[PDF] Top 20 Mathematical Excalibur, Volume 14, Number 1

... 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 ... 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 ... 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

... 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

... Right: A photo of the six members of the Hong Kong Team and one of the editors (far right) taken at the Shatin Town Hall after the closing ceremony of the 35th International [r] ... See full document

... If the starting point lies within a distance of 1 from the origin, the subsequent points will get closer and closer to the origin.. If the intial point is more than a distance o[r] ... See full document

... FOK (Homantin Government Secondary School), Michael LAM Wing Young (St. Paul's College), LIN Kwong Shing (University of Illinois). and LIU Wai Kwong (Pui Tak Canossian[r] ... 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

... 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

... 例 11 2n名選手參加象棋循環賽，每 一輪中每個選手與其他 2 n − 1 人各賽 一場，勝得 1 分，平各得 1 2 分，負得 0 分．證明：如果每個選手第一輪總 分與第二輪總分至少相差n分，那麼 每個選手兩輪總分恰好相差n分． ... See full document

... (continued from page 2) For the proof of the second step, we follow the approach in J. Michael Steele’s book The Cauchy-Schwarz Master Class, MAA-Cambridge, 2004. For a n×n matrix M, we will denote its entry in the j-th ... See full document

... C 1 , C 2 , ⋯, C n in a plane such that for each 1 ≤ i < n, the center C i is on the circumference of C i+1 , and the center of C n is on the circumference of C 1 ...the number of pairs ... 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

... Problem 2. As it turned out, this problem caused quite a bit of trouble and many students didn’t know how to tackle the problem at all. More sophisticated inequalities such as Muirhead do not work, since the expression ... 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 Selection By the end of March 2014, the host country (South Africa) received 141 problem proposals from 43 countries. I don’t know when the problem selection group started to work, but surely, it took them more ... See full document

... polynomials in problem 1 and numbers in problem 2. Like vectors expressed in coordinates, the v i 's are objects that may take on different values at different positions. So functions corresponding to solutions of ... See full document