第三屆丘成桐中學數學獎獲獎論文集（簡體書）

《第三屆丘成桐中學數學獎獲獎論文集》收錄了獲得第三屆丘成桐中學數學獎和丘成桐中學應用數學科學獎金獎、銀獎、鋼獎和優勝獎的論文。《第三屆丘成桐中學數學獎獲獎論文集》是丘成桐中學數學獎的推薦參考書，也可供廣大數學愛好者閱讀。．

《丘成桐中學數學獎推薦參考書:第3屆丘成桐中學數學獎獲獎論文集》介紹了丘成桐中學數學獎由國際著名華人數學家丘成桐先生與泰康人壽保險股份有限公司聯合設立。本獎項旨在激發和提升全球華人中學生對于數學研究的興趣和創新能力，培養和發現年輕的數學天才，增進海內外華人中學生的了解和友誼。為了提升中學生對應用數學科學的興趣及研究能力，從2010年開始增加了應用數學科學獎。第一、二屆丘成桐中學數學獎總決賽和頒獎典禮已分別于2008年10月、2009年12月在北京隆重舉行。第三屆丘成桐中學數學獎暨第一屆丘成桐應用數學科學獎頒獎儀式于2010年12月在北京舉行。

丘成桐中學數學獎競賽手冊（S.—T.Yau High School Mathematics Competition Manual）
丘成桐中學數學獎開幕詞
歡迎辭
丘成桐中學數學獎組織構架
丘成桐中學應用數學科學獎組織構架
丘成桐中學數學獎
歷屆丘成桐中學數學獎成果
丘成桐教授
泰康人壽保險股份有限公司
美國坦普頓基金會
2010丘成桐中學數學獎各賽區入圍決賽隊伍
2010丘成桐中學應用數學科學獎各賽區入圍決賽隊伍
2010丘成桐中學數學獎獲獎名單
2010丘成桐中學應用數學科學獎獲獎名單
第三屆丘成桐中學數學獎工作日程安排
致謝
Welcome Letters
Organization of S.—T.Yau High School Mathematics Awards
Organization of S.—T.Yau High School Applied Mathematical Sciences Awards
S.—T.Yau High School Mathematics Awards
Achievement of the YHMA
Professor Shing—Tung Yau
Taikang Life Insurance Company
John Templeton Foundation
2010 S.—T.Yau High School Mathematics Awards Finalists
2010 S.—T.Yau High School Applied Mathematical Sciences Awards Finalists
Schedule for the Third S.—T.Yau High School Mathematics Awards
Acknowledgements
獲獎論文（Award—winning Research Reports）
A Proof on the Non—differentiability of Weierstrass Function in an Uncountable Dense Set Weierstrass函數在不可列的稠密集上不可導的一種證明
A Research on a Kind of Special Points Inside Convex對凸形內部一類特殊點的研究
From the Hinge Device to Curves of Linkages從畫正多邊形的鉸鏈到連桿軌跡
Extension，Enhancement and Analogue of the Vasiliev Inequality瓦西列夫不等式的推廣、加強與類似
A New Definition of Distance for Graphs圖上距離的一種新定義
On the Minimum Number of Convex Quadrilaterals in Point Sets of Given Numbers of Points關于凸四邊形最小個數問題
The Distribution of the First Digit in Polynomials with Positive Integer Coefficients正整系數多項式的首位數字分布
Study on Inscribed Ellipse of Triangle三角形內切橢圓及其性質的研究
On Supersingular Elliptic Curves and Hypergeometric Functions關于超奇橢圓曲線和超幾何函數
Entries of Random Matrices隨機矩陣的元
The Study and Application of Velocity Transmission in a Queue變速隊伍中速度傳遞問題的研究及其實際應用
Understanding Flocking Dynamics in Natrue理解自然界中的群聚動力學
Actuarial Modeling of a Children's Protection Insurance兒童防護保險的精算模型
The Mathematical Model of the Oil Spilling in the Gulf of Mexico Based on the Extended Fay Formulas墨西哥灣原油泄漏事件在推廣Fay公式基礎上的建模

During the study,we have found quite a few interesting characteristics regarding inscribed ellipse in triangle,but we have also left some questions that we cannot answer.
The first one is about the relationship between the center and the foci of an inscribed ellipse.In Chapter Ⅱ we proved that for any given point O within the medial triangle of △ABC,there exists an ellipse centered at O inscribed in △ABC.Yet up till now we cannot determine the two foci directly by O,nor do we know much about the relationship of P,Q and O.
The second one is about the fact that the largest inscribed ellipse of △ABC is tangent to each side of △ABC at the midpoint.Reference provides an ingenious proof based on projective transformation,but we haven't found a direct proof based on merely Euclidean geometry.
The last one involves complex function.We know that the complex numbers corresponding to the two foci of the largest inscribed ellipse of △ABC are the two roots of the equation(w-z1)(w-z2)+(w-z2)(w-z3)+(w-z3)(w-z1)=0,or the two zeroes of the derivative of the complex function f(w)=(w-z1)(w-z2)(w-z3).So quite naturally we will ask whether there is any connection between these facts,or whether we can extend it to more points.Unfortunately,limited by our knowledge,we cannot delve further into this question now.
There is another thing that we need to mention.Though the whole study is original,a small part of the results has already been published and we were not aware of it.One of the judges,Professor Pan Jianzhong from Chinese Academy of Sciences,told us that he had found a paper On Inscribed and Escribed Ellipses of a Triangle(Keisuke MATSUMOTO,Kazunori FUJITA and Hiroo FUKAISHI,Mem.Fac.Educ.,Kagawa Univ.Ⅱ,59(2008),1-10),part of which coincides with our Lemma 1.So we hereby explain the case.