[PDF] Top 20 Mathematical Excalibur, Volume 14, Number 4

... 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

... Olympiad Corner The following were the problems of the 2009 Asia-Pacific Math Olympiad. Problem 1. Consider the following operation on positive real numbers written on a blackboard: Choose a number r written on ... See full document

... Solution. LAM Cho Ho (CUHK Math Year 1). Take a circle of radius r so that all intersection points of the n lines are inside the circle and none of the n lines is tangent to the circle. Now each line intersects the ... 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

... Yet another generalization of the van der Waerden Theorem (which says that ( , ) W r k exists for all r, k) is the Hales-Jewett Theorem. The exact statement of the theorem is rather technical, but we can look at an ... 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

... 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

... 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

... of number theory problems, namely problems on integers that have to do with the set of perfect squares 1, 4, 9, 16, 25, 36, ...many Mathematical Olympiads from different countries for over 50 ... See full document

... problem 6 is more complicated, but there is a nice and not too complicated complex number solution. In short, leaders generally agreed that those problems are do-able. If one understands what is going on, one ... See full document