Contents 0.1 Preface 0.2 Notations and Symbol Chapter Introduction 1.1 Notations and Symbols in mathematics 1.1.1 1.2 1.3 1.4 Some differences in the math symbols in English (Eng.) and Vietnamese (Vie.) 1.1.2 Geometry 1.1.3 Abbreviations and Notations 1.1.4 Notations for Numbers, Sets and Logic Relations 1.1.5 The Greek alphabet is commonly used in mathematics 1.1.6 Mathematical Symbols Pronunciation of mathematical expressions 1.2.1 Logic and Set 1.2.2 Real numbers and operations 10 1.2.3 Functions 11 1.2.4 Some notation shortcuts are used in written English 12 1.2.5 Some notation shortcuts are often used in mathematics 13 Some Common Mathematical Symbols and Abbreviations (with History) 14 1.3.1 Binary Relations 14 1.3.2 Some Symbols from Mathematical Logic 14 1.3.3 Some Notation from Set Theory 16 1.3.4 Some Important Numbers in Mathematics 17 1.3.5 Appendix: Common Latin Abbreviations and Phrases 18 Skills Needed for Mathematical Problem Solving 20 1.4.1 Introduction 20 1.4.2 Mathematical problem solving as a process 21 1.4.3 Factors and skills involved in problem solving 22 i ii Contents 1.4.4 Conclusion 1.5 Mathematical Writing 32 32 1.5.1 Some notes when writing mathematical 32 1.5.2 A Guide to Writing Mathematics 39 1.5.3 Mathematical Ideas into Writing 43 Chapter Basic of Mathematics 49 2.1 Sets and Relations 50 2.1.1 Notation and Set Theory 50 2.1.2 Relations and Functions 54 2.1.3 Equivalence Relations and Classes 56 2.1.4 Natural Numbers, Integers, and Rational Numbers 58 2.2 Infinity and Induction 61 2.2.1 Countable Infinity 61 2.2.2 Uncountable Infinity 64 2.2.3 The Principle of Induction 67 2.2.4 The Real Number System 71 2.3 Logic 74 2.3.1 Negation of a Statement 74 2.3.2 Conjunction 76 2.3.3 Disjunction 79 2.3.4 Conditional Statements 80 2.3.5 Compound Statements 83 2.3.6 Biconditional Statements 85 2.3.7 Tautologies 89 2.3.8 Equivalence 91 Chapter Methods of mathematical proof 92 3.1 Mathematical Induction - Problems With Solutions 93 3.2 The Pigeonhole Principle 100 3.2.1 Introduction 100 3.2.2 Applications of the Pigeonhole Principle 101 3.2.3 Examples 111 3.2.4 More difficult examples and exercises 113 3.2.5 Conclusion 118 3.3 Direct Proof 118 iii Contents 3.3.1 3.3.2 3.4 3.5 Theorems Definitions 119 120 3.3.3 Direct Proof 123 3.3.4 Exercises 131 Contrapositive Proof 132 3.4.1 Contrapositive Proof 133 3.4.2 Congruence of Integers 3.4.3 Exercises 138 136 Proof by Contradiction 140 3.5.1 Proving Statements with Contradiction 141 3.5.2 Exercises 148 Chapter Number Theory 150 4.1 Elementary properties of integers 151 4.1.1 Fundamental Notions and Laws 151 4.1.2 Definition of Divisibility The Unit 152 4.1.3 Prime Numbers The Sieve of Eratosthenes 154 4.1.4 The Number of Primes is Infinite 155 4.1.5 The Fundamental Theorem of Euclid 157 4.1.6 Divisibility by a Prime Number 157 4.1.7 The Unique Factorization Theorem 158 4.1.8 The Divisors of an Integer 160 4.1.9 The Greatest Common Factor of Two or More Integers 161 4.1.10 The Least Common Multiple of Two or More Integers 164 4.1.11 Scales of Notation 165 4.1.12 Highest Power of a Prime p Contained in n! 168 4.1.13 Remarks Concerning Prime Numbers 172 4.2 4.3 On the indicator of an integers 173 4.2.1 Definition Indicator of a Prime Power 173 4.2.2 The Indicator of a Product 173 4.2.3 The Indicator of any Positive Integer 175 4.2.4 Sum of the Indicators of the Divisors of a Number 177 Elementary properties of congruences 179 4.3.1 Congruences Modulo m 179 4.3.2 Solutions of Congruences by Trial 181 iv Contents 4.4 4.3.3 Properties of Congruences Relative to Division 182 4.3.4 Congruences with a Prime Modulus 183 4.3.5 Linear Congruences 185 The theorems of Fermat and Wilson 187 4.4.1 Fermat’s General Theorem 187 4.4.2 Euler’s Proof of the Simple Fermat Theorem 188 4.4.3 Wilson’s Theorem 189 4.4.4 The Converse of Wilson’s Theorem 191 4.4.5 Impossibility of ⋅ ⋅ 3⋯n − + = nk for n > 191 4.4.6 Extension of Fermat’s Theorem 192 4.4.7 On the Converse of Fermat’s Simple Theorem 195 4.4.8 Application of Previous Results to Linear Congruences 196 4.4.9 Application of the Preceding Results to the Theory of Quadratic Residues 197 Chapter Other Topics 200 5.1 Complex Numbers 201 5.1.1 Square Roots 201 5.1.2 Complex Numbers 201 5.1.3 Cube Roots of Unity 203 5.1.4 Geometrical Representation 204 5.1.5 Product 204 5.1.6 Quotient 205 5.1.7 De Moivre’s Theorem 205 5.1.8 Cube Roots 206 5.1.9 Roots of Complex Numbers 207 5.1.10 Roots of Unity 208 5.1.11 Primitive Roots of Unity 210 5.2 Polynomials 211 5.2.1 Definitions and basic operations 212 5.2.2 Some applications 218 5.2.3 Analytic behavior 222 5.2.4 More identities 225 5.2.5 Integers and rationals 230 5.2.6 Polynomial problems with detailed solutions 236 v Contents 5.2.7 5.3 5.4 Exercises 242 Functional Equations 245 5.3.1 Introduction 245 5.3.2 Examples 245 5.3.3 Basic Methods For Solving Functional Equations 248 5.3.4 Cauchy Equation and Equations of the Cauchy type 250 5.3.5 Problems with Solutions 250 5.3.6 More Examples 270 5.3.7 Problems for Independent Study 279 A Tour of Triangle Geometry 285 5.4.1 Introduction 285 5.4.2 Isogonal conjugates 286 5.4.3 Simson line and line of reflections 295 5.4.4 Rectangular circum-hyperbolas 298 5.4.5 Conics 302 5.4.6 Further examples of reflections 309 5.4.7 A metric relation and its applications 313 5.4.8 The Apollonian Circles and Isodynamic Points 316 Chapter Exercises with Solutions and Answers 336 6.1 6.2 Exercises 337 6.1.1 Grade 10 algebra excercises 337 6.1.2 Grade 10 math word exercises 339 6.1.3 Grade 10 geometry excercises 6.1.4 Grade 10 trigonometry excercises 342 6.1.5 Grade 10 math algebra excercises (advanced) 344 6.1.6 Grade 10 math word excercises (advanced) 347 6.1.7 Grade 10 geometry excercises (advanced) 349 6.1.8 Grade 10 trigonometry excercises (advanced) 354 6.1.9 Grade 10 excercises on complex (advanced) 357 340 Solutions and Answers to the Above Excercises 359 6.2.1 Grade 10 algebra excercises 359 6.2.2 Grade 10 math word exercises 360 6.2.3 Grade 10 geometry excercises 6.2.4 Grade 10 trigonometry excercises 364 362 vi Contents 6.2.5 Grade 10 math algebra excercises (advanced) 367 6.2.6 Grade 10 math word excercises (advanced) 373 6.2.7 Grade 10 geometry excercises (advanced) 379 6.2.8 Grade 10 trigonometry excercises (advanced) 387 6.2.9 Grade 10 excercises on complex (advanced) 395 Chapter Examples and Exercises on Mathematical Training 397 7.0.10 Introduction 398 7.1 Operations on Rational Numbers 399 7.1.1 Basic Rules on Addition, Subtraction, Multiplication, Division 399 7.1.2 Rule for Removing Brackets 399 7.1.3 Ingenious Ways for Calculating 399 7.1.4 Examples 400 7.1.5 Exercises 403 7.2 Monomials and Polynomials 405 7.2.1 Definitions 405 7.2.2 Operations on Polynomials 406 7.2.3 Examples 407 7.2.4 Exercises 409 7.3 Linear Equations of Single Variable 410 7.3.1 Usual Steps for Solving Equations 410 7.3.2 Examples 411 7.3.3 Exercises 414 7.4 System of Simultaneous Linear Equations 416 7.4.1 Examples 417 7.4.2 Exercises 421 7.5 Multiplication Formulae 424 7.5.1 Basic Multiplication Formulae 424 7.5.2 Generalization of Formulae 424 7.5.3 Derived Basic Formulae 424 7.5.4 Examples 425 7.6 Some Methods of Factorization 429 7.6.1 Basic Methods of Factorization 429 7.6.2 Examples 429 7.6.3 Exercises 432 Contents 7.7 7.8 7.9 vii Absolute Value and Its Applications 434 7.7.1 Basic Properties of Absolute Value 434 7.7.2 Examples 434 7.7.3 Exercises 438 Linear Equations with Absolute Values 439 7.8.1 Examples 439 7.8.2 Exercises 443 Sides and Angles of a Triangle 445 7.9.1 Basic Knowledge 445 7.9.2 Examples 446 7.9.3 Exercises 449 7.10 Pythagoras’ Theorem and Its Applications 450 7.10.1 Examples 451 7.10.2 Exercises 456 7.11 Congruence of Triangles 458 7.11.1 Basic Criteria for Congruence of Two Triangles 458 7.11.2 Examples 458 7.11.3 Exercises 464 7.12 Divisions of Polynomials 465 7.12.1 Examples 467 7.12.2 Exercises 471 7.13 Congruence of Integers 472 7.13.1 Basic Properties of Congruence 473 7.13.2 Examples 474 7.13.3 Exercises 477 7.14 Decimal Representation of Integers 478 7.14.1 Decimal Expansion of Whole Numbers with Same Digits or Periodically Changing Digits 478 7.14.2 Examples 478 7.14.3 Exercises 482 7.15 Perfect Square Numbers 484 7.15.1 Basic Properties of Perfect Square Numbers 484 7.15.2 Examples 485 7.15.3 Exercises 488 viii Contents 7.16 [x] and {x} 489 7.16.1 Some Basic Properties of x and{x} 490 7.16.2 Examples 491 7.16.3 Exercises 496 7.17 Diophantine Equations (I) 497 7.17.1 Definitions 497 7.17.2 Examples 499 7.17.3 Exercises 503 7.18 Diophantine Equations (II) 505 7.18.1 Basic Methods for Solving Quadratic Equations on Z 505 7.18.2 Examples 505 7.18.3 Exercises 511 7.19 Pigeonhole Principle 512 7.19.1 Basic Forms of Pigeonhole Principle 512 7.19.2 Examples 513 7.19.3 Exercises 517 7.20 Geometric Inequalities 519 7.20.1 Examples 519 7.20.2 Exercises 526 7.21 Solutions to given exercises 528 Bibliography 534 Preface 0.1 Preface In recent years, due to the requirements of international integration, the demand for the knowledge of English in some professional field is becoming increasingly urgent, especially for the class teachers, high school students and college students In Viet Nam, The Ministry of Education and Training has planned to set up the programs for class teaching in bilingual Vietnamese - English, first for students of natural science subjects and then for students of social science ones in Specializing Upper Secondary Schools However, this is an extremely hard work because the knowledge of English of most teachers in this professional field is not good enough to carry out the task They need to be trained again to meet the demand Students, also, need to be taught in such a way that they can be able to understand the lessons in English Another obstacle is that teachers and students’ ability of English listening, speaking and writing is rather poor which has been considered an inherent weakness of foreign language learning and teaching in our country today To make some contribution to the ambitious program, I have decided to have the lectures written and designed in English in order to help students specializing in math understand and know the technical terms of solving exercises in English, so that their general knowledge of English will be improved as well Some of the beginning chapters and sections have been made directly by the teachers who are teaching in class, but most of the content of this research is for students to read and practice under the help and guidance of the teachers This book includes chapters and is divided into two parts Chapter I Introduction: Provide the knowledge needed to understand the book: The system of notation, Greek alphabet and the rules of mathematical word in English Chapter II Basic of Mathematics: Present the basic knowledge of mathematics including set theory, logic, relations and functions Chapter III Number Theory: Present some basic knowledge of number theory This chapter is only for students specializing in math Chapter IV Methods of mathematical proof : Provide some common methods of proof in mathematics: Direct and indirect proof, Contradiction and Induction etc Preface Chapter V Other topics: Present the individual subjects in the school program as well as for students of math Chapter VI Exercises with Solutions and Answers: Offer some simple exercises as rehearsals for students to and explain exercises in English Chapter VII Examples and Exercises on Mathematical Training: Offer some questions for students to practise, and test students’ ability to apply their knowledge in solving real competition questions These examples and exercises are taken from a range of countries, e.g China, Russia, the USA and Singapore, e.t.c The books shown in the Bibliography are mostly sent to the author from students who are studying abroad and hard to find in Vietnam With any luck, readers can find a pdf- files online, but if possible, you can order at amazon.com However, because of our limited level of writing, the limitation of time and the length of the research, there are still some dissatisfaction in the discussion as the following: Some parts missing, such as: inequality, the transcendental equations, inequations, and systems of equations (exponential, logarithmic and trigonometric), survey plot functions, applications of derivatives, etc Some topics for gifted students have not been put in, such as (Invariant theory, game theory, extreme theory, combinatorial mathematics, etc.) or only superficial presented in this textbook (Graph Theory, Principle Direchlet, etc.) Also lack the classical geometry (plane and space), Vector and applications, Transformation, Method coordinates in Space, etc In addition, the desired book is applied to three characters: mathematics teachers, students specializing in math, and students of Specializing Upper Secondary Schools in general Hence the book style is not consistent While sections for students of math and for teacher are written with style accurately and scientifically, while other parts of text are freely written With full of hope, the author’s colleagues will continue to improve he research on the shortcomings and with the sincere comments of your readers, the next version will be better The book is written by software Viettex 2.5 and PCTEXv5.2 The picture drawn by WinTpic, WinFig, Sketchpad and Graph4.3 523 7.20 Geometric Inequalities Figure 7.12: Example 7.176 (USAMO/1996) Let ABC be a triangle Prove that there is a line L (in the plane of triangle ABC ) such that the intersection of the interior of triangle ABC and the interior of its refiection A′ B ′ C ′ in L has area more than the area of triangle ABC Solution: Let BC = a, CA = b, AB = c Without loss of generality we may assume that a ⩽ b ⩽ c Let AD be the angle bisector of the ∠BAC, B ′ , C ′ be the symmetric points of B, C in the line AD, respectively, then C ′ is on the segment AB and B ′ is on the extension of AC,as shown in the figure 7.13 Figure 7.13: From [BDC ′ ] = BC ′ BD c−b c c−b × × [ABC ] = × × [ABC ] = × [ABC ] AB BC c b+c b+c 524 Chapter 7.Examples and Exercises on Mathematical Training 2b > a + b > c implies that b c+b [AC ′ DC ] = (1 − > b 2b + b = , so c−b 2b ) [ABC ] = [ABC ] > [ABC ] c+b c+b Thus, the line AD satisfies the requirement (⊠) Example 7.177 (CHINA/1978) Through the center of gravity G of the ∆ABC introduce a line to divide ABC into two parts Prove that the difference of areas of the two parts is not greater than of area of the ABC Solution: Suppose that an arbitrary line passing through G intersects AB and AC at D and E respectively When the line DE is parallel to BC, then [ADE ] = ( DE ) [ABC ] = [ABC ], so BC ∣[ADE ] − [BDCE ]∣ = ( − ) [ABC ] = [ABC ], 9 the conclusion is true When DE is not parallel to BC, say D is between P and B and E is between Q and N, where P Q ∥ BC and N is the midpoint of AC, as shown in the figure, we show that Figure 7.14: ∣[DBCE ] − [ADE ]∣ < ∣[P BCQ] − [AP Q]∣ = [ABC ] 525 7.20 Geometric Inequalities below Since ∠DP G > ∠P QA, we can introduce P S ∥ AC , intersecting DE and BN at R and S respectively Then P RG ≅ QEGand RSG ≅ ENG, so that [P RG] = [QEG] and [RSG] = [ENG] Therefore [DBCE ] = [P BCQ] − [P DG] + [QEG] < [P BCQ] and [ADE ] = [AP Q] + [P DG] − [QEG] > [AP Q], so that [DBCE ] − [ADE ] < [P BCQ] − [AP Q] = [ABC ] It suffices to show that [DBCE ] > [ADE ] For this notice that [DBCE ] = [BCN ] − [ENG] + [DBG] > [BCN ] − [ENG] + [RSG] = [BCN ] = [ABC ], ⇒ [ADE ] < [ABC ] < [DBCE ] Thus, the conclusion is proven (⊠) Example 7.178 (CMO/1969) Let ABC be the right-angled isosceles triangle whose equal sides have length P is a point on the hypotenuse, and the feet of the perpendiculars from P to the other sides are Q and R Consider the areas of the triangles AP Q and P BR, and the area of the rectangle QCRP Prove that regardless of how P is chosen, the largest one of these three areas is at least Solution: Let BR = x, then BR = P R = QC = x and RC = P Q = AQ = − x (i) When x ⩾ , then [P BR] = x2 ⩾ ⋅ (ii) When x ⩽ ,then − x ⩾ , so that (iii) When [AP Q] = (1 − x)2 ⩾ 1 < x < , then − ⩽ x − ⩽ , so that 3 6 1 [QCRP ] = x(1 − x) = − (x − ) + > − + = ⋅ 36 Thus, the conclusion is proven (⊠) 526 Chapter 7.Examples and Exercises on Mathematical Training Figure 7.15: 7.20.2 Exercises Exercise 7.270 Given that R is an inner point of ∆ABC Prove that (AB + BC + CA) < RA + RB + RC < AB + BC + CA Exercise 7.271 (BMO/1967) In ∆ABC, if ∠C > ∠B, and BE, CF are the heights on CA and AB respectively Prove that AB + CF > AC + BE Exercise 7.272 In ∆ABC, AB > BC, AD ⊥ BC at D.P is an arbitrary point on AD different from A and D, prove that P B − P C > AB − AC Exercise 7.273 For an acute triangle ABC, let a = BC, b = CA, c = AB, and the lengths of height on BC, CA, AB are , hb , hc respectively, prove that (a + b + c) < + hb + hc < a + b + c Exercise 7.274 (MOSCOW/1974) Prove that if three segments with lengths a, b, c can form a triangle, then the three segments of lengths , can form a+b ; b+c ; c+a a triangle also 527 7.20 Geometric Inequalities Exercise 7.275 Given that BB1 and CC1 are two medians of ∆ABC Prove that BB12 + CC12 > BC Exercise 7.276 (MOSCOW/1972) A straight line intersects the sides AB and BC of ∆ABC at the point M and K respectively, such that the area of MBK and the area of the quadrilateral AMKC are equal Prove that MB + BK ⩾ ⋅ AM + CA + KC Exercise 7.277 (CHINA/1994) There are n straight lines in a plane, such that every two intersect with each other Prove that among the angles formed there is at least one angle which is not greater than 180o ⋅ n Exercise 7.278 (RUSMO/1983) In ∆ABC, D is the midpoint of AB, E and F are on AC and BC respectively Prove that the area of ∆DEF is not greater than sum of areas of ∆ADE and ∆BDF Exercise 7.279 (CHNMOL/1979) Given that ∆ABC is an acute triangle, a = BC, b = CA, c = AB and a > b > c Among its inscribed squares, which one has the maximum area? Exercise 7.280 (KIEV/1969) Is there a triangle with the three altitudes of √ √ lengths 1, 5, + 5? Exercise 7.281 (RUSMO/1981) The points C1 , A1 , B1 belong to sides AB, BC, CA, respectively, of the ∆ABC AC1 BA1 CB1 = = = ⋅ C1 B A1 C B1 A Prove that the perimeter P of the ∆ABC and the perimeter p of the ∆A1 B1 C1 satisfy inequality P < p < P Exercise 7.282 (KIEV/1966) Let a, b, c be the lengths of three sides of ∆ABC, and I = a + b + c, S = ab + bc + ca Prove that 3S ⩽ I ⩽ 4S Exercise 7.283 (RUSMO/1989) If a, b, c denote the lengths of sides of a triangle, satisfying a + b + c = 1, prove that a2 + b2 + c2 + 4abc < ⋅ 528 Chapter 7.Examples and Exercises on Mathematical Training Exercise 7.284 (PUTNAM/1973) (i) Given that ∆ABC is an arbitrary triangle, the points X, Y, Z are on the sides BC, CA, AB respectively If BX ⩽ XC, CY ⩽ Y A, AZ ⩽ ZB, prove that [XY Z ] ⩾ [ABC ] (ii) Given that ∆ABC is an arbitrary triangle, the points X, Y, Z are on the sides BC, CA, AB respectively (but there is no any assumptions to the ratio of distances, like BX XC ) Please use the method used in (i) or otherwise to show that among ∆AZY, ∆BXZ, ∆CY Z there must be one with area not greater than that of ∆XY Z Exercise 7.285 (IREMO/2003) Let T be a triangle of perimeter 2, and a, b, c be the lengths of its three sides Prove that (i) abc + 28 ⩾ ab + bc + ca; 27 (ii) ab + bc + ca ⩾ abc + 7.21 Solutions to given exercises For subsection 1 -1 5 ⋅ 17 ⋅ 15 − 2007 ⋅ 4018 25 ⋅ 26 n(n + 1) ⋅ 2019045 10 2222222184 11 49 49 ⋅ 100 12 5049 13 2575 ⋅ 10302 14 1275 ⋅ 2551 15 For subsection 16 (B) and (C) 19 1.2 17 (D) 20 (C) 18 9x2 + 21 n = 10 26 (a + b)(b + c)(c + a) = 27 x2 + y + z = (ab)2 + (ca)2 + (bc)2 abc 22 2009 23 (C) 28 29 = { 24 ⋅ 25 P (7) = −17 for even n for odd 529 7.21 Solutions to given exercises For subsection 30 x = 402 31 x = 32 x = − 14 ⋅ 37 n ≠ m ∶ x = n2 n−m 38 a + b = ∶ S = R ; 74 ⋅ 167 33 (D) 35 (B) 34 36 n = m ∶ S = ∅ ; a + b ≠ 0, a = ∶ S = ∅ ; a + b ≠ 0, a ≠ ∶ x = a+b ⋅ a 39 −1 40 m ≠ 0; ∶ x = ; m 41 < k < ∨ k > 42 x = − 2a ⋅ m=1∶ S=R ; 45 x = 300 ; mmin = 43 x = − ⋅ 44 x = m = ∶ S = ∅ 46 x = ⋅ 47 2008 ⋅ 2009 For subsection 48 (C) 49 (A) 50 k ≠ −2 ; k = −2 ; ∅ 51 ( 40 40 , , −40) 11 52 (7, 5, −3) 54 −1 55 m2 = 56 12 53 (0, 6, 7, 3, −1) 57 (2, 1, 3, 4) 58 (−9, −6) ; (−6, −4) ; (−3, −2) ; (0, 0) ; (3, 2) ; (6, 4) ; (9, 6) 59 ( 23 23 23 , , ) 10 60 (4, 5, 6) or (−4, −5, −6) For subsection 63 37 64 28 61 (4, 7) 62 (0, 1, 3, 5, 7) 68 (a2 − 2)3 − 3(a2 − 2) 69 Do it yourself 65 (ac + bd)2 + (ad − bc)2 70 Do it yourself 66 a ∶ b ∶ c = ∶ ∶ 67 a2 10a2 − 6a + 73 (A) 74 Do it yourself 75 Do it yourself 71 (x − 2)(y − 2)(z − 2) = 076 71 ⋅ 72 Do it yourself 77 a + b = or a + b = −2 530 Chapter 7.Examples and Exercises on Mathematical Training For subsection (ii) − 6(x − y )(x − y + 2) 78 (i) (x3 + y )(x4 + 6x3 y + y ) (ii) (2x − y + 3z )2 (iii) (3x + 4)(3x + 1)(12x2 + 20x + 11) (v ) (x − 2)(x2 − x + 2a) (vi) (a + b + c)(a2 + b2 + c2 ) (iii) (x2 + 4x − 1)2 (iv ) (x + 1)(x + 6)(x2 + 4x + 6) (iv ) 2(2x2 − 4x − 1)(x2 − 2x − 2) (v ) − 6(x2 − 4x)(x2 − 4x + 2)(x2 − 3x + 1) (vi) (x + 1)(x2 − x + 1)× ×(x2 + x + 1)(x6 + 1) 83 (i) a = 11 or −19 (ii) (x2 + x + 1)(x2 − 2x + 5) 79 (i) (x − a − b)(x + a + b)× ×(x − a + b)(x + a − b) 84 m = −15, n = −36 (ii) (ab + b + 1)(ab + a + 1) 85 9(x2 + 4x + 1)(x + 1)2 80 The expression has a factor 45 86 a + b = 31 ⎞ ⎛ ⎟ ⎜ 81 ⎜66⋯66⎟ ⎟ ⎜ ⎝ n digits ⎠ 87 (a − d)(b + c)(a + b − c + d)(a − b + c + d) 88 3abxy (a + b)(x + y ) 82 (i) x(x + 1)(x2 + x − 3) 89 Do it yourself For subsection 90 = { for x < for x > 91 max = ; = − ⋅ 11 (D) for x < or x > 92 The answer is { (A), (B ), (D) for ⩽ x ⩽ 93 = 96 ab = 99 x1 = a, x2 = a+6, x3 = a−6 94 = −x 95 S = ∣c − b∣ + ∣d − a∣ 97 100 n = 50 98 4019 101 S = an + an−1 + ⋯ + a n+1 +1 − a n−1 − a n−3 − ⋯ − a2 − a1 102 ⩽ x ⩽ ⋅ For subsection 105 x = or x = ⋅ 103 4013 2 104 10000 531 7.21 Solutions to given exercises 106 (B) 109 (B) 107 (C) 110 −2008 < a < 2008 108 (C) 111 (A) 115 (x, y ) ∈ {(−1, −1) ; ( , ) ; (1 − 1) ; (− , − )} 3 116 (A) 117 (B) 112 (x, y ) = (1, 2) 113 (C) 114 x = , y = 118 Do it yourself.119 Do it yourself For subsection 120 Prove ∠DAB + ∠DBA < 90o 121 Perimetre is132 123 (A) 122 max n = 13 124 ∠ACF = 45o 125 ∶ (8, 8, 1) ; (8, 7, 1) ; (7, 7, 3) ; (6, 6, 3) ; (7, 6, 4) ; (8, 5, 4) ; (6, 6, 5) ; (7, 5, 5) 126 129 (C) 132 (B) 127 70 130 29 133 ∠A5 = 3o 128 ∠EAD = 10o 131 (D) 135 ∠B = 30o 134 Do it yourself 136 ∠NMB = 30o For subsection 10 137 Do it yourself 142 Do it yourself 138 5.12, 13 143 Do it yourself 139 4.5cm 144 400 140 4.5cm √ 20 cm 141 145 Do it yourself √ 146 150 (E) 152 ∠CHD = 45o 155 Do it yourself 158 Do it yourself 153 ∠BP D = 39o 156 Do it yourself 159 P C = 154 ∠P CQ = 45o 157 ∠NMC = 30o 147 25cm √ 148 149 There is no a triangle 151 Do it yourself For subsection 11 160 [ABCDE ] = 532 Chapter 7.Examples and Exercises on Mathematical Training 161 Do it yourself 163 Do it yourself 165 Do it yourself 162 Do it yourself 164 Do it yourself 166 Do it yourself For subsection 12 167 q (x) = 3x3 + 6x + ; r = 174 f (x) = (x + 1)(x + 2)(x + 4) 169 k = − 177 f (x) = x2 − 3x + 168 q (x) = (−24x3 − 12x2 − 34x + 15) ;175 2(x2 + y + xy )2 87 r= 176 (x + y + z )(x − y )(y − z )(x − z ) 83 ⋅ 170 a = 6, b = 178 Do it yourself 171 r = x 172 q = 16, r = 32 173 f (x) = − x3 − 3x2 + 179 r(x) = x 11 x − For subsection 13 180 (x − y )(y − z )(z − x)(x + y + z + xyz ) 181 g (x) = x2 + x + 1, h(x) = x + 182 16 187 875 192 36 183 188 1731 193 29 184 Do it yourself 189 Do it yourself 185 190 (E) 195 Do it yourself 186 191 88 196 6k + or 6k + 202 n = 1977 207 312 194 min{n} = 101 For subsection 14 197 37 198 625 × 10n−2 199 x = 489 200 An = (2 66⋯66 7)2 203 S = 17901 208 (3, 2, 1) ; (6, 8, 4) ; (8, 3, 7) 204 1979 209 52 205 29995 210 7744 206 Do it yourself 211 1681 n−1 201 142587 533 7.21 Solutions to given exercises For subsection 15 212 k does not exist 217 Do it yourself 222 98 213 (D) 218 Do it yourself 223 p = 214 421 219 Do it yourself 224 Do it yourself 215 1045 220 Do it yourself 225 Do it yourself 216 1972 221 Do it yourself 226 Do it yourself 227 Do it yourself 232 −2 237 Do it yourself 228 x = −1 233 x = 5026 ; 5027 238 Do it yourself 234 600 239 10; ; ; ; ⋅ For subsection 16 229 x = ; x = 230 x = 1; √ √ 1+ 33; 231 n2 − n + √ 41; 235 1486 240 08 241 −9.2 236 298 For subsection 17 242 (D) 244 246 248 1984 243 (C) 245 1997 247 541 249 mn = 84 250 (0, 25, 75) ; (4, 28, 78) ; (8, 11, 81) ; (12, 4, 84) 251 (667, 666, 665) 254 Do it yourself 252 60 × 1g ; 39 × 2g and × 50g 255 Do it yourself 253 Do it yourself 256 Do it yourself For subsection 18 257 a = 3, 4, 6, 9, 11 259 2187 261 260 x = 3, y = 262 267 {(1, 1)} ∪ {(k, 2k ) ∶ ∀ k ∈ N} 258 (x, y ) ∈ {(4, 2); (4, −2); (−4, −2); (−4, 2)} 263 265 x = y = z = 264 p = 29 266 (B) 268 x = m(m + n)t, y = ±n(m + n)t, z = mnt, where m, n, t are arbitrary integers 534 Chapter 7.Examples and Exercises on Mathematical Training 269 p = 76 270 (p, q ) ∈ {(2, 3), (3, 2), (2, 2), (1, 5), (5, 1)} 271 (m, n) ∈ {(2, 2), (2, 1), (3, 1), (5, 2), (5, 3), (1, 2), (1, 3), (2, 5), (3, 5)} For subsection 19 Do it yourself For subsection 20 Do it yourself Bibliography [1] Dusan Djukic- Vladimir Jankovic- Ivan Matic-Nikola Petrovic, The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads, 1959-2004 Springer, 2006 [2] Titu Andreescu and Zuming Feng, 102 Combinatorial Problems from the Training of the USA IMO Team Bikhauser, 2002 [3] Titu Andreescu and Zuming Feng, A path to combinatorics for undergraduates counting strategies Bikhauser, 2004 [4] Titu Andreescu, Razvan Gelca, Mathematical Olympiad Challenges Bikhauser, 2000 [5] Titu Andreescu and Zuming Feng, Mathematical Olympiads 1998 - 1999 Problems and Solutions From Around the World Published and distributed by The Mathematical Association of America, 2000 [6] Titu Andreescu and Zuming Feng, Mathematical Olympiads 1999 - 2000 Problems and Solutions From Around the World Published and distributed by The Mathematical Association of America, 2002 [7] Arthur Engel, Problem - Solving Strategies Springer, 1998 [8] Loren C Larson, Problem - Solving Through Problems Springer, 1983 [9] Walter Mientka et al., Mathematical Olympiads 1996 - 1997, Problems and Solutions From Around the World, MMA 1997 [10] Walter Mientka et al., Mathematical Olympiads 1997 - 1998, Problems and Solutions From Around the World, MMA 1998 [11] Ball, W W R A Short Account of the History of Mathematics London, 1888, 2nd edition, 1893 [12] Acz’el, J.: 1966, Lectures on Functional Equations and Their Applications New York, Birkhaăuser 535 536 Bibliography [13] Acz’el, J.: 1984, On history, applications and theory of functional equa- tions In Functional Equations: History, Applications and Theory, J Acz’el (ed.), Mathematics and Its Applications Boston, Reidel [14] Alexanderson, G L., Klosinski, L F., and Larson, L C.: 1985, The William Lowell Putnam Mathematical Competition Problems and Solutions: 1938-1964 Mathematical Association of America, Washington, DC [15] Gleason, A M., Greenwood, R E and Kelly, L M.: 1980, The William Lowell Putnam Mathematical Competition Problems and Solutions: 1965- 1984 Mathematical Association of America, Washington, DC [16] Kedlaya, K S., Poonen, B., and Vakil, R.: 2002, The William Lowell Putnam Mathematical Competition 1985-2000 Problems, Solutions and Commentary Mathematical Association of America, Washington, DC [17] Kuczma, M., Choczewski, B and Ger, R.: 1990, Iterative Functional Equations Cambridge, UK, Cambridge University Press [18] Young, G S.: 1958, The linear functional equation Amer Math Monthly 65, pp 37-38 [19] A T Benjamin and J J Quinn, Proofs That Really Count: The Art of Combinatorial Proof, The Mathematical Association of America, 2003 Modern [20] Keith Devlin, Mathematics, The New Golden Age, Columbia Univ Press, NY, 1999 [21] P J Eccles, An Introduction to Mathematical Reasoning: Numbers, Sets and Functions, Cambridge Univ Press, Cambridge, UK, 1997 [22] P D Magnus, Forall x: An Introduction to Formal Logic, version 1.22, 2005, [23] T Nagell, Introduction to Number Theory, Wiley, New Your, 1951 [24] Robin J Wilson, Four Colors Suffice: How the Map Problem was Solved, Princeton University Press, Princeton, NJ, 2002 [25] Leonard Gillman, Writing mathematics well, The Mathematical Association of America, Washington, D.C (1987) [ISBN: 0-88385-443-0] [26] K Houston, How to think like a mathematician, Cambridge University Press, Cambridge (2009) [27] V K Balakrishnan, Theory and Problems of Combinatorics, Schaum’s Outline Series, McGraw-Hill, 1995 [28] A Engel, Problem-Solving Strategies, Springer Verlag, 1998 537 Bibliography [29] R Graham, D Knuth, O Patashnik, Concrete Mathematics, 2nd edition, AddisonWesley, 1994 [30] K Kuratowski and A Mostowski, Set Theory, North-Holand, Amsterdam, 1967 [31] S Savchev, T Andreescu, Mathematical Miniatures, MAA, 2003 [32] A History of Mathematical Notations by Florian Cajori [33] Article on "What is Pigeonhole Principle?" by Alexandre V Borovik, Elena V Bessonova [34] Article on "The Puzzlers’ Pigeonhole" by Alex Bogomolny, Mathematical Association of America [35] Article on "Pigeonhole Principles" by Dmitri Fomin, Sergey Genkin and Ilia Itenberg, Mathematical Circles (Russian Experience) [36] MathLinks topic, 17th Junior Tournament of the Towns 1995 Autumn problems [37] Howard Anton Calculus A new Horizon Sixth Edition John wiley & Sons, inc 1999 [38] P.E Danko Higher Mathematics in Problems and Exercises Mir Publishers Moscow.1983 [39] Websites of mathemathics: http://diendantoanhoc.net http://mathforum.org http://mathnfriend.net http://www.math.com/ www.kalva.co.uk http://www.bymath.com/ http://www.ams.org/ http://lib.mexmat.ru/ http://www.math.ac.vn/ http://sms.math.nus.edu.sg/ http://math.ca/crux/ http://thesaurus.maths.org/ http://kvant.mccme.ru/ http://www.mathlinks.ro/ http://www.mathematicsmagazine.com/ http://mathworld.wolfram.com/ [...]... comments on the content and the format of the manuscript, and Doctor, Associate Professor Nguyen Vu Luong for his strong support The teachers: MA Bach Dang Khoa, MA Nguyen Anh Tuan, MA Tran Thi Ha Phuong, BA Nguyen Van Thao and MA Tran Anh Duc for reading and editing this manuscript In particular, thanks to MA Tran Thi Ha Phuong and a group of students of math in mathematics courses K17, K18, K19, K20,... the manuscript And, the most sincere thanks to some teachers of English in Bac Giang Specializing Upper Secondary school, Ms Do Thi Minh Hong, Ms Mai Thu Giang, Ms Vu Thi Hue and especially Mr Nguyen Danh Hao, who have checked the text carefully Without timely support and help from these characters, the research could not be completed successfully The author would like to receive feedback and contributions ... for his strong support The teachers: MA Bach Dang Khoa, MA Nguyen Anh Tuan, MA Tran Thi Ha Phuong, BA Nguyen Van Thao and MA Tran Anh Duc for reading and editing this manuscript In particular, thanks... Secondary school, Ms Do Thi Minh Hong, Ms Mai Thu Giang, Ms Vu Thi Hue and especially Mr Nguyen Danh Hao, who have checked the text carefully Without timely support and help from these characters,