## 2010年10月17日 星期日

### Polynomial roots

An elementary problem seen from other site.

Let be a polynomial with odd integral coefficients,

show that it cannot have rational root.

## 2010年8月22日 星期日

### Two Interesting Questions

It is a long while since I wrote here last time. I have been to Europe and visited the Deutsche Museum in Munich. There is a small corner about Mathematics there and I took some photos. Maybe I post it next time.

Recently I read two interesting questions. Let me share them here.

1)

Find a smallest finite set of integers, such that every integer in the set is the sum of two other distinct elements in the set. There is one another rule: if x is an element of the set, then -x must NOT be an element of the set.

2)

A square matrix is said to be doubly stochastic if the entries in each row and in each column sum to the same value.

A permutation matrix is a square matrix where each entry in the matrix is either 0 or 1, and there is one and only one entry in each row and in each column which is 1.

Prove that any doubly stochastic matrix is a convex linear combination of permutation matrices.

Comment: The first question is interesting by itself. The second question is considered interesting because an easy solution by induction is possible, after knowing a combinatorial theorem called Hall's Theorem. The question itself may mislead you to some linear algebra arguments.

## 2010年8月15日 星期日

### 多買了，反而便宜了？

換季了。時裝店裏滿是新一季的秋裝，而未賣完的夏裝則大減價促銷。七折，五折，甚至三折，都不是新鮮事。為求盡快賣掉存貨，不少商店都提供「遞進式折扣」，那就是買得愈多，折扣愈大。我見過以下的折扣表：

「一件五折，兩件四折，三件或以上三折」

看過這樣的優惠後，我的第一個反應是：很少人會只買兩件吧？

在大多數情況下，很多人應該寧買三件也不買兩件。那是因為多買了，可能反而便宜了。試想想：假設我買了兩件衣服，原價共值　\$300。四折後則為　\$300 × 40% = \$120。如果我多買一件原價　\$50 的衣服，三件三折，折扣後為　\$350 × 30% = \$105。即使這件多買的衣服完全不管用，多買它也立即省了　\$15。事實上，簡單的代數運算告訴我們，只要第三件衣服不超過　\$100（也就是原來的總價格的三分之一），我們都可以免費拿走它，更可能省下了金錢。

## 2010年8月1日 星期日

### 新網頁測試

http://silver.mathdb.org:8080/silverstripe-test/

## 2010年7月23日 星期五

### Abstract Algebra and Famous Impossibilities

Abstract Algebra and Famous Impossibilities

By Patrick Wong, HKUST

Date: July 27, 2010. (Tuesday)
Time: 6:00pm-7:30pm
Venue: Room 4504 (near Lifts 25 & 26) , HKUST
Target Audience: Students with basic algebra background

Abstract: The famous problems of squaring the circle, doubling the cube, and trisecting the angle have not yielded to purely geometrical methods. It was, however, the development of abstract algebra in the nineteenth century which enabled mathematicians to conclude that these constructions are not possible. We are going to look at the way how algebra solve the geometric problems.

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## 2010年7月9日 星期五

### 馬會可能通殺？買中了也輸錢？

世界盃是國際矚目的體育盛事，不少球迷一邊在電視前搖旗吶喊外，一邊下注，考考自己的眼光。投注項目五花八門，除了每場比賽的賽果外，亦有以世界盃為整體的專項，例如競猜小組首名、冠軍等。以上各種項目裏，只要你猜中了正確的結果（例如哪隊取得冠軍），便肯定獲得彩金，而彩金亦肯定比投注本金多。（如果猜對了也要輸，誰會下注？）可是這種想法未必一定正確。例如競猜誰是「神射手」（世界盃所有比賽裏入球最多的球員）便是一例。在某些情況下，猜錯了要輸掉本金，但猜對了也贏不了！

為甚麼會這樣荒謬呢？這是因為「神射手」與世界盃冠軍不同：冠軍只有一個，但「神射手」卻可以超過一個（見註）。簡單來說，假如有兩名球員都射入五球，而沒有人射得六球或以上，這兩名球員都是「神射手」。如果「神射手」多於一個，香港賽馬會如何計算彩金？原來在這種情況下，投注本金會先除以神射手的數量才計算彩金。這就是買中了也要賠錢的關鍵。看看以下的現實例子：

準決賽結束後，荷蘭的史奈達 (Wesley Sneijder) 和西班牙的韋拿 （David Villa) 暫時以五球領先。撰寫本文時，韋拿的「神射手」賠率為 1.85 倍。假設現在某人以 1000 元本金下注韋拿為「神射手」。如果決賽和季軍戰都沒有球員入球，史奈達和韋拿將同為「神射手」。雖然這位投注者買中了，但他可得的「彩金」卻只有 1000 ÷ 2 × 1.85 = 925 元，輸了 75 元。如果同射得五球的球員有三個或四個，他將更倒楣，輸得更多。

## 2010年7月7日 星期三

### CMPC Student Seminar Series

A group of people from HKUST are holding a series of seminar in different area in undergraduate mathematics.

The topics are mainly from undergraduate,
students who are interested in are welcome.

The first two seminars will start on the next week.

The location are taken place at the Hong Kong University of Science and Technology.
One can go to the HKUST by minibus at Choi Hung MTR Station (Exit C)

Symmetry, Group Theory and Art
By Hoi Luk, (HKU Space)

Date: July 14, 2010. (Wednesday)
Time: 6:00pm-7:30pm
Venue: Room 4504 (near Lifts 25 & 26) , HKUST
Target Audience: Undergraduate who is interested in exploring more about Abstract Algebra

Abstract: This is an introductory talk on Abstract Algebra. Group is a central concept in Abstract Algebra and it plays important roles in Geometry, Topology, ODE and etc. The talk will include the mathematical concept of symmetry, the use of Group in studying symmetry and Wallpaper Groups in Art. It is designed for undergraduate students who are interested in exploring more about Abstract Algebra and it may serve as the motif for those going to take MATH 311 (Algebra I).

Financial Mathematics - The Art of Compromise

By Jeff Tam, (Tokyo Metropolitan University)

Date: July 15, 2010. (Thursday)
Time: 6:30pm-8:00pm
Venue: Room 4504 (near Lifts 25 & 26) , HKUST
Target Audience: Undergraduate with basic quantitative background who is interested in financial mathematics

Abstract: Financial Mathematics has taken the world by storm since its inception during the 80s. In light of the recent financial crisis, the purpose of this talk is to unveil financial mathematics, divulging vital information such as its theoretical foundation, industrial practice, research frontier, etc. Most importantly, where does the "model world" and "reality" meet? What kind of compromise can bring these two together while (hopefully) not sacrificing too much reliability.

For more details on the coming seminars,

## 2010年7月6日 星期二

### Solving IMO Q6 collaboaratively

There's been this proposal on this blog that seeks to solve Q6 of IMO 2010 collaboratively. The proposed starting time is July 8 16:00 UTC (i.e. July 9 00:00 Hong Kong time). Further details can be found in the following blog post.

## 2010年6月22日 星期二

=====================================================

《密碼攻防戰》

War of Encryption

We have to exchange data with many people and websites every day, in which they contain a lot of secrets and confidential information. How can we protect our transferred data with mathematics and only let authorized people read them? At the same time, how can they be cracked? We will discuss the use of mathematics in encryption from the ancient times to the modern world.

******

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## 2010年6月6日 星期日

### 寫在高考放榜前（序）

******

******

A-Level的物理和大學物理有甚麼分別？

Pure Maths處理3x3的matrix，大學再難也不外乎就是需處理4x4甚至更大的matrix吧？

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## 2010年5月23日 星期日

I post here not to blame any particular person, but wanna to let ordinary readers, which may not be in academia, sees another view of the academia, which may have been phenotyped by media.

I decided to pursue PhD study in America, one main reason due to my interest in the subject of Math and CS, and another reason is although there can be such sad stories in academia, as far as I know, in most cases, the truth is still ultimately respected in academia.

## 2010年5月18日 星期二

### MAC2010 決賽作品感想

(1) 「C Drive 大法」 (有的作品甚至乎連用戶名稱也放在連結上..)
(2) 全形’ 和 半形'
(3) Flash 亂碼問題 (UTF-8 , BIG-5 與 Locale 的設定)
(4) 放錯 link (如 main.html 及 X theory main.html 的分別"
(5) htm 與 html 檔名不分

(如把上述因素計算在內....

(html 絕不能確保排版, 完全取決於評判的電腦)

## 2010年5月9日 星期日

### 直角三角形的中線（四）

BD2 = (2x cos y)2 + x2 - 2 (2x cos y) (x) cos y = x2

AB2 = DA2 + DB2 - 2 (DA) (DB) cos y
BC2 = DC2 + DB2 - 2 (DC) (DB) cos (180o - y)

4 DA2 = 2 DA2 + 2 DB2

\100dpi \begin{align*} \overrightarrow{BA}\cdot\overrightarrow{BC}&=0\\ (\overrightarrow{BD}+\overrightarrow{DA})\cdot(\overrightarrow{BD}+\overrightarrow{DC})&=0\\ (\overrightarrow{BD}+\overrightarrow{DA})\cdot(\overrightarrow{BD}-\overrightarrow{DA})&=0\\ \overrightarrow{BD}\cdot\overrightarrow{BD}-\overrightarrow{DA}\cdot\overrightarrow{DA}&=0\\ |\overrightarrow{BD}|^2-|\overrightarrow{DA}|^2&=0\\ |\overrightarrow{BD}|&=|\overrightarrow{DA}| \end{align*}

## 2010年4月30日 星期五

### S.S.A.

(試由上圖假設 ABC_1 , ABC_2 兩組三角形全等)

A=(0,0)
X=(10,0)
a=Line[A,X]
X'=Angle[X,A,50°]
Segment[A,X']
d=Circle[X',8]
Intersect[d,a]

## 2010年4月23日 星期五

### [轉貼]由三分之二變成0.66

At Issue In a Massachusetts Town, the Value of Two-Thirds

BTW，這件事會教一些人想起那道培正數學邀請賽第一屆的經典題目吧。

## 2010年4月11日 星期日

### Algebra and Algorithm

A quote from the book "A History of Abstract Algebra" by Israel Kleiner:

Islamic mathematicians attained important algebraic accomplishments between the ninth and fifteenth centuries AD. Perhaps the foremost among them was Muhammad ibn-Musa al-Khwarizmi, dubbed by some "the Euclid of agebra" because he systematized the subject and made it into an independent field of study. He did this in his book al-jabr w al-muqabalah.

A small game: the book says that the words "Algebra" and "Algorithm" are derived from two words in the quoted paragraph. Can you find them?

## 2010年4月4日 星期日

### 一道概率問題（下）

6/3125 顯然是因為把以上的「120」當成了「1」，而這顯然是不正確的，因為在數算出 3125 個可能結果的過程中，那「1」個可能結果（即 5 個球成一水平線）是被數算了 120 次的。而 2/3125 則顯然是把「5」和「120」都當成了「1」，也自然是不正確的。

1/21 呢？相信這是從 6/126 化簡而來的。「6」個勝出的結果自然是「5 直 1 橫」。如果「成一水平線」的結果只算一次的話，那麼可能結果的總數是多少？（也就是說我們只關心每條坑道中球的數目，例如 (1,1,1,1,1) 只算一次，這個在之前的解法中是算了 120 次的；而 (5,0,0,0,0) 和 (0,5,0,0,0) 則算作兩個不同的結果。這裡 (0,5,0,0,0) 表示 5 個球都被射進第二條坑道，如此類推。）這個總數就是方程 a+b+c+d+e=5 的非負整數解的數目，即 H(5,5) = C(9,5) = 126。

## 2010年3月31日 星期三

### 一道概率問題（上）

符合Ｅ的結果的數目
事件Ｅ的概率　＝　－－－－－－－－－
可能結果的總數

## 2010年2月28日 星期日

=====================================================

Finite Geometry and Combinatorics

Finite Geometry is a kind of Geometry which is often used in computer programming and design (a branch of statistics).

In Euclidean Geometry, we seldom use "counting" to solve a problem. However, in finite geometry, combinatorics plays an important role. Also, finite geometry helps to solve problems in combinatorics, such as the Kirkman's schoolgirl problem.

In this talk, we will introduce two important objects in Finite Geometry: finite projective planes and finite generalized quadrangles and use counting to see some interesting structures of them.

## 2010年2月27日 星期六

### 圓周率日活動

3 月 14 日（星期日）是圓周率日，也是數學資料庫的生日！如此大日子，我們安排了一連串的活動，大家萬勿錯過！

· 當天晚上是數學資料庫的週年晚宴暨生日會，安排如下：

其他　 -- 118 元
預先登記每位減收 10 元*

* 需於 3 月 12 日（星期五）下午 8 時前連同姓名、人數（中學生和非中學生）和聯絡電話電郵至 mathdb.fomd@gmail.com，我們將以電話回覆作實。

## 2010年2月22日 星期一

### 骰戰

1) 若A1比B1大，甲得一分；若B1比A1大，乙得一分。若A2比B2大，甲得一分；若B2比A2大，乙得一分。誰分數較多為勝。問誰贏的機會較大？

2) 規矩同1)，但再加一條：若A3是1、2或3，乙得一分；若A3是4、5或6，甲得一分。現在誰贏的機會較大？

3) 規矩同1)，但若A1=B1，乙得一分；若A2=B2，乙得一分。（在1)時，若A1=B1或A2=B2，甲乙都沒有分數。）現在誰贏的機會較大？

## 2010年1月25日 星期一

### Nice Example on Schutte Problem

Some background first. A tournament of N players mean a competition that every pair of players play against each other exactly once. In our case, no draw is allowed; either one wins or the other wins.

A tournament is with Schutte property of order k if every set of k players are all defeated by one of the other players.

Using probabilistic method, it is easy to show that for any k, there exists sufficiently large N such that a tournament with Schutte property of order k is possible.

My focus here is a cute example of tournament with Schutte property of order 2: when N=7, name the players by 0,1,2,...,6, then a tournament with Schutte property of order 2 is given by:

i defeats j if and only if (i-j) is a quadratic residue of 7.

## 2010年1月17日 星期日

### 神奇教練

（聲明：以上計算的假設毫不嚴謹，亦無任何統計數據backup支持，純為筆者吃飽飯沒事幹（現為紐約時間晚上九點左右）發表的文章。）

## 2010年1月13日 星期三

### 培正數學邀請賽：最新消息

1. 准考證已寄給各參賽學校及個人報名的參賽者。
2. 參賽者必須帶備准考證及身分證應考。
3. 參賽者應使用大會提供的 HB 鉛筆作答（答題紙樣本），惟需自備橡皮擦及其他文具。
4. 如答案小於 1000，須「補 0」以湊足四位，例如：如答案為 39，應填「0039」。
5. 本年將不會派發答案予領隊老師。試題及答案將於初賽後盡快上載至數學資料庫（屆時會在本網誌公佈），亦會於一星期內上載至比賽網頁