[PDF] Top 20 Mathematical Excalibur, Volume 5, Number 1

... 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

... 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

... 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 ... 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 right or ... See full document

... First Day Problem 1. On the control board of a nuclear station, there are n electric switches (n > 0), all in one row. Each switch has two possible positions: up and down. The switches are connected to each other ... 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

... exist at least two who have shaken hands with each other. Find the greatest possible value of N. Solution. The answer is 4949. We first show that N = 4949 is possible: suppose there are 49 groups of 101 people each, and ... See full document

... digits 1, 2, 3, …, 7 appears in the decimal expansion of n ten times (and 8, 9 and 0 do not ...no number of this form can divide another number of this ... See full document

... Below are the problems of the 2011 IMO Team Selection Contest from Estonia. Problem 1. Two circles lie completely outside each other. Let A be the point of intersection of internal common tangents of the circles ... 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

... < 1. Prove that a/b+c/d < 1 −1/n 3 ...from 1 to 100 on the table. Andriy and Nick took the same number of cards in a way that the following condition holds: if Andriy has a card with a ... 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

... Below we will outline the cleverest solution due to Dan Carmon, the leader of Israel. It suffices to consider the case n=p t with p prime, t ≥1. By multiplying the denominator and translating, we may assume the ... See full document

... Mathematical Excalibur, Vol. 21, No. 5, ...the number of different starting arrangements for which the books will finally be ordered ... 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

... 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

... Acknowledgment: Thanks to Janet Wong, 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

... Problem 1. Two circles Γ 1 and Γ 2 intersect at M and ...Γ 1 and Γ 2 so that M is closer to " than N is. Let " touches Γ 1 at A and Γ 2 at ...Γ 1 again at C and the circle Γ 2 at ... See full document