[PDF] Top 20 Mathematical Excalibur, Volume 4, Number 3

... 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

... − 3 的性質，證實了方程 x 3 + y 3 = z 3 無解。但由於歐拉在他的證明中，在沒 有足夠論據的支持下，認為複數 a + b − 3 的立方根必定可以再次寫成 a + b − 的形式， 因此他的證明 未算圓 3 滿。 歐拉證明的缺憾，又過了近半個 世 紀 ， 才 由 德 國 數 學 家 高 斯 （ Carl Friedrich Gauss, ... See full document

... Time Allowed: 4 1 2 hours. Problem 1. In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white (as on a chessboard). For any pair of ... 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

... At a certain party, there are two or more 3-cliques, but no 5-clique. Every pair of 3-cliques has at least one person in common. Prove that there exist at least one, and not more than two persons at 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

... 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

... Problem 3. Consider a circle S. The point P lies outside S and a line is drawn through P, cutting S at distinct points X and Y. Circles S 1 and S 2 are drawn through P which are tangent to S at X and Y ... 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

... Problem 3. Let AB and AC be two distinct rays not lying on the same line, and let ω be a circle with center O that is tangent to ray AC at E and ray AB at F. Let R be a point on segment EF. The line through O ... 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

... Problem 4. Let ABC be a triangle and D the foot of the altitude from A. Let E and F be on a line passing through D such that AE is perpendicular to BE, AF is perpendicular to CF, and E and F are different from D. ... See full document

... x 3 + ax 2 + bx + c 的點所組成的曲線，其中 a , b, c 為有理數使 x 3 + ax 2 + bx + c 有 不同的根。在曲線上定一個有理點 O 。不難證明，當直線穿過兩個曲 線上的有理點 A, B 後，該直線必定 與曲線再相交於第三個有理點 C 。 由 C 和 O 再得一點 D ... See full document