[PDF] Top 20 Mathematical Excalibur, Volume 8, Number 4

... The 44 th International Mathematical Olympiad (IMO) was held in Tokyo, Japan during the period 7 - 19 July 2003. Because Hong Kong was declared cleared from SARS on June 23, our team was able to leave for Japan as ... See full document

... Lines 50 and 60 of the program set the ranges of X and Y Then line 70 computes the real and imaginary parts of the corresponding co... Converting these computations to codes yie[r] ... See full document

... page 4) 大約在 1637 年，當法國業餘數學 家費馬 (Pierre de Fermat, 1601-1665) 閱 讀古希臘名著《算術》時，在書邊的空 白地方，他寫下了以下的一段說話：「將 個立方數分成兩個立方數，一個四次冪 分成兩個四次冪，或者一般地將一個高 於二次冪的數分成兩個相同次冪，這是 不可能的。 我對這個命題有一個美妙 的證明，這裏空白太小，寫不下。」換 成現代的數學術語，費馬的意思就即 是：「當整數 ... See full document

... positive integers, [x] denote the greatest integer less than or equal to x and min{x,y} denote the minimum of x and y.. Prove or disprove that.[r] ... See full document

... However, this will repeat numbers, for example, 1/1 and 2/2 are the same. So to avoid repetitions, we will write down only numbers whose numerators and denominators are relatively prime. This will not omit any positive ... See full document

... C 4 6 > C 5 6 > C 6 6 ， C m 6 在 m = 3 時最大值。）利用這個 關係，讀者可以証明，如果在容斥原 則的右邊，略去一個正項及它以後各 項，則式的左邊大於右邊，這是因為 x 對於右邊的貢獻非正，或者被略去的 貢獻非負。同理，如果在容斥原則的 右邊略去一個負項及它以後各項，則 式的左邊變為小於右邊。這是一個有 ... See full document

... Solution. SIU Ho Chung (Queen’s College, Form 5). Let O, G and H be the circumcenter, centroid and orthocenter of triangle ABC respectively. Let T and R be the projections of G and L onto line S. From the Euler line ... See full document

... For any polynomial Q(x) with real coefficients, leading coefficient 1 and a non-zero constant term, we group consecutive terms of the same signs together to express [r] ... See full document

... On-line: http://www.math.ust.hk/mathematical_excalibur/ The editors welcome contributions from all teachers and students. With your submission, please include your name, address, school, email, telephone ... See full document

... On-line: http://www.math.ust.hk/mathematical_excalibur/ The editors welcome contributions from all teachers and students. With your submission, please include your name, address, school, email, telephone ... See full document

... On-line: http://www.math.ust.hk/mathematical_excalibur/ The editors welcome contributions from all teachers and students. With your submission, please include your name, address, school, email, telephone ... See full document

... On-line: http://www.math.ust.hk/mathematical_excalibur/ The editors welcome contributions from all teachers and students. With your submission, please include your name, address, school, email, telephone ... See full document

... On-line: http://www.math.ust.hk/mathematical_excalibur/ The editors welcome contributions from all teachers and students. With your submission, please include your name, address, school, email, telephone ... See full document

... On-line: http://www.math.ust.hk/mathematical_excalibur/ The editors welcome contributions from all teachers and students. With your submission, please include your name, address, school, email, telephone ... See full document

... On-line: http://www.math.ust.hk/mathematical_excalibur/ The editors welcome contributions from all teachers and students. With your submission, please include your name, address, school, email, telephone ... See full document

... Apart from the first person in the line, every person indicates one of those before him and declares either “this person is a villain” or “this person is a knight”. It is known that the number of villains is ... See full document

... of mathematical induction: when we use induction, our task is essentially to prove the original statement about an arbitrary positive integer but equipped with an additional tool – the assumption that the ... See full document

... Problem 4. Let ΔABC be a triangle with |AC|=2|AB| and let O be its circumcenter. Let D be the intersection of the angle bisector of ∠A and BC. Let E be the orthogonal projection of O on AD and let F≠D be a point ... See full document

... Number of Participating Teams: 75 Informal Rank for the Hong Kong Team: 25 Medals for the Hong Kong Team: 1 silver and 4 bronze medals. Below: A photo of the Hong Kong Team taken at the Kai Tak Airport ... See full document

... On-line: http://www.math.ust.hk/mathematical_excalibur/ The editors welcome contributions from all teachers and students. With your submission, please include your name, address, school, email, telephone ... See full document