15 squares are to be filled in using exactly once each of the number 1,2,…,15, so that the sum of the four number in each row, each column and each diagonal is the same. Find the maximum value of the sum of the six numbers in the shaded squares shown in the diagram below.

Problem 2. All but one of the numbers from 1 to 21 are put into the squares of a  $4\times 5$ table, one number in each square, such that the sum of all the numbers in each row is constant, and the sum of all the numbers in each column is also constant. Find the number which is left out.

Problem 3. The diagram below show ten circles in a triangular array. Place each of the numbers 0 to 9 in a different circles so that for each of the six right-side up triangles marked with plus signs, the sum of the numbers in the three circles at its vertices is the same.

Problem 4. Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers a,b,c,d are replaced by a-b,b-c,c-d,d-a.  Is it possible after 2016 such to have numbers a,b,c,d such the numbers |bc-ad|, |ca-bd|, |ab-cd|  are primes?

Problem 5. Assume an  $8\times 8$ chessboard with the usual coloring. You may repaint all squares

1 – Of a row or column;

2 – Of a $2\times 2$ square.

The goal is to attain just one black square. Can you reach the goal?

Problem 6. A rectangular floor is covered by  $2\times 2$ and $1\times 4$  tiles. One tile got smashed. There is a tile of the other kind available. Show that the floor cannot be covered by rearranging the tiles.

Problem 7. A beetle sits on each square of a  $9\times 9$ chessboard. At a signal each beetle crawls diagonally onto a neighboring square. Then it may happen that several beetles will sit on some squares and none on others. Find the minimal possible number of free squares.

Problem 8. $10\times 10$ chessboard cannot be covered by 25 T-tetrominoes. Continue reading “IMC training 2016 (4)”

## IMC training 2016 (3)

Methods of Counting (2)

Problem 1. Find the number of pairs (x;y) of integers such that $|x|+|y|\le 1000$.

Problem 2. How many positive integers not exceeding 2001 are multiples of 3 or 4 but not 5?

Problem 3. Let x=.1234567891011…998999, where the digits are obtained by writing the integers 1 through 999 in order. Find the ${{1983}^{rd}}$ digit to the right of the decimal point.

Problem 4. A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?

Problem 5.  Find the number of sets {a,b,c} of three distinct positive integers with the property that the product of a,b, and c is equal to the product of 11,21,31,41,51, and 61.

Problem 6. Find the number of five-digit positive integers, n, that satisfy the following conditions:

(a) the number n is divisible by 5,

(b) the first and last digits of n are equal, and

(c) the sum of the digits of n is divisible by 5.

Problem 7. Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people such that exactly one person receives the type of meal ordered by that person. Continue reading “IMC training 2016 (3)”

## IMC training 2016 (2)

Methods of Counting

Problem 1. How many subsets are there in a set of size $10$?

Problem 2. Find the number of squares contained in an $10\times 10$ squares array.

Problem 3. A team is to be chosen from 4 girls and 6 boys. The only requirement is that it must contain at least 2 girls. Find the number of different teams that may be chosen.

Problem 4. You wish to give your Aunt Mollie a basket of fruit. In your refrigerator you have six oranges and nine apples. The only requirement is that there must be at least one piece of fruit in the basket (that is, an empty basket of fruit is not allowed). How many different baskets of fruit are possible?

Problem 5. Find the number of ways 30 identical pencils can be distributed among three girls so that each gets at least 1 pencil.

Problem 6. There 7 boys and 3 girls in a gathering. In how many ways can they be arranged in a row so that

1) the 3 girls form a single block?

2) the two end-possitions are occupied by boys and no girls are adjacent?

Problem 7. Between 20000 and 70000, find the number of even integers in which no digit is repeated. Continue reading “IMC training 2016 (2)”

## IMC training 2016 (1)

The arithmetic of integers

Problem 1. The positive integers from 1 to 12 have been divided into six pairs so that the sum of the two numbers in each pair is a distinct prime number. Find the largest of these prime number.

Problem 2. The sum of the squares of three prime numbers is 5070. Find the product of these three prime numbers.

Problem 3. A number is said to be strange if in its prime factorization, the power of each prime number is odd. For istance, 22,23 and 24 form a block of three consecutive strange numbers because $22={{2}^{1}}\times {{11}^{1}},23={{23}^{1}},24={{2}^{3}}\times {{3}^{1}}$. Find the greatest length of a block of consecutive strange numbers.

Problem 4. Find the number of consecutive 0s at the end of $2003!=1\times 2\times 3\times ...\times 2003.$

Problem 5. Find the smallest positive integer which is $2$ times the square of some positive integer and also $5$ times the fifth power of some other positive integer.

Problem 6. Sum of seven consecutive positive integer is the cube of an integer and the sum of the middle three numbers is the square of an integer. Find the smallest possible value of the middle number.

Problem 7. Some factors in the product $1\times 2\times 3\times 4\times ...\times 27$ are to be removed so that the product of the remaining factors is the square of an integer. Find the minimum number of factors that must be removed. Continue reading “IMC training 2016 (1)”