Hong Kong Joint School Mathematics Society HKJSMS Newsletter
Issue 3
14-15
August 2015
From the editors: Dear readers, it’s hot summer and everyo n e o f you a re en jo yin g th e summer holidays! Feeling bored at home? Don’t forget to participate our big activity — The Day Camp on 24/8 and 25/8. We will have an enjoyable day in these two days! See you there.
Content: The Spiritual History P.2-4 Background Behind Calculus
Mandelbrot Set
Solve Mathematics Module 2 Problems by Using Euler’s Formula ● Summation of Cosines P.5-6 ●
Integration of Power of P.6-7 sinx or cosx
More About Geometric Construction Brain Teaser Corner
P.8-10 P.10-11
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The Spiritual History Background Behind Calculus By Tai Wai Ting
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Solve Mathematics Module 2 Problems by Using Euler’s Formula By Ken Leung
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More About Geometric Construction By Christopher Fok In HKMO, the mathematics competition held by the Hong Kong Education Bureau, there is a part testing competitors’ ability of geometric construction. In this part, the only tools provided are compasses and rulers without scales. The questions are quite straightforward for those who have read books about geometric construction. In this article, more advanced techniques and questions about geometric construction are introduced. Neusis construction The process involves finding a straight line L with a particular length d between 2 lines C 1, C2 (either straight of curved), with L passing through a particular point P. Neusis construction is done by the neusis ruler. The ruler consists of a point about which the ruler can rotate. A marked point M on the ruler is moved along C1 until another point N on the ruler d units from M coincides with C2. Neusis construction can be used to construct a regular heptagon, trisect any given angle. However, it popularity faded away as compasses and rulers as well as other means for geometric construction are preferred to it if the former can be used to solve a construction problem. Trisecting any given angle Given an angle CAB, trisect the angle using compass, ruler and neusis ruler.
Fig. 1 1, Draw a circle centered at A with radius AB which cut the extension of AB. 2, Using a neusis ruler, find a point E on AB and a point F on the circle such that EF=AB and E, F, C are collinear. Angle FAE is the angle which equals one-third of angle CAB. Can you prove this fact? In fact, it is very trivial. Construction of Heptadecagon using compass and ruler When Gauss was 19, he constructed a heptadecagon using compass and ruler, which shocked the mathematics society. 5 years later, he claimed that a regular polygon with number of edges being Fermat number is constructible. Fermat numbers are numbers in the form of Firstly, let’s discuss the mathematics background of such a construction.
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Prerequisite On the complex plane, the 17th roots of unity form a regular heptadecagon. Let the points be All these numbers satisfy the equation
When Let
where Let
Therefore, x1, x2 are the roots of
Let
U, u are the roots of V, v are the roots of
Let
W, w are the roots of
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Construction method Given a unit circle centered at O, 1, Draw x1, x2. 2, Draw U, V. 3, Draw W. 4, Draw a diameter through the unit circle which cut circle at A, B. Draw an arc centered at O with radius W such that it cut the diameter at M (M is closer to A than B). 5, Draw the perpendicular bisector of OW, which cut the circle at P, Q. AP is the length of the edge of the heptadecagon.
Brain Teaser Corner By Christopher Fok Level 1 Conan, the famous child detective, disappeared while investigating a case of international smuggling. While inspecting his last-known location, you find a note: 710 57735 34 5508 51 7718 Currently there are 3 suspects: Bill, John, and Todd. Can you break the detective's code and find the criminal's name? Hint: This is not a maths question.
Level 2 Two old friends, Chris and Bill, meet after a long time. Chris: Hey, how are you man? Bill: Not bad, got married and I have three kids now. Chris: That’s awesome. How old are they? Bill: The product of their ages is 72 and the sum of their ages is the same as your birth date. Chris: Cool… But I still don’t know. Bill: My eldest kid just started taking piano lessons. Chris: Oh now I get it. How old are Bill’s kids?
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Solutions 1. If you read the message upside down, the chain of numbers will resemble the sentence ‘Bill is boss. He sells oil.” So Bill is the criminal. 2. The product of their ages equals 72. Possible choices are: 2+2+18 → 22 2+4+9 → 15 2+6+6 → 14 2+3+12 → 17 3+4+6 → 13 3+3+8 → 14 1+8+9 → 18 1+3+24 → 28 1+4+18 → 23 1+2+36 → 39 1+6+12 → 19 The sum is a birth date, so it is larger than 0 and smaller than 32. Besides that, Chris cannot find their ages only with the provision of the sum, so more than 1 combination gives the sum. They are 2, 6, 6 and 3, 3, 8. Since one kid is the eldest, the only possible combination that fits all conditions is 3, 3, and 8.
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