high-school-questions
2014-2015 Math League Contests, Grades 9 – 11
Second-Round, Feb 2015
Instructions:
1.This second-round contest consists of two parts. Part 1 is math questions. Part 2 is English writing.
2.This document contains 22 pages in total, including this page.
3.Please write all your answers on a separate sheet, high-school-answer-sheet.doc, downloadable
together with this document at https://www.wendangku.net/doc/226414343.html,, not on this document.
4.In Part 1, you are asked to read a math subject, The Geometry of Fractal Shapes, first. Then you have
15 questions to work on. You will need to give precise, unambiguous answers to Questions 1-12. You
do not have to write your detailed solutions unless otherwise specified in the question, answers will suffice. Question 13-15 are Projects and Papers, which means you need to do your research and write
a paper for each question. There is no word limit on each of your papers, but it doesn’t necessary
mean the more words the better. The best paper is precise and succinct. Please don’t feel frustrated at all if you can’t write a paper, as the topics, fractal dimension, fractals and music, and fractals and
antenna design, are very hard for a high school student, even for an adult. Please don’t feel frustrated even if you can’t finish all Questions 1-12, as they are not trivial questions and it requires a lot
reading and thinking. Students who can work out a few questions should be commended.
5.The more questions you answered correctly, the more credit you will get.
6.You can seek help by reading books, searching the Internet, asking an expert, and etc. But you can’t
delegate this to someone else and turn in whatever he/she wrote for you. To make it clear, the purpose of the second-round contest is to test your ability to read and research. You need to be the one who
understand the topics and solve the problems. You will be caught if it is not the case during the
interview.
7.For Part 1, you can write in either English or Chinese.
8.In Part 2, you are asked to write an English essay. You have to write in English in Part 2.
9.If you have any questions regarding the contest, please contact us at once at
INFO@https://www.wendangku.net/doc/226414343.html,
10.Submission of your answers:
a)Your answers should be on a separate document, high-school-answer-sheet.doc, downloadable
together with this document at https://www.wendangku.net/doc/226414343.html,, not on this document.
b)You can either submit high-school-answer-sheet.doc, or scan the pages of
high-school-answer-sheet.doc, and submit the scanned pages in one zip file.
c)Please submit your answers online at https://www.wendangku.net/doc/226414343.html,. You need to submit your answers no
later than 12:00AM, Feb 23, 2015, Beijing Time. Later submission will not be accepted.
11.答案提交：
a)请将你的答案写在文档high-school-answer-sheet.doc里(high-school-answer-sheet.doc可以和
此文档共同下载)，请不要将答案写在此文档里。请一定要将答案写在
high-school-answer-sheet.doc里, 不能写在其他文档里。写在其他文档里的答案组委会有权
拒绝接受。
b)请在high-school-answer-sheet.doc的每一页上正楷写上你所在的城市、区、学校、姓名、你
现在的年级。请务必在high-school-answer-sheet.doc的每一页上都写上，以免判卷时弄混了。
c)两种方式，一种是你在电脑上将你的答案写在文档high-school-answer-sheet.doc里，然后上
传文档high-school-answer-sheet.doc。
d)或者你将high-school-answer-sheet.doc打印出来，将答案写在high-school-answer-sheet.doc
上，上传答案时扫描或拍照(必须清晰)，将扫描或拍照的图形文件zip成一个文件上传。
e)在https://www.wendangku.net/doc/226414343.html,上传你的答案。请注意在2015年2月23日凌晨12点之前上传你的
答案，2015年2月23日凌晨12点之后不接受上传。
Part 1 – The Geometry of Fractal Shapes
The following is an excerpt from some math book. The purpose of the excerpt is to introduce the basic ideas behind fractals and their geometry – a geometry very different from the traditional Euclidean geometry of triangles, squares, and circles.
The word fractal (from the Latin fractus, meaning “broken up or fragmented”) was coined by Benoit Mandelbrot in the mid-1970s to describe objects as diverse as the Koch snowflake, the Sierpinski gasket, as well as many shapes in nature, such as clouds, mountains, trees, rivers, a head of cauliflower, and the vascular system in the human body.
These objects share one key characteristic: They all have some form of self-similarity. (Self-similarity is not the only defining characteristic of a fractal. Others include fractal dimension, and so on.) There is a striking visual difference between the fractal geometry of self-similar shapes and the traditional geometry of lines, circles, spheres, and so on. This visual difference is most apparent when we compare the look and texture of natural objects with those of man-made objects.
Geometry as we have known it in the past was developed by the Greeks about 2000 years ago and passed on to us essentially unchanged. It was (and still is) a great triumph of the human mind, and it has allowed us to develop much of our technology, engineering, architecture, and so on. As a tool and a language for modeling and representing nature, however, Greek geometry has by and large been a failure. The discovery of fractal geometry seems to have given science the right mathematical language to overcome this failure and thus promises to be one of the great achievements of twentieth-century mathematics. Today fractal geometry is used to study the patterns of clouds and how they affect the weather, to diagnose the pattern of contractions of a human heart, to design more efficient antennas, and to create some of the otherworldly computer graphics that animate many of the latest science fiction movies.
Question 1:
Question 2: (In this question, “in” stands for “inch”)
Question 3: (Questions 3 & 4 refer to a variation of the Koch snowflake called the Koch anti-snowflake.)
Question 4: (In this question, “in” stands for “inch”)
Question 5:
Question 6:
Question 7:
Question 8:
Question 9:
Question 10:(Questions 10 & 11 refer to the Sierpinski ternary gasket, a variation of the Sierpinski gasket defined by the following recursive replacement rule.)
Question 11:
Question 12: (Question 12 refers to the Menger sponge, a three-dimension cousin of the Sierpinski gasket.)