丘成桐数学竞赛(分析与方程)
丘成桐数学竞赛2020年笔试真题probability_and_statistics_20
S.-T.Yau College Student Mathematics Contests2020Probability and StatisticsSolve every problem.Part I:ProbabilityProblem2.Letλbe a positive number.Suppose that X is a random variable with E|X|<∞. Suppose thatλE f(X+1)=E{X f(X)}for all bounded smooth functions.Show that X has the Poisson distribution Poisson(λ).Problem3.Consider the random walkS n=a+X1+X2+···+X n,where a is a positive integer and{X i}are independent and identically distributed random variables with a common distributionP{X i=1}=p,P{X i=−1}=1−p.Letτ0=inf{n:S n=0}be thefirst time the random walk reaches the state x=0.For all p∈[0,1]find the probability P a{τ0<∞}that the random walk will eventually hit the state x=0.Problem4.Let Z=(X,Y)be an R2-valued random variable such that(1)X and Y are independent;(2)both X and Y have mean zero andfinite(nonvanishing)second moments;(3)the distribution of Z is invariant under the rotation counter-clockwise around the origin by an angleθnot a multiple of 90degrees.Show that X and Y must be normal random variables with the same variance.Part II:StatisticsThe following collection of questions concerns the design of a randomized experiment where the N units to be randomized to drug A or drug B are people,for whom we have a large number of background covariates,collectively labelled X(e.g.,age,sex,blood pressure,height,weight, occupational status,history of heart disease,family history of heart disease).The objective is to assign approximately half to drug A and half to drug B where the means of each of the X variables (and means of non-linear functions of them,such as squares or products)are close to equal in the two groups.Instead of using classical methods of design,such as blocking or stratification,the plan is to use modern computers to try many random allocations and discard those allocations that are considered unacceptable according to a pre-determined criterion for balanced X means,in particularan affinely invariant measure such as the Mahalanobis distance between the means of X in the two groups.After an acceptable allocation is found,outcome variables will be measured,and their means will be compared in group A and group B to estimate a treatment effect.Problem5.Prove that if the two groups are of the same size(i.e.,N/2for even N),this plan will result in unbiased estimates of the A versus B casual effect based on the sample means of Y in groups A and B,where Y is any linear function of X.Problem6.Provide a counter-example to the assertion that Problem5is true in small samples with odd N.。
丘成桐大学生数学竞赛奖项设置
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第三届“丘成桐中学数学奖”获奖名单
第三届“丘成桐中学数学奖”获奖名单丘成桐中学数学奖金奖(1名)上海市市北中学参赛队员:陈波宇指导老师:金荣生论文题目:Weierstrass函数在不可列的稠密集上不可导的一种证明丘成桐中学数学奖银奖(1名)中国人民大学附属中学参赛队员:刘頔、于伦指导老师:阳庆节论文题目:对凸形内部一类特殊点的研究丘成桐中学数学奖铜奖(3名)杭州第二中学参赛队员:干悦、陈宇戈、孙璐璐指导老师:金洁论文题目:从画正多边形的铰链到连杆轨迹广东广雅中学参赛队员:吴俊熹、熊奥林、刘哲指导老师:杨志明论文题目:瓦西列夫不等式的推广、加强与类似美国Milton Academy参赛队员:Farzan Vafa 指导老师:Yong Lin论文题目:A new definition of distance for graphs丘成桐中学数学奖优胜奖(5名)绍兴市第一中学参赛队员:胡煜中、陈璐萍、朱晖指导老师:凌晓锋论文题目:关于凸四边形最小个数问题绍兴市第一中学参赛队员:卢尔涵、林志成指导老师:凌晓锋论文题目:正整系数多项式的首位数字分布中国人民大学附属中学参赛队员:靳兆融、曾力玮指导老师:唐晓苗论文题目:三角形内切椭圆及其性质的研究美国Hazleton Area High School参赛队员:Keenan Monks 指导老师:Kenneth G. Monks论文题目:On supersingular elliptic curves and hypergeometric functions美国Liberty High School参赛队员:Benjamin Kraft 指导老师:Gregory Minton 论文题目:Entries of random matrices丘成桐中学数学奖鼓励奖(8名)华东师范大学第二附属中学参赛队员:景琰杰、顾韬指导老师:施洪亮论文题目:高次剩余理论研究北师大实验中学参赛队员:施燕捷指导老师:黎栋材论文题目:汽车行驶最短时间武昌实验中学参赛队员:迟浩指导老师:瞿双清论文题目:关于京广铁路武汉枢纽列车最大通过能力的研究华中师范大学第一附属中学参赛队员:路康奕指导老师:王维佳论文题目:交通路口拥堵程度的量化与评价广东实验中学参赛队员:张舒瑶、张陈灵、陈明蕙指导老师:郭卫东论文题目:论Wilcoxon-Mann-Whitney检验的容许性华南师范大学附属中学参赛队员:霍泽恩指导老师:李维民论文题目:Heron 三角形的有理角平分线问题及与完全长方体的联系华南师范大学附属中学参赛队员:梁昊、张可天指导老师:黄毅文论文题目:金融投资中的复利期望模型深圳市外国语学校参赛队员:郭梦绮、袁可馨、萧桐桐指导老师:袁智斌论文题目:对选拔性考试量分法的实证性研究丘成桐中学应用数学科学奖金奖(1名)杭州外国语学校参赛队员:朱陶元敏、张允宜指导老师:潘俊论文题目:变速队伍中速度传递问题的研究及其应用丘成桐中学应用数学科学奖银奖(1名)新加坡NUS High School of Mathematics and Science参赛队员:Li Ang, Lim Sung Hyun, Wang Qi 指导老师:Chai Ming Huang 论文题目:Understanding flocking dynamics in nature丘成桐中学应用数学科学奖铜奖(2名)江门市第一中学参赛队员:张姝、李璟、李健斌指导老师:李凌山论文题目:Actuarial modeling on a children’s protection insurance广东广雅中学参赛队员:黎骁旸、蔡宇指导老师:杨志明论文题目:墨西哥湾原油泄漏事件在推广Fay公式基础上的建模丘成桐中学应用数学科学奖鼓励奖(2名)郑州市第八中学参赛队员:汪晗、李晨、宋晨指导老师:刘正峰论文题目:二次拟合灰色预测模型在黄河水质发展趋势预测中的应用越南High School for the Gifted- Vietnam University of Hochiminh City 参赛队员:Nguyen Manh Tien, Tu Nguyen Thai Son 指导老师:Tran Nam Dung 论文题目:A stone-picking game丘成桐中学数学奖组织奖(3名)中国人民大学附属中学绍兴市第一中学广东广雅中学第四届丘成桐中学数学奖获奖名单金奖学校:美国Sidwell Friends Upper School题目:Shock Profile For Gas Dynamics in Thermal Nonequilibrium 参赛队员:Wang Xie指导老师:Luo Tao银奖学校:深圳中学题目:存在一个非VAN DOUWEN极大几乎不相交族队员:何卓东指导老师:张承宇铜奖(共2项,排名不分先后)学校:北京十一学校题目:三次函数切割线的斜率关系参赛队员:曾文远指导老师:潘国双学校:安徽省合肥市168中学题目:裴蜀定理的加强证明参赛队员:丁宇堃指导老师:孙文海优胜奖(共5项,排名不分先后)学校:华东师范大学第二附属中学题目:Euler-Maclaurin公式的推广及其应用参赛队员:景琰杰指导老师:施洪亮学校:河南省实验中学题目:曲线的拆解参赛队员:汪晗刘婧孜宋晨指导老师:李新德学校:东北育才中学题目:经典趣题“老鼠与毒药问题”推广研究参赛队员:白天衣项思陶指导老师:张雷学校:辽宁省鞍山市第一中学题目:拓扑和的推广参赛队员:夏铭辰指导老师:张继红学校:广东实验中学题目:树的模染色数参赛队员:王家林源洁莹指导老师:郭卫东鼓励奖(共4项,排名不分先后) 学校:浙江大学附属中学题目:数独的计数、分类与图案设计参赛队员:李梦鸽指导老师:李刚豪学校:江苏省苏州第十中学校题目:关于错排问题的思考与讨论参赛队员:钱炘祺于浩佳指导老师:吉剑锋学校:郑州第八中学题目:树枝的分形参赛队员:王佳瑞李明哲梁栋指导老师:刘正峰学校:郑州一高题目:基于防雷接地需求的土壤结构模型研究参赛队员:李豆豆指导老师:曹恒阁获得组织奖的学校(共2项,排名不分先后)杭州外国语学校广东省实验中学第五届丘成桐中学数学奖获奖名单金奖学校:清华大学附属中学题目:论两个函数方程解析解的渐近性质参赛队员:邵城阳指导老师:杨利军银奖学校:杭州第二中学题目:数学物理中的一个丢番图问题参赛队员:任之杨东辰指导老师:斯理炯铜奖(共3项,排名不分先后)学校:杭州外国语学校题目:一类离散最值问题的探究参赛队员:蔡煜晟赵海洲指导老师:张传鹏学校:广东实验中学题目:含Euler数和Bernounlli数的恒等式新探参赛队员:魏锐波储岸均指导老师:张俊杰学校:华南师范大学附属中学题目:莫比乌斯带分割的结构与拓扑性质参赛队员:樊润竹李想指导老师:罗碎海优胜奖(共3项,排名不分先后)学校:清华大学附属中学题目:扫雷游戏中数字和的最大值探究参赛队员:张益深指导老师:王殿军学校:北京十一学校题目:差分方程与微分方程间的关系及其解的性质的研究指导老师:潘国双学校:华南师范大学附属中学题目:N线坐标体系及应用参赛队员:余欣航指导老师:李兴怀获得组织奖的学校(共3项,排名不分先后)华南师范大学附属中学清华大学附属中学杭州外国语学校第三届丘成桐中学应用数学科学奖获奖名单银奖学校:南京外国语学校题目:最优交通拥堵费定价研究队员:陈宗灿杜楠指导老师:龚强铜奖(共3项,排名不分先后)学校:南京外国语学校题目:一种工业上测量椭圆截面积和椭球体积的新方法参赛队员:尤宸超田汉指导老师:朱胜强学校:Weston High School, USA题目:Computationally Determining the Dimensions of the Homology Groups of Directed Graphs 参赛队员:Ariya R. Shajii指导老师:Gabor Lippner学校:清华大学附属中学题目:CG图和形独基本性质探究指导老师:杨青明优胜奖(共5项,排名不分先后)学校:江苏省锡山高级中学题目:基于整数型延时差分方程组的基因表达调控的数学模型研究参赛队员:高竹指导老师:高军晖学校:清华大学附属中学题目:太阳时钟——计算时间的方法参赛队员:薛宇皓指导老师:周建军学校:澳门培正中学题目:云深不知处——代数学在云端储存上的应用参赛队员:谭知微指导老师:黄灿霖学校:福建省厦门第一中学题目:用微积分和微分方程解决家用热水器节能问题参赛队员:林梓楠谭天琪钱坤儿指导老师:徐小平学校:NUS High School, Singapore题目:Generalized Quantum Tic-Tac-Toe参赛队员:Ananya Kumar, ?Ang Yan Sheng指导老师:Chai Ming Huang金奖学校:St. Gregory College Preparatory School, USA题目:3D Surface Fabrication using Conformal Geometry参赛队员:Yuanqi Zhang指导老师:Xianfeng Gu银奖(共2项,排名不分先后)学校:北京市中关村中学题目:足球弧线球的数学分析方法参赛队员:杨祚儒富宏远计润达指导老师:潘凤易学校:NUS High School, Singapore题目:A General Algorithm of Flattening Convex Prismatoids 参赛队员:Chenglei Li 、Jingqi Zhou指导老师:Minghuang Chai铜奖(共4项,排名不分先后)学校:北京市十一学校题目:一类Pell方程的可解性研究参赛队员:王嘉琦蔡立德指导老师:贾祥雪学校:South Brunswick High School, USA题目:q-Symmetric Polynomials and nilHecke Algebras参赛队员:Ritesh Ragavender指导老师:Alexander Ellis学校:吉林大学附属中学题目:Quantum Watermarking in M-band Wavelet Domain参赛队员:刘通徐旋指导老师:任玉莲学校:Arcadia High School, USA题目:Feature Identification for Colon Tumor Classification 参赛队员:Anthony Hou指导老师:Ernie Esser优胜奖(共4项,排名不分先后)学校:澳门培正中学题目:A New Secure Distributed Storage Scheme for Cloud- Geometric and Algebraic design and Implementation参赛队员:张颢霆谭知微指导老师:黄灿霖学校:广东广雅中学题目:700阶以内有限群单性的探究参赛队员:邵芷茵赵子琪谢瑞恒指导老师:徐敏学校:清华大学附属中学题目:关于“青蛙跳几次,一米徘徊”概率问题的研究参赛队员:齐天博张胤泰指导老师:李劲松学校:杭州外国语学校题目:高层建筑安全疏散问题的研究及泛函背景下疏散方案设计的尝试参赛队员:周大桐王雨菡鼓励奖(共8项,排名不分先后)学校:华东师范大学第二附属中学题目:形如ax+bx=cx的简单指数方程解的无理性判定参赛队员:钮敏学蔡偌箐指导老师:戴中元学校:南京外国语学校题目:基于基本体格检查提出综合肥胖指数的分析与应用参赛队员:崔珈铭王英之杰王子南指导老师:黄文龙学校:南京外国语学校题目:Leslie修正模型在南京市人口预测中的应用参赛队员:王伯文陈思齐楼嘉钰指导老师:王刚学校:华东师范大学第二附属中学题目:三角形中神奇的点参赛队员:潘星宇指导老师:戴中元学校:清华大学附属中学题目:地图投影变换在全球一张图量算中的应用参赛队员:马浩程李寅晓指导老师:周建军学校:福建省南安第一中学题目:Analysis and Comparison between the Algorithm Time Efficiency of Dijkstra and SPFA 参赛队员:谢新锋指导老师:林建源学校:广东实验中学题目:稳健的对数最优策略理论研究与实践参赛队员:郭屹峰黄辰光指导老师:刘江宁学校:广州市第六中学题目:“三国杀”中的数学问题分析参赛队员:杨卓潇李思聪伍思航获得组织奖的学校(共4项,排名不分先后)广东广雅中学南京外国语学校华东师范大学第二附属中学清华大学附属中学。
丘成桐英才班考试范围
丘成桐英才班考试范围全文共四篇示例，供读者参考第一篇示例：一、数学1. 初等数论：包括整数、有理数等基本概念的考察，以及一些中级数论题目的解答。
2. 数列与数学归纳法：包括等差数列、等比数列等常见数列的求和公式，以及数列与数学归纳法在解题中的应用。
3. 平面几何：包括角度、三角形、四边形、圆等几何图形的性质和相关定理的考察。
4. 进阶数学：包括微积分、线性代数等高阶数学概念和定理的考察。
5. 竞赛数学：包括奥赛数学中的高难度题目和解题技巧的练习。
二、物理1. 力学：包括牛顿三大定律、摩擦力、弹簧力等力学知识和题目。
2. 热学：包括热力学、温度、热平衡等热学基础知识和问题。
3. 电磁学：包括电场、磁场、电流等电磁学基础知识和问题。
4. 光学：包括光的传播、反射、折射等光学知识和问题。
5. 现代物理：包括相对论、量子力学等现代物理领域的知识。
三、信息学1. 基本算法：包括排序算法、查找算法等常见算法的实现和应用。
2. 数据结构：包括链表、树、图等数据结构的基本概念和应用。
3. 计算机原理：包括计算机组成原理、操作系统、编程语言等计算机基础知识。
4. 算法设计：包括贪心算法、动态规划、回溯法等高级算法设计和分析。
5. 程序设计：包括编程能力、程序调试、算法实现等计算机编程技能的练习。
以上是丘成桐英才班考试范围的主要内容，学生们需要在这些领域取得一定的基础才能进入这个特殊的班级学习。
通过参加丘成桐英才班的学习，学生们将能够更好地提高自己的数学、物理和信息学能力，为未来参加奥赛比赛和科研工作打下坚实基础。
希望学生们在这个班级的学习过程中，不断努力，不断挑战自己，取得更好的成绩。
【2000字】第二篇示例：【丘成桐英才班考试范围】丘成桐英才班作为国内著名的数学培训机构，向来以其严格的教学标准和高质量的教育服务而闻名。
对于学生来说，通过丘成桐英才班的培训，不仅可以提高数学水平，更可以为未来的学业和职业发展打下坚实的基础。
2016 丘成桐大学生数学竞赛获奖名单
2016丘成桐大学生数学竞赛获奖名单丘成桐大学生数学竞赛华罗庚奖，即分析与微分方程方向获奖者是:铜奖清华大学邵城阳铜奖清华大学朱晶泽铜奖复旦大学钱列铜奖清华大学李阳垟铜奖清华大学张志宇银奖清华大学王昊宇银奖清华大学徐凯银奖中国科技大学马明辉银奖北京大学李艺轩金奖复旦大学张页丘成桐大学生数学竞赛陈省身奖，即几何与拓扑方向获奖者是:铜奖清华大学王高明铜奖北京大学沈澈铜奖清华大学白少云铜奖清华大学邵城阳铜奖清华大学熊昊仁银奖北京大学李艺轩银奖清华大学王志涵银奖中国科技大学马翘楚金奖清华大学徐凯金奖北京大学黄开丘成桐大学生数学竞赛周炜良，即代数、数论与组合方向奖获奖者是:铜奖清华大学王浩旭铜奖中国科技大学钱舰铜奖清华大学张志宇铜奖台湾大学羅啟恒铜奖台湾大学趙庭偉银奖北京大学陈成银奖复旦大学孟凡君银奖台湾大学吴博生金奖北京大学吕世极金奖清华大学徐凯丘成桐大学生数学竞赛林家翘奖，即应用数学与计算数学方向获奖者是:铜奖清华大学刘冠华铜奖北京大学金晨子铜奖清华大学王昊宇银奖武汉大学黄旷银奖北京大学金辉金奖清华大学李阳垟丘成桐大学生数学竞赛许宝騄奖，即概率统计方向获奖者是:铜奖北京大学王飞骋铜奖北京大学付伟龙铜奖北京大学顾超铜奖复旦大学唐博浩银奖北京大学刘浩然金奖清华大学王昊宇丘成桐大学生数学竞赛丘成桐奖，即个人全能奖获奖者是:银奖清华大学张志宇银奖北京大学李艺轩银奖清华大学王昊宇金奖清华大学李阳垟金奖清华大学徐凯丘成桐大学生数学竞赛团体赛获奖者是:铜奖复旦大学钱列、周易铖、石佳、陈小帖、陈品翰铜奖复旦大学孟凡君、邹嘉骅、缪欣晨、金正中、唐博浩铜奖中国科技大学何声、马翘楚、马明辉、高英瓒、袁望钧铜奖北京大学王翔、沈澈、孙成章、金辉、肖非依银奖清华大学徐凯、王志涵、赵瑞屾、贾楸烨、李林骏银奖清华大学王浩旭、白少云、李阳垟、郭怡辰、徐则驰银奖北京大学顾超、黄开、李艺轩、袁宏霖、段雅琦金奖清华大学秦翊宸、王怡、邵城阳、杨羽轩、王昊宇。
丘成桐领军计划数学水平测试申请理由
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2011年丘成桐大学生数学竞赛——计算与概率卷
S.-T.Yau College Student Mathematics Contests 2011Applied Math.,Computational Math.,Probability and StatisticsIndividual6:30–9:00pm,July 9,2011(Please select 5problems to solve)1.Given a weight function ρ(x )>0,let the inner-product correspond-ing to ρ(x )be defined as follows:(f,g ):= baρ(x )f (x )g (x )d x,and let f :=(f,f ).(1)Define a sequence of polynomials as follows:p 0(x )=1,p 1(x )=x −a 1,p n (x )=(x −a n )p n −1(x )−b n p n −2(x ),n =2,3,···wherea n =(xp n −1,p n −1)(p n −1,p n −1),n =1,2,···b n =(xp n −1,p n −2)(p n −2,p n −2),n =2,3,···.Show that {p n (x )}is an orthogonal sequence of monic polyno-mials.(2)Let {q n (x )}be an orthogonal sequence of monic polynomialscorresponding to the ρinner product.(A polynomial is called monic if its leading coefficient is 1.)Show that {q n (x )}is unique and it minimizes q n amongst all monic polynomials of degree n .(3)Hence or otherwise,show that if ρ(x )=1/√1−x 2and [a,b ]=[−1,1],then the corresponding orthogonal sequence is the Cheby-shev polynomials:T n (x )=cos(n arccos x ),n =0,1,2,···.and the following recurrent formula holds:T n +1(x )=2xT n (x )−T n −1(x ),n =1,2,···.(4)Find the best quadratic approximation to f (x )=x 3on [−1,1]using ρ(x )=1/√1−x 2.1Applied Math.Prob.Stat.,2011-Individual 22.If two polynomials p (x )and q (x ),both of fifth degree,satisfyp (i )=q (i )=1i,i =2,3,4,5,6,andp (1)=1,q (1)=2,find p (0)−q (0)y aside m black balls and n red balls in a jug.Supposes 1≤r ≤k ≤n .Each time one draws a ball from the jug at random.1)If each time one draws a ball without return,what is the prob-ability that in the k -th time of drawing one obtains exactly the r -th red ball?2)If each time one draws a ball with return,what is the probability that in the first k times of drawings one obtained totally an odd number of red balls?4.Let X and Y be independent and identically distributed random variables.Show thatE [|X +Y |]≥E [|X |].Hint:Consider separately two cases:E [X +]≥E [X −]and E [X +]<E [X −].5.Suppose that X 1,···,X n are a random sample from the Bernoulli distribution with probability of success p 1and Y 1,···,Y n be an inde-pendent random sample from the Bernoulli distribution with probabil-ity of success p 2.(a)Give a minimum sufficient statistic and the UMVU (uniformlyminimum variance unbiased)estimator for θ=p 1−p 2.(b)Give the Cramer-Rao bound for the variance of the unbiasedestimators for v (p 1)=p 1(1−p 1)or the UMVU estimator for v (p 1).(c)Compute the asymptotic power of the test with critical region |√n (ˆp 1−ˆp 2)/ 2ˆp ˆq |≥z 1−αwhen p 1=p and p 2=p +n −1/2∆,where ˆp =0.5ˆp 1+0.5ˆp 2.6.Suppose that an experiment is conducted to measure a constant θ.Independent unbiased measurements y of θcan be made with either of two instruments,both of which measure with normal errors:fori =1,2,instrument i produces independent errors with a N (0,σ2i )distribution.The two error variances σ21and σ22are known.When ameasurement y is made,a record is kept of the instrument used so that after n measurements the data is (a 1,y 1),...,(a n ,y n ),where a m =i if y m is obtained using instrument i .The choice between instruments is made independently for each observation in such a way thatP (a m =1)=P (a m =2)=0.5,1≤m ≤n.Applied Math.Prob.Stat.,2011-Individual 3Let x denote the entire set of data available to the statistician,in this case (a 1,y 1),...,(a n ,y n ),and let l θ(x )denote the corresponding log likelihood function for θ.Let a =n m =1(2−a m ).(a)Show that the maximum likelihood estimate of θis given by ˆθ= n m =11/σ2a m −1 n m =1y m /σ2a m.(b)Express the expected Fisher information I θand the observedFisher information I x in terms of n ,σ21,σ22,and a .What hap-pens to the quantity I θ/I x as n →∞?(c)Show that a is an ancillary statistic,and that the conditional variance of ˆθgiven a equals 1/I x .Of the two approximations ˆθ·∼N (θ,1/I θ)and ˆθ·∼N (θ,1/I x ),which (if either)would you use for the purposes of inference,and why?。
丘成桐大学生数学竞赛试卷
S.-T.Yau College Student Mathematics Contests 2010Analysis and Differential EquationsTeam(Please select 5problems to solve)1.a)Let f (z )be holomorphic in D :|z |<1and |f (z )|≤1(z ∈D ).Prove that|f (0)|−|z |1+|f (0)||z |≤|f (z )|≤|f (0)|+|z |1−|f (0)||z |.(z ∈D )b)For any finite complex value a ,prove that 12π 2π0log |a −e iθ|dθ=max {log |a |,0}.2.Let f ∈C 1(R ),f (x +1)=f (x ),for all x ,then we have ||f ||∞≤ 10|f (t )|dt + 10|f (t )|dt.3.Consider the equation¨x +(1+f (t ))x =0.We assume that ∞|f (t )|dt <∞.Study the Lyapunov stability of the solution (x,˙x )=(0,0).4.Suppose f :[a,b ]→R be a L 1-integrable function.Extend f to be 0outside the interval [a,b ].Letφ(x )=12h x +h x −hf Show thatb a |φ|≤ b a |f |.5.Suppose f ∈L 1[0,2π],ˆf (n )=12π 2π0f (x )e −inx dx ,prove that 1)∞ |n |=0|ˆf(n )|2<∞implies f ∈L 2[0,2π],2)n |n ˆf (n )|<∞implies that f =f 0,a.e.,f 0∈C 1[0,2π],where C 1[0,2π]is the space of functions f over [0,1]such that both f and its derivative f are continuous functions.126.SupposeΩ⊂R3to be a simply connected domain andΩ1⊂Ωwith boundaryΓ.Let u be a harmonic function inΩand M0=(x0,y0,z0)∈Ω1.Calculate the integral:II=−Γu∂∂n(1r)−1r∂u∂ndS,where 1r=1(x−x0)2+(y−x0)2+(z−x0)2and∂∂ndenotes theout normal derivative with respect to boundaryΓof the domainΩ1.(Hint:use the formula∂v∂n dS=∂v∂xdy∧dz+∂v∂ydz∧dx+∂v∂zdx∧dy.)S.-T.Yau College Student Mathematics Contests 2010Applied Math.,Computational Math.,Probability and StatisticsTeam(Please select 5problems to solve)1.Let X 1,···,X n be independent and identically distributed random variables with continuous distribution functions F (x 1),···,F (x n ),re-spectively.Let Y 1<···<Y n be the order statistics of X 1,···,X n .Prove that Z j =F (Y j )has the beta (j,n −j +1)distribution (j =1,···,n ).2.Let X 1,···,X n be i.i.d.random variable with a continuous density f at point 0.LetY n,i =34b n (1−X 2i /b 2n )I (|X i |≤b n ).Show that n i =1(Y n,i −EY n,i )(b n n i =1Y n,i )1/2L −→N (0,3/5),provided b n →0and nb n →∞.3.Let X 1,···,X n be independently and indentically distributed ran-dom variables with X i ∼N (θ,1).Suppose that it is known that |θ|≤τ,where τis given.Showmin a 1,···,a n +1sup |θ|≤τE (n i =1a i X i +a n +1−θ)2=τ2n −1τ2+n −1.Hint:Carefully use the sufficiency principle.4.The rules for “1and 1”foul shooting in basketball are as follows.The shooter gets to try to make a basket from the foul line.If he succeeds,he gets another try.More precisely,he make 0baskets by missing the first time,1basket by making the first shot and xsmissing the second one,or 2baskets by making both shots.Let n be a fixed integer,and suppose a player gets n tries at “1and 1”shooting.Let N 0,N 1,and N 2be the random variables recording the number of times he makes 0,1,or 2baskets,respectively.Note that N 0+N 1+N 2=n .Suppose that shots are independent Bernoulli trails with probability p for making a basket.(a)Write down the likelihood for (N 0,N 1,N 2).12(b)Show that the maximum likelihood estimator of p is ˆp =N 1+2N 2N 0+2N 1+2N 2.(c)Is ˆp an unbiased estimator for p ?Prove or disprove.(Hint:E ˆp is a polynomial in p ,whose order is higher than 1for p ∈(0,1).)(d)Find the asymptotic distribution of ˆp as n tends to ∞.5.When considering finite difference schemes approximating partial differential equations (PDEs),for example,the scheme(1)u n +1j =u n j −λ(u n j −u n j −1)where λ=∆t ∆x ,approximating the PDE (2)u t +u x =0,we are often interested in stability,namely(3)||u n ||≤C ||u 0||,n ∆t ≤T for a constant C =C (T )independent of the time step ∆t and the spa-tial mesh size ∆x .Here ||·||is a given norm,for example the L 2norm orthe L ∞norm,of the numerical solution vector u n =(u n 1,u n 2,···,u n N ).The mesh points are x j =j ∆x ,t n =n ∆t ,and the numerical solutionu n j approximates the exact solution u (x j ,t n )of the PDE (2)with aperiodic boundary condition.(i)Prove that the scheme (1)is stable in the sense of (3)for boththe L 2norm and the L ∞norm under the time step restriction λ≤1.(ii)Since the numerical solution u n is in a finite dimensional space,Student A argues that the stability (3),once proved for a spe-cific norm ||·||a ,would also automatically hold for any other norm ||·||b .His argument is based on the equivalency of all norms in a finite dimensional space,namely for any two norms ||·||a and ||·||b on a finite dimensional space W ,there exists a constant δ>0such thatδ||u ||b ≤||u ||a ≤1δ||u ||b .Do you agree with his argument?If yes,please give a detailed proof of the following theorem:If a scheme is stable,namely (3)holds for one particular norm (e.g.the L 2norm),then it is also stable for any other norm.If not,please explain the mistake made by Student A.6.We have the following 3PDEs(4)u t +Au x =0,(5)u t +Bu x =0,3 (6)u t+Cu x=0,C=A+B.Here u is a vector of size m and A and B are m×m real matrices. We assume m≥2and both A and B are diagonalizable with only real eigenvalues.We also assume periodic initial condition for these PDEs.(i)Prove that(4)and(5)are both well-posed in the L2-norm.Recall that a PDE is well-posed if its solution satisfies||u(·,t)||≤C(T)||u(·,0)||,0≤t≤Tfor a constant C(T)which depends only on T.(ii)Is(6)guaranteed to be well-posed as well?If yes,give a proof;if not,give a counter example.(iii)Suppose we have afinite difference schemeu n+1=A h u nfor approximating(4)and another schemeu n+1=B h u nfor approximating(5).Suppose both schemes are stable in theL2-norm,namely(3)holds for both schemes.If we now formthe splitting schemeu n+1=B h A h u nwhich is a consistent scheme for solving(6),is this scheme guar-anteed to be L2stable as well?If yes,give a proof;if not,givea counter example.S.-T.Yau College Student Mathematics Contests2010Geometry and TopologyTeam(Please select5problems to solve)1.Let S n⊂R n+1be the unit sphere,and R n⊂R n+1the equator n-plane through the center of S n.Let N be the north pole of S n.Define a mappingπ:S n\{N}→R n called the stereographic projection that takes A∈S n\{N}into the intersection A ∈R n of the equator n-plane R n with the line which passes through A and N.Prove that the stereographic projection is a conformal change,and derive the standard metric of S n by the stereographic projection.2.Let M be a(connected)Riemannian manifold of dimension2.Let f be a smooth non-constant function on M such that f is bounded from above and∆f≥0everywhere on M.Show that there does not exist any point p∈M such that f(p)=sup{f(x):x∈M}.3.Let M be a compact smooth manifold of dimension d.Prove that there exists some n∈Z+such that M can be regularly embedded in the Euclidean space R n.4.Show that any C∞function f on a compact smooth manifold M (without boundary)must have at least two critical points.When M is the2-torus,show that f must have more than two critical points.5.Construct a space X with H0(X)=Z,H1(X)=Z2×Z3,H2(X)= Z,and all other homology groups of X vanishing.6.(a).Define the degree deg f of a C∞map f:S2−→S2and prove that deg f as you present it is well-defined and independent of any choices you need to make in your definition.(b).Prove in detail that for each integer k(possibly negative),there is a C∞map f:S2−→S2of degree k.1S.-T.Yau College Student Mathematics Contests 2010Algebra,Number Theory andCombinatoricsTeam(Please select 5problems to solve)1.For a real number r ,let [r ]denote the maximal integer less or equal than r .Let a and b be two positive irrational numbers such that 1a +1b = 1.Show that the two sequences of integers [ax ],[bx ]for x =1,2,3,···contain all natural numbers without repetition.2.Let n ≥2be an integer and consider the Fermat equationX n +Y n =Z n ,X,Y,Z ∈C [t ].Find all nontrivial solution (X,Y,Z )of the above equation in the sense that X,Y,Z have no common zero and are not all constant.3.Let p ≥7be an odd prime number.(a)Evaluate the rational number cos(π/7)·cos(2π/7)·cos(3π/7).(b)Show that (p −1)/2n =1cos(nπ/p )is a rational number and deter-mine its value.4.For a positive integer a ,consider the polynomialf a =x 6+3ax 4+3x 3+3ax 2+1.Show that it is irreducible.Let F be the splitting field of f a .Show that its Galois group is solvable.5.Prove that a group of order 150is not simple.6.Let V ∼=C 2be the standard representation of SL 2(C ).(a)Show that the n -th symmetric power V n =Sym n V is irre-ducible.(b)Which V n appear in the decomposition of the tensor productV 2⊗V 3into irreducible representations?1。