[PDF] Top 20 Mathematical Excalibur, Volume 5, Number 4

... (continued on page 4) In comparing two similar expressions, often they involve a common function. To see which expression is greater, the shape of the graph of the function on an interval is every important. A ... 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

... a 0 = 1 and a n = kn + (− 1 ) n a n-1 for each n ≥ 1. (continued on page 4) 我們知道，圓錐曲線是㆒些所謂㆓ 次形的曲線，即㆒條圓錐曲線會滿足 以㆘的㆒般㆓次方程：Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0，其㆗ A、B 及 C 不 會同時等於 0。 假設 A ≠ 0，那麼我們 可以將㆖式除以 A，並化簡成以㆘模 式： ... 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

... hours. Letf be an odd prime. The display initially shows 0. Given any positive rational number q, show that pressing some finite sequence of buttons will yield q. Assume that [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

... = CA and CY = BA. The line XY meets the perpendicular bisector of side BC at P. Show that ∠ BPC + ∠ BAC = 180 o . Problem 4. An exam consisting of six questions is sat by 2006 children. Each question is marked ... 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

... 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

... Below are the problems of the 2011 Canadian Math Olympiad, which was held on March 23, 2011. Problem 1. Consider 70-digit numbers n, with the property that each of the digits 1, 2, 3, …, 7 appears in the decimal ... 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

... same number of cards in a way that the following condition holds: if Andriy has a card with a number n then Nick has a card with a number ...maximal number of cards could be taken by the two ... See full document

... In 1980, Kiev, Moscow and Riga participated in a mathematical problem solving contest for high school students, later called the Tournament of the Towns. At present thousands of high school students from dozens of ... See full document

... With an even parity code, the receiver can detect one transmission error, but unable to correct it... Mathematical Excalibur, Vol.[r] ... See full document

... The rest of the solution follows by induction on the number of sides of the polygon and the two claims. Problem 4. A set of positive integers is called fragrant if it contains at least two elements and each ... See full document

... Example 4 (2000 Russian Math Olympiad). Two pirates divide their loot, consisting of two sacks of coins and one diamond. They decide to use the following rules. On each turn, one pirate chooses a sack and takes 2m ... See full document

... Acknowledgment: Thanks to Elina Chiu, MATH Dept, HKUST for general assistance. On-line: http://www.math.ust.hk/mathematical_excalibur/ The editors welcome contributions from all teachers and students. With ... See full document

... 國科學院學部委員。 1984 年證實患㆖ 了「帕金遜症」，直至 1996 年 3 月 19 日，終於不治去世。 其實除了對「哥德巴赫猜想」的 證 明 有 貢 獻 外 ， 陳 景 潤 的 另 ㆒ 個 成 就，就是對「孿生質數猜想」證明的 貢獻。在質數世界㆗，我們不難發現 有時有兩個質數，它們的距離非常接 近，它們的差祇有 2，例如：3 和 5、5 和 7 、 11 和 13 … 10016957 和 ... See full document