A study on the english vietnamese translation of the mathematical puzzles for national team of maths at vinhphuc gifted secondary school

VIETNAM NATIONAL UNIVERSITY, HANOI COLLEGE OF FOREIGN LANGUAGES POST GRADUATE DEPARTMENT LÊ THỊ LIÊN A STUDY ON THE ENGLISH-VIETNAMESE TRANSLATION OF THE MATHEMATICAL PUZZLES FOR NATIONAL TEAM OF MATHS AT VINHPHUC GIFTED SECONDARY SCHOOL Nghiên cứu cách dịch Anh-Việt toán dùng cho học sinh đội tuyển toán quốc gia Trường THPT chuyên Vĩnh Phúc MA THESIS FIELD: ENGLISH LINGUISTICS CODE : 60.22.15 HANOI-2009 VIETNAM NATIONAL UNIVERSITY, HANOI COLLEGE OF FOREIGN LANGUAGES POST GRADUATE DEPARTMENT LÊ THỊ LIÊN A STUDY ON THE ENGLISH-VIETNAMESE TRANSLATION OF THE MATHEMATICAL PUZZLES FOR NATIONAL TEAM OF MATHS AT VINHPHUC GIFTED SECONDARY SCHOOL Nghiên cứu cách dịch Anh-Việt toán dùng cho học sinh đội tuyển toán quốc gia Trường THPT chuyên Vĩnh Phúc MA THESIS FIELD: ENGLISH LINGUISTICS CODE: 60.22.15 SUPERVISOR: ASSOC PROF PHD LÊ HÙNG TIẾN HANOI-2009 TABLE OF CONTENTS Acknowledgements i Abstract ii Abbreviations iii Table of contents iv PART A: INTRODUCTION Rationale Aims Scope Methods PART B: DEVELOPMENT CHAPTER I: LITERATURE REVIEW I Translation theory I.1.1 Definition I.1.2 Translation equivalence I.1.3 Translation methods and procedure I.13.1 Word-for word translation I.1.3.2 Faithful translation I.1.3.3 Semantic translation I.1.3.4 Idiomatic translation I.1.3.5 Shift or transposition translation I.1.3.6 Modulation I Mathematical texts I.2.1.Mathematical terminology I.2.2 Syntactic features 10 I.2.2.1.Relative clauses as post modifiers 10 I.2.2.2.–ed participle clause and passive structure 11 CHAPTER II: THE STUDY 14 II Subject of the study and collection of data 14 II.1.1 Subjects of the study 14 II.1.2 Collection of data 14 II Findings and discussions 15 II.2.1 Translation of mathematical terms by transposition procedure 15 II.2.2 Translation of relative clauses by transposition procedure 17 II.2.2.1 Translation of relative clauses by replacing word class 17 II.2.2.2 Translation of relative clauses by zero linking devices 20 II.2.2.3 Translation of relative clauses by means of apposition 22 II.2.2.4 Translation of relative clauses by syntagmatic change 23 II.2.3 Translation of –ed clause and passive structure 24 II.2.3.1 Translation of –ed clause and passive structure by transposition 25 procedure II.2.3.2 Translation of –ed clause and passive structure by either transposition or modulation procedure II.2.4 Summary 26 29 PART C: CONCLUSION I Major findings 31 I.1 Translation of mathematical terms 31 I.2 Translation of relative clauses 31 I Translation of –ed participle clauses and passive structures 32 II Implications for translation of English mathematical problems 33 REFERENCES ABBREVIATIONS SL: source language TL: target language PART A: INTRODUCTION Rationale Vinhphuc gifted secondary school is the most prestigious high school in Vinhphuc province During more than ten years since its establishment in 1997, the school has gained significant achievements in teaching and training gifted students in the province and has gradually emerged as one of top gifted schools in Vietnam thanks to its praiseworthy results in international contests like IMO, PhO,… In deed, subjects of natural sciences, especially mathematics, have always received great attention of management board and teachers because since the school’s infancy, mathematics has gradually bought fame and reputation for the school Many students have gained medals in international mathematic contests Besides achievements, it is impossible not to the mention the great difficulties encountered by teachers and students during the process of teaching and learning mathematics Specifically, teachers and students in national team of mathematics have constantly had difficulties in exploring materials from abroad and the Internet due to their poor knowledge of English They often fail to give a smooth translation to international math puzzles written in English, which places them in a great disadvantage The reason is that a poor translation is surely to prevent or mislead the understanding of both teachers and students As an English teacher at this high school, I am fully aware of this problem and I also realize the necessity for a study to be conducted on the way of translation math puzzles Many Vietnamese mathematicians have acknowledged this problem and in fact they have been trying to translate the math puzzles to help improve the quality of math teaching and learning It can be seen that some books which include Vietnamese version of the international math problems have been published in recent years, like “A collection of IMO problems” by Le Hai Chau, “40 years of international mathematical Olympiad” Strongly motivated by the desire to help my colleagues and my students as well as the existing publications, I have decided to carry out a study on the translation of mathematic problems as the thesis for my M.A attainment Aims of the study The study is aimed at:  Pinpointing the prominent factors affecting the translation of mathematic problems in English, that is mathematical terminologies and certain syntactic features of mathematical problems such as relative clauses, -ed participle clauses and passive structure  Identifying the translation strategies applied in the translation of mathematical terminologies and some main syntactic features of mathematical problems by observing and investigating a great number of mathematical problems and their translation In general, it is expected that the thesis would provide math teachers and students specializing in mathematics with some ideas about ways of dealing with English math problems, thus reducing the pitfalls or problems that may arise during their teaching and learning Scope of the study This study is carried out on the basis of what has been explored in the book entitled “40 years of international mathematical Olympiad” by Dr Vu Duong Thuy and Ms Nguyen Van Nho (2006) Within the scope of this thesis, only typical discourse features of math problems are put under investigation, which comprise the terminologies, syntactic features such as complex sentences with relative clauses, -ed participle clause and passive structure Finally, suggestions for translating math problems will be only made on the basis of the results of the study Methods of the study This is a descriptive study since the aim of the study is to analyze and pinpoint the common strategies used in the translation of English math puzzles into Vietnamese In order to accomplish the thesis, a flexible combination of methods is employed, which embraces reference to publications, collection, categorizing and analyzing data, and description of result PART B: DEVELOPMENT CHAPTER I: LITERATURE REVIEW I.1 TRANSLATION THEORY I.1.1 Definition The study of translation has been dominated by the debate about its status as an art or a science Different linguists have put the definition of translation in various ways, among which the followings stand out “Translation is the expression in another language (or target language) of what has been expressed in another, source language, preserving semantic and stylistic equivalents” (Marlone, 1988) Translation means “the replacement of text in one language (SL) by an equivalent in another language (TL)” (Catford, 1965) Translation is a process based on the theory that it is possible to abstract the meaning of a text from it its form and reproduce that meaning with the very different form of a second language Translation, then, consists of studying the lexicon, grammatical structure, communication situation, and cultural context of the source language text, analyzing it in order to determine its meaning, and then reconstructing this same meaning using the lexicon and grammatical structure which are appropriate in the receptor language and its cultural context (Larson l998, p 3) In spite of the differences in these definitions, there are still common features which can be realized as the notion of movement between languages, content and the responsibility to find equivalents that preserve the attributes or characteristic features of the original text It is such an idea of equivalence that we are going to discuss hereafter 1.1.2 Translation equivalence Equivalence is well-recognized as a central concept in translation theory, which postulates a relation between SL text and TL text The following are some elaborate approaches to translation equivalence Nida (1964) strongly advocates dynamic equivalence rather than formal equivalence "Formal equivalence" corresponds to "metaphrase," and "dynamic equivalence", to "paraphrase." As he puts it, formal equivalence means closest possible match of form and content between SL and TL, or a means of providing some degree of inside into the lexical, grammatical or structure form of a source text Meanwhile, dynamic equivalence is the principle of equivalence of effect on readers of target text or the same effect on the TL receivers as the source text has on the SL receivers Newmark, P (1995) terms Nida’s dynamic equivalence as “ equivalence response” or “ equivalent effect”, and holds that “ the overriding purpose of any translation should be to achieve “equivalence effect ,i e to produce the same effect ( or one as close as possible) on the readership of the translation as was obtained on the readership of the original” This, according to Newmark, should be consider the desirable result, rather than the aim of any translation except for two cases: (1) if the purpose of the SL text is to affect and the TL translation is to inform (or vice versa); (2) if there is a pronounced cultural gap between the SL and the TL Koller (1979) presents five types of equivalence as follows Denotative equivalence: This orients towards the extra-linguistic content transmitted by a text Connotative equivalence: This respect indicates that individual expressions in the textual context not only have a denotative meaning but also additional values which mean various or synonymous ways of expressions Text-normative equivalence: This has to with text-type and specific features or text and language norms for given text types To put it another way, the SL and TL words are used in the same or similar context in their respective languages Pragmatic equivalence: This means translating the text for a particular readership, i.e the receiver to whom the translation is directed, and to whom the translation is turned in order to achieve a given effect Formal equivalence: This aim to produce an “analogy of form” in the translation by exploiting the formal possibilities of the TL or even by creating new forms if necessary Baker, M (1992) approaches the concept of equivalence differently by discussing the notion of non-equivalence at word level and above word level, grammatical equivalence, textual equivalence and pragmatic equivalence  Non-equivalence at word level means that the target language has no direct equivalent for a word which occurs in the source text Common problems of nonequivalence then involve such cases as culture-specific concepts, the SL concepts is not lexicalized in the target language, the SL word is semantically complex, the SL and TL make different distinctions in meaning, the TL lack a super-ordinate, a specific term (hyponym), differences in physical or interpersonal perspective, differences in expressing meaning, difference in form, the use of loan word in the source text  Non-equivalence above word level is closely related to the differences in the  collocational patterning of the SL and TL, which create potential pitfalls and can pose various problems in translation  Grammatical equivalence is more concerned with the differences in the grammatical structures of the SL and TL, which often result in some change in the information content of the message during the process of translation This may change may take the form of adding to the target text information which is not expressed in the source text This can happen when TL has a grammatical category which the SL lacks Likewise, the change in information content of the message may be in the form of omitting information specified in the source text If the TL lacks a grammatical category which exists in the SL, the information expressed by that category may have to be ignored e.g.2 On an infinite chessboard a game is played as follows At the start, n pieces are arranged in an n  n block of adjoining squares, one piece on each square (p 293) Trên bàn cờ có vơ hạn người ta quy ước trò chơi sau: Đầu tiên, n mảnh xếp thành khối n  n hình vng kề e.g.3 For each positive integer n, S(n) is defined as the greatest integer such that for every positive integer, k≤ S(n), n can be rewritten as the sum of k positive squares.(p290) Với số nguyên dương n, ta kí hiệu S(n) số nguyên dương lớn cho với k ngun dương, k≤ S(n), n viết thành tổng bình phương k số nguyên dương e.g.4 100 cards are numbered to 100 (each card different) and placed in boxes ( at least one card in each box) How many ways can this be done so that if two boxes are selected and a card is taken from each, then the knowledge of their sum alone is always sufficient to identify the third box? ( p325) Có 100 thẻ khác đánh số từ đến 100 toàn 100 thẻ đặt hộp phân biệt ( hộp phải chứa thẻ Hỏi có bao nhi cách sếp 100 thẻ vào hộp nói ta chọn ngẫu nhiên hộp, rút lại ngẫu nhiên thẻ hộp chọn, từ tổng hai thẻ ( tổng số đ ược đánh chúng, ta suy hộp thứ ba không chọn? As we see it, sometimes the change in sentence structure, that is, passive in SL to active in TL is not very necessary if the SL sentence can be transferred into TL without any loss of passive meaning that is intended For instance, in example (2), “a game is played as follows” can also be translated as “một trò chơi chơi sau” Similarly in (1), “one medal and of the remaining medals were awarded” can be translated as “một huy chương phần bảy số huy chương lại trao” However, in some cases, a change in sentence structure is necessary because if the SL sentence structure is preserved, it would sound unnatural in the TL For example, e.g.1 “Triangle ABC has sides a,b,c Tangents to the inscribed circle are constructed parallel to the sides” ( p.43) “Cho tam giác ABC có độ dài cạnh a,b,c Ta vẽ tiếp tuyến với đường trịn nơị tiếp tam giác song song với cạnh tam giác.” e.g.2 An arbitrary M is taken in the interior of the segment AB Squares AMCD and MBEF are constructed on the same side of AB.( p.22) “Cho điểm M tuỳ ý nằm đoạn thẳng AB Dựng hình vng AMCD MBEF phía AB” In the above examples, if the SL structure is maintained, the translated versions would sound not naturally Vietnamese For example, - “Cho tam giác ABC có độ dài cạnh a,b,c Các tiếp tuyến với đường trịn nơị tiếp tam giác dựng song song với cạnh tam giác.” - “Một điểm M tuỳ ý lấy đoạn thẳng AB Các hình vng AMCD MBEF dựng phía với AB” Obviously, the translator’s expertise of Vietnamese language has an important role to play in deciding when to adopt a passive or active structure for an –ed clause or passive structure in English II.2.4 Summary We have so far looked into the translation of some typical syntactic features and lexical features of mathematical problems in English into Vietnamese to identify the strategies used As far as the translation of mathematical terms is concerned, transposition procedure is the most frequently used It is applied in translating both single and compound terms Regarding the translation of relative clauses, a significant syntactic feature in mathematical problems, transposition procedure stands out Though the notion of relative clauses doesn’t exist in Vietnamese, the translation of such clauses has been successfully done by various means under transposition procedure such as by replacing word class i.e replace the absence of a relative clause pronoun in TL with other linking devices or linking words like mà, đó, với to create the ideas between the main clause and relative or subordinate clauses; by means of zero linking device i.e the absence of a relative pronoun is taken for granted in the TL and the relative clause is translated as post-modification to the noun it modifies without any referring to any inking device and any loss of meaning Besides, apposition and syntagmatic change are sometimes used here and there in the translation of relative clauses Similarly, -ed participle clauses and passive structure are translated into Vietnamese by means of transposition procedure in some cases, in either transposition and modulation procedure in other cases The transposition procedure is reflected in the appearance of passive words such as “được, bị” which come before the verb in the sentence PART C: CONCLUSION So far we have made an effort to study the translation of some mathematical problems, especially the translation of the most outstanding syntactic and lexical features of English mathematical problems: relative clauses, -ed participle clauses and passive structure as well as mathematical terms The main concepts in translation theory that are relevant to the study have been revised in the literature review to make ground for the research Specifically, translation methods and procedures are studied in detail,then the notion of terms in mathematics as well as of relative clauses, -ed clauses and passive structure in English and Vietnamese are also discussed rather deeply for later analysis of the data Now, we would like to summarize briefly the main findings and bring about some implications for the translation of mathematical problems Major findings I Translation of mathematical terms The translation of both single and compound terms is supported by transposition procedure which is the automatic change in word order from SL to TL Transposition is realized to be the most common strategy applied to almost all types of terms appearing in mathematical texts I Translation of relative clauses As the concept of relative clauses doesn’t exist in Vietnamese, translators have to resort to other means to translate this grammatical category The first realization of translation of transposition procedure is replacing word class, which means the use of liking words like “mà, đó, với.” to create the connection between ideas represented by the main clause and its subordinate or relative clause in a complex sentence There are also cases when a sentence containing a relative clause is translated without reference to the relative pronoun present i.e without any liking words (zero liking devices); the relative clause in English is now recognized as a verb phrase, adjective phrase or noun modifier in the TL sentence The second category is the use apposition which is seen as the repetition of a noun or noun phrase that modifies the preceding noun Apposition is the most common in translating non-defining relative clauses, which provides extra information to the noun it modifies The application of transposition procedure in the translation of relative clauses is also seen in syntagmatic change of the TL sentences Particularly, it is the division of a sentence containing a non-defining relative clause, one beginning with a relative pronoun to say something about the whole situation described in the main clause, into simple independent sentences or two independent clauses of a coordinating compound sentence, provided that there is a loosen structure between clauses in the sentences and the intention of the SL sentence stays the same after separation The change in the structure of the TL sentences is sometimes necessary as it may result in an absurb translation if the SL structure is preserved This strategy, in my opinion, depends much on the translator’s expertise and his/her master of the Vietnamese language I The translation of –ed participle clauses and passive structures It is observed that –ed participle clause is in fact a passive The translation of such grammatical categories has made use of transposition and modulation procedure The former is reflected in the use of passive words like “bị, được”, sometimes just by the meaning of the past participle or the verb itself when passive words are omitted The later involves a change of view point view point from passive to active voice, thus resulting in a change of structure from SL to TL This modulation procedure, to my view, also depends much on the translator’s expertise and proficiency of both SL and TL II Implication for translation of mathematical problems From the investigation above about translation procedures it is necessary that there be further consideration for the translation of mathematical problems Admittedly, it is prerequisite that the translator has good knowledge of the language he is dealing with so as to manage them successfully, avoiding awkward mistakes in translation which may arise from common pitfalls of the language such as multi-meanings, complex phrases or sentence structures etc because of the differences in the expressions between languages In-depth understanding of the TL certainly helps the translator with his choice of words or structures or ways of expressions to transfer the ideas from the SL to the TL smoothly and successfully In the context of mathematics, it is important that the translator have professional knowledge of this field before embarking on translating specialized texts Professional understanding of mathematics helps translators easily follow the flow of ideas, comprehend it better, and thus producing translation of good quality Besides, the translator is recommended to have knowledge of term categories such as single terms versus compound terms and the strategies applied in translating such categories into Vietnamese Last but not least, the translator of mathematical problems has to be aware of some other typical discourse features of English mathematical problems, such as the use of relative clauses and –ed participle clauses and passive structures, which help to convey the specialized message to all readers He/she has to be flexible because during the course of his/her work he/she often has to resort to more than one translation procedure for different grammatical or vocabulary categories present in a text At one time, it may be word-for-word translation, or transposition; at another time, it could turn out to be modulation The translator, thus, has to be careful in making his decision on an appropriate translation strategy so as not to produce unsound sentences which would hinder readers’ comprehension when they deal with the TL version In general, as translation involves a number of skills which include text analysis, vocabulary use, grammatical formulation etc of both SL and TL, the translator has to be a master of such skills, which is considered the most important condition In addition, in dealing with the specialized field of English mathematical problems, the translator himself should accumulate professional knowledge in order to manage these specialized materials successfully without deviating from the intention of the SL texts This study is only my first attempt to an investigation into the translation field, but I hope that the study would provide some ideas on the translation of English mathematical problems for me and other teachers and students, especially those interested in Mathematics Due to the limit of time and knowledge, the shortcomings are inevitable in this study, consequently, I wish to receive the comments and suggestions from other teachers and those concerned REFERENCES English Baker, M (1992) In other words- A course book on Translation Routledge Baker, M (1998) Routledge Encyclopedia of Translation Studies, Routledge Bassnette-Mc Guire, S (1980) Translation Studies London: Methuen Bell, R.T (1991) Translation and Translating Routledge Catford, J.C (1965) A linguistic Theory of Translation London: OUP Collin Cobuild (1990) English Grammar Rupa &Co Hatim, B & Mason, I (1990) Discourse and the Translator Longman Koller, W (1979) Equivalence in Translation Theory, in Chesterman, A (1989), Reading in Translation Theory Larson, M L (1984) Meaning-based Translation University Press America 10 Marlone, J L (1988).The Science of Linguistics in the Art of Translation, in Bell, R.T (1991), Translation and Translating, Longman 11 Newmark, P (1995) A text book of translation Prentice Hall International 12 Newmark, P (1988a) Approaches to Translation Oxford Pergamon 13 Nida, E A (1964) Toward a Science of Translation Brill Leiden 14 Nida, E.A (1969) The Theory and Practice of Translation Leiden 15 Nunan, D (1993) Introducing to Discourse Analysis Penguin English Vietnamese Lương, T Nguyễn 2005 Câu tiếng Việt Nhà xuất ĐHSP Hà Nội Thắng, T Lý 2004 Lý thuyết trật tự từ cú pháp Nxb ĐHQG HN Hà Nội Giáp, T Nguyễn 1996 Từ nhận diện từ tiếng Việt Nxb Giáo Dục Hà Nội Thêm, N Trần 1985 Hệ thống liên kết văn tiếng Việt Nxb Khoa học xã hội Hà Nội Uỷ Ban KHXH Việt Nam 1983 Ngữ pháp tiếng Việt Nxb KHXH Hà Nội APPENDIX COMMON SENTENCE PATTERNS AND EXPRESSIONS IN MATHEMATICAL PROBLEMS AND THEIR VIETNAMESE EQUIVALENTS English Vietnamese equivalents Let… be/denote………… Gọi/Cho………là……… Examples: VD - Let a, b, c be the sides of a triangle - Gọi a,b,c cạnh tam giác and A be its area.(1961) A diện tích - Let P denote the perimeter of the -Gọi P chu vi đa giác hexagon - Let a, b , c be real numbers - Cho số thực a,b,c - Let n be the number of distinct - Gọi n tất nghiệm nguyên phân integers roots to P(x)= or P(x) = -1 biệt phương trình P(x)= 1và P(x) = -1 - Let A be the sum of the decimal digits - Gọi A tổng chữ số hệ thập of 4444 (209) phân - Let A and E be opposite vertices of an - Gọi A E đỉnh đối octagon bát giác - Let F(n, r) be the arithmetic means of - Gọi F(n, r) trung bình cộng tất the smallest elements phần tử bé - Let S be a square with sides length 100 - Gọi S hình vng có cạnh dài 100 (đơn vị) - Let ABC be an equilateral triangle and - Cho tam giác ABC S tập hợp S the set of all points contained in the tất điểm nằm cạnh AB, three segments AB, BC and CA BC CA - Let a,b and c be the lengths of sides of - Cho a,b,c độ dài cạnh a triangle tam giác - Let d be the sum of the lengths of all - Gọi d tổng độ dài đường chéo the diagonals of a plane convex polygon đa giác lồi mặt phẳng có n with n>3 Let p be its perimeter đỉnh, n > Gọi p chu vi đa giác - Let n and k be relatively prime - Cho n số tự nhiên, k số positive integers with k2 points in the plane - Given 100 coplanar points, no collinear đường tròn K - Cho n điểm mặt phẳng với n>2 - Trong mặt phẳng cho 100 điểm phân biệt, khơng có điểm thẳng hàng Construct +Object Dựng…………………… Ex: Construct a triangle ABC Vd: Dựng tam giác ABC Construct point M Dựng điểm M Construct points B Dựng điểm B Prove that/Show that S+V Chứng minh rằng……… Ex: - Chứng minh đường thẳng EF tiếp - Prove that the line EF is tangent to(C2) xúc với đường tròn C2 - Show that three new lines form a - Chứng minh đường thẳng triangle with vertices on the incircle tạo thành tam giác có đỉnh nằm đường tròn nội tiếp - Find the locus of the midpoints of the Tìm/ xác định quỹ tích ……… segments PQ as M varies Vd: Tìm quỹ tích trung điểm đoạn - What is the locus of …… PQ M thay đổi - Determine the locus of all points B in space PA intersects BD in D PA cắt BD tạiD AF and BC intersect at N AF BC cắt N Take +O/ S+ be+ taken Lấy……………… Ex: Vd: - Take arbitrary points A,B,C in (P) Lấy điểm mặt phẳng P -Take any points K,L,M Lấy điểm K,L, M - An arbitrary point M is taken in the Lấy điểm M đoạn thẳng AB interior of the segment AB Determine……………………… Hãy xác định………………… -Determine the possible dimensions of Vd: Hãy xác định giá trị hình the box hộp -Determine all possible lengths for the - Hãy xác định độ dài other diagonal đường chéo khác 10 Lines M1S1,M1S2, M1S3 are concurrent Các đường thẳng M1S1,M1S2, M1S3 đồng quy - AP, BD, CD meet at a point 11 BC and BD are perpendicular 12 Given that/Suppose that/ It is given BC v BD vng góc với Giả sử rằng…………………………… that S+ V Ex: Vd: - It is given that (C1) passes - Giả sử C1 qua tâm C2 - Giả sử số điểm có through the center of (C2) - Suppose that exactly k of these points are to be colored black - Given that (n+1) >f(f(n)) k điểm tô màu đen - Giả sử với n ta có (n+1) >f(f(n)) 13 Be divisible by Chia hết cho Ex: M is divisible by 1979 n The sum C k 0 k 1 n 1 Vd: M chia hết cho 1979 3k is not divisible by 14 15 n Tổng C k 0 k 1 n 1 3k không chia hết cho Where S + be+ Cs ……………… Với……………… Ex: where n is a natural number Vd: Với n số tự nhiên Solve the following equations for x,y,z Giải hệ ph ương trình biến x,y,z với a,b where a, b are given số cho trước Suppose that S + be + Cs Giả sử…………… là…………… Ex: - Suppose that a,b, c are sides of a - Giả sử a,b,c cạnh tam giác triangle - Suppose that a,b,n are integers which - Giả sử a,b,n số nguyên lớn are greater than 16 Find ……which satisfy/ satisfying… Tìm ……… thỏa mãn………… Ex: - Find all pairs (a, b) of positive integers - Tìm tất cặp số nguyên b2 dương (a,b) Ø thoả mãn: a b  b a that satisfy: a  b ( p312) a - Find all polynomials P(x,y) in too - cho P(1,0)=1 variables such that P(1,0)=1 - - Find all real x satisfying 3 x - x 1 > Tìm tất đa thức biến Tìm tất số thực x thoả mãn 3 x - x 1 > 17 18 For + noun phrases Với………… Ex: Vd: -For any natural real number n - Với n số tự nhiên -For every natural number n/for every - Với số tự nhiên n/ Với số integer k nguyên k - For positive reals a, b,c, d - Với số thực dương a,b,c,d Find ………… such that…… Tìm ……… cho……… Ex: Vd: - Find all functions f, defined on the - Tìm tất cá hàm f xác định non-negative real number and taking tập số thực không âm nhận giá non-negative real values such that trị thực không âm cho f(2)=2 f(2)=0 - Tìm bốn số thực cho số - Find all sets of four real number such cộng với tích số cịn lai that the sum of any one and the product of the other three is - - Find all pairs a, such that q2+r = 1997 - Find m, n such that m + n has the least q2+r = 1997 - possible value - Find one pair of positive integers a, b Tìm tất cặp số cho Tìm m v n cho tổng m+n có giá trị nhỏ - such that ab(a+b) is not divisible by Tìm hai số nguyên dương a, b thỏa mãn ab(a+b) không chia hết cho - Let n be an integer greater than or equal to 19 - Cho n số nguyên lớn Determine when equality occurs When we have equality? Xác định dấu xảy When does the equality hold? 20 Let + ……+ be+ ……such that …… Cho …… là……….sao cho…… Vd: - Let k,m,n be natural numbers such that - Cho k, m, n số tự nhiên cho m+k+1 is a prime greater than n+1 m+k+1 số nguyên tố lớn n+1 - Let m and n be positive integers such - Cho m n số nguyên dương that cho m 1 1        n 1318 1319 m 1 1        n 1318 1319 - Let a, b, c, d be odd integers such that - Cho a, b,c, d số nguyên lẻ 0