第2届丘成桐大学生数学竞赛试题
第2届丘成桐大学生数学竞赛试题

S.-T.Yau College Student Mathematics Contests 2011Analysis and Differential EquationsIndividual2:30–5:00pm,July 9,2011(Please select 5problems to solve)1.a)Compute the integral: ∞−∞x cos xdx (x 2+1)(x 2+2),b)Show that there is a continuous function f :[0,+∞)→(−∞,+∞)such that f ≡0and f (4x )=f (2x )+f (x ).2.Solve the following problem: d 2u dx 2−u (x )=4e −x ,x ∈(0,1),u (0)=0,dudx(0)=0.3.Find an explicit conformal transformation of an open set U ={|z |>1}\(−∞,−1]to the unit disc.4.Assume f ∈C 2[a,b ]satisfying |f (x )|≤A,|f(x )|≤B for each x ∈[a,b ]and there exists x 0∈[a,b ]such that |f (x 0)|≤D ,then |f (x )|≤2√AB +D,∀x ∈[a,b ].5.Let C ([0,1])denote the Banach space of real valued continuous functions on [0,1]with the sup norm,and suppose that X ⊂C ([0,1])is a dense linear subspace.Suppose l :X →R is a linear map (not assumed to be continuous in any sense)such that l (f )≥0if f ∈X and f ≥0.Show that there is a unique Borel measure µon [0,1]such that l (f )= fdµfor all f ∈X .6.For s ≥0,let H s (T )be the space of L 2functions f on the circle T =R /(2πZ )whose Fourier coefficients ˆf n = 2π0e−inx f (x )dx satisfy Σ(1+n 2)s ||ˆf n |2<∞,with norm ||f ||2s =(2π)−1Σ(1+n 2)s |ˆf n |2.a.Show that for r >s ≥0,the inclusion map i :H r (T )→H s (T )is compact.b.Show that if s >1/2,then H s (T )includes continuously into C (T ),the space of continuous functions on T ,and the inclusion map is compact.1S.-T.Yau College Student Mathematics Contests2011Geometry and TopologyIndividual9:30–12:00am,July10,2011(Please select5problems to solve)1.Suppose M is a closed smooth n-manifold.a)Does there always exist a smooth map f:M→S n from M into the n-sphere,such that f is essential(i.e.f is not homotopic to a constant map)?Justify your answer.b)Same question,replacing S n by the n-torus T n.2.Suppose(X,d)is a compact metric space and f:X→X is a map so that d(f(x),f(y))=d(x,y)for all x,y in X.Show that f is an onto map.3.Let C1,C2be two linked circles in R3.Show that C1cannot be homotopic to a point in R3\C2.4.Let M=R2/Z2be the two dimensional torus,L the line3x=7y in R2,and S=π(L)⊂M whereπ:R2→M is the projection map. Find a differential form on M which represents the Poincar´e dual of S.5.A regular curve C in R3is called a Bertrand Curve,if there existsa diffeomorphism f:C→D from C onto a different regular curve D in R3such that N x C=N f(x)D for any x∈C.Here N x C denotes the principal normal line of the curve C passing through x,and T x C will denote the tangent line of C at x.Prove that:a)The distance|x−f(x)|is constant for x∈C;and the angle made between the directions of the two tangent lines T x C and T f(x)D is also constant.b)If the curvature k and torsionτof C are nowhere zero,then there must be constantsλandµsuch thatλk+µτ=16.Let M be the closed surface generated by carrying a small circle with radius r around a closed curve C embedded in R3such that the center moves along C and the circle is in the normal plane to C at each point.Prove thatMH2dσ≥2π2,and the equality holds if and only if C is a circle with radius √2r.HereH is the mean curvature of M and dσis the area element of M.1S.-T.Yau College Student Mathematics Contests 2011Algebra,Number Theory andCombinatoricsIndividual2:30–5:00pm,July 10,2011(Please select 5problems to solve)For the following problems,every example and statement must be backed up by proof.Examples and statements without proof will re-ceive no-credit.1.Let K =Q (√−3),an imaginary quadratic field.(a)Does there exists a finite Galois extension L/Q which containsK such that Gal(L/Q )∼=S 3?(Here S 3is the symmetric group in 3letters.)(b)Does there exists a finite Galois extension L/Q which containsK such that Gal(L/Q )∼=Z /4Z ?(c)Does there exists a finite Galois extension L/Q which containsK such that Gal(L/Q )∼=Q ?Here Q is the quaternion group with 8elements {±1,±i,±j,±k },a finite subgroup of the group of units H ×of the ring H of all Hamiltonian quaternions.2.Let f be a two-dimensional (complex)representation of a finite group G such that 1is an eigenvalue of f (σ)for every σ∈G .Prove that f is a direct sum of two one-dimensional representations of G3.Let F ⊂R be the subset of all real numbers that are roots of monic polynomials f (X )∈Q [X ].(1)Show that F is a field.(2)Show that the only field automorphisms of F are the identityautomorphism α(x )=x for all x ∈F .4.Let V be a finite-dimensional vector space over R and T :V →V be a linear transformation such that(1)the minimal polynomial of T is irreducible;(2)there exists a vector v ∈V such that {T i v |i ≥0}spans V .Show that V contains no non-trivial proper T -invariant subspace.5.Given a commutative diagramA →B →C →D →E↓↓↓↓↓A →B →C →D →E1Algebra,Number Theory and Combinatorics,2011-Individual2 of Abelian groups,such that(i)both rows are exact sequences and(ii) every vertical map,except the middle one,is an isomorphism.Show that the middle map C→C is also an isomorphism.6.Prove that a group of order150is not simple.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?S.-T.Yau College Student Mathematics Contests 2011Analysis and Differential EquationsTeam9:00–12:00am,July 9,2011(Please select 5problems to solve)1.Let H 2(∆)be the space of holomorphic functions in the unit disk ∆={|z |<1}such that ∆|f |2|dz |2<∞.Prove that H 2(∆)is a Hilbert space and that for any r <1,the map T :H 2(∆)→H 2(∆)given by T f (z ):=f (rz )is a compact operator.2.For any continuous function f (z )of period 1,show that the equation dϕdt=2πϕ+f (t )has a unique solution of period 1.3.Let h (x )be a C ∞function on the real line R .Find a C ∞function u (x,y )on an open subset of R containing the x -axis such that u x +2u y =u 2and u (x,0)=h (x ).4.Let S ={x ∈R ||x −p |≤c/q 3,for all p,q ∈Z ,q >0,c >0},show that S is uncountable and its measure is zero.5.Let sl (n )denote the set of all n ×n real matrices with trace equal to zero and let SL (n )be the set of all n ×n real matrices with deter-minant equal to one.Let ϕ(z )be a real analytic function defined in a neighborhood of z =0of the complex plane C satisfying the conditions ϕ(0)=1and ϕ (0)=1.(a)If ϕmaps any near zero matrix in sl (n )into SL (n )for some n ≥3,show that ϕ(z )=exp(z ).(b)Is the conclusion of (a)still true in the case n =2?If it is true,prove it.If not,give a counterexample.e mathematical analysis to show that:(a)e and πare irrational numbers;(b)e and πare also transcendental numbers.1S.-T.Yau College Student Mathematics Contests2011Applied Math.,Computational Math.,Probability and StatisticsTeam9:00–12:00am,July9,2011(Please select5problems to solve)1.Let A be an N-by-N symmetric positive definite matrix.The con-jugate gradient method can be described as follows:r0=b−A x0,p0=r0,x0=0FOR n=0,1,...αn= r n 22/(p TnA p n)x n+1=x n+αn p n r n+1=r n−αn A p nβn=−r Tk+1A p k/p TkA p kp n+1=r n+1+βn p nEND FORShow(a)αn minimizes f(x n+αp n)for allα∈R wheref(x)≡12x T A x−b T x.(b)p Ti r n=0for i<n and p TiA p j=0if i=j.(c)Span{p0,p1,...,p n−1}=Span{r0,r1,...,r n−1}≡K n.(d)r n is orthogonal to K n.2.We use the following scheme to solve the PDE u t+u x=0:u n+1 j =au nj−2+bu nj−1+cu njwhere a,b,c are constants which may depend on the CFL numberλ=∆t ∆x .Here x j=j∆x,t n=n∆t and u njis the numerical approximationto the exact solution u(x j,t n),with periodic boundary conditions.(i)Find a,b,c so that the scheme is second order accurate.(ii)Verify that the scheme you derived in Part(i)is exact(i.e.u nj =u(x j,t n))ifλ=1orλ=2.Does this imply that the scheme is stable forλ≤2?If not,findλ0such that the scheme is stable forλ≤λ0. Recall that a scheme is stable if there exist constants M and C,which are independent of the mesh sizes∆x and∆t,such thatu n ≤Me CT u0for all∆x,∆t and n such that t n≤T.You can use either the L∞norm or the L2norm to prove stability.1Applied Math.Prob.Stat.,2011-Team2 3.Let X and Y be independent random variables,identically dis-tributed according to the Normal distribution with mean0and variance 1,N(0,1).(a)Find the joint probability density function of(R,),whereR=(X2+Y2)1/2andθ=arctan(Y/X).(b)Are R andθindependent?Why,or why not?(c)Find a function U of R which has the uniform distribution on(0,1),Unif(0,1).(d)Find a function V ofθwhich is distributed as Unif(0,1).(e)Show how to transform two independent observations U and Vfrom Unif(0,1)into two independent observations X,Y fromN(0,1).4.Let X be a random variable such that E[|X|]<∞.Show thatE[|X−a|]=infE[|X−x|],x∈Rif and only if a is a median of X.5.Let Y1,...,Y n be iid observations from the distribution f(x−θ), whereθis unknown and f()is probability density function symmetric about zero.Suppose a priori thatθhas the improper priorθ∼Lebesgue(flat) on(−∞,∞).Write down the posterior distribution ofθ.Provides some arguments to show that thisflat prior is noninforma-tive.Show that with the posterior distribution in(a),a95%probability interval is also a95%confidence interval.6.Suppose we have two independent random samples{Y1,i=1,...,n} from Poisson with(unknown)meanλ1and{Y i,i=n+1,...,2n}from Poisson with(unknown)meanλ2Letθ=λ1/(λ1+λ2).(a)Find an unbiased estimator ofθ(b)Does your estimator have the minimum variance among all un-biased estimators?If yes,prove it.If not,find one that has theminimum variance(and prove it).(c)Does the unbiased minimum variance estimator you found at-tain the Fisher information bound?If yes,show it.If no,whynot?S.-T.Yau College Student Mathematics Contests2011Geometry and TopologyTeam9:00–12:00am,July9,2011(Please select5problems to solve)1.Suppose K is afinite connected simplicial complex.True or false:a)Ifπ1(K)isfinite,then the universal cover of K is compact.b)If the universal cover of K is compact thenπ1(K)isfinite.pute all homology groups of the the m-skeleton of an n-simplex, 0≤m≤n.3.Let M be an n-dimensional compact oriented Riemannian manifold with boundary and X a smooth vectorfield on M.If n is the inward unit normal vector of the boundary,show thatM div(X)dV M=∂MX·n dV∂M.4.Let F k(M)be the space of all C∞k-forms on a differentiable man-ifold M.Suppose U and V are open subsets of M.a)Explain carefully how the usual exact sequence0−→F(U∪V)−→F(U)⊕F V)−→F(U∩V)−→0 arises.b)Write down the“long exact sequence”in de Rham cohomology as-sociated to the short exact sequence in part(a)and describe explicitly how the mapH kdeR (U∩V)−→H k+1deR(U∪V)arises.5.Let M be a Riemannian n-manifold.Show that the scalar curvature R(p)at p∈M is given byR(p)=1vol(S n−1)S n−1Ric p(x)dS n−1,where Ric p(x)is the Ricci curvature in direction x∈S n−1⊂T p M, vol(S n−1)is the volume of S n−1and dS n−1is the volume element of S n−1.1Geometry and Topology,2011-Team2 6.Prove the Schur’s Lemma:If on a Riemannian manifold of dimension at least three,the Ricci curvature depends only on the base point but not on the tangent direction,then the Ricci curvature must be constant everywhere,i.e.,the manifold is Einstein.S.-T.Yau College Student Mathematics Contests 2011Algebra,Number Theory andCombinatoricsTeam9:00–12:00pm,July 9,2011(Please select 5problems to solve)For the following problems,every example and statement must be backed up by proof.Examples and statements without proof will re-ceive no-credit.1.Let F be a field and ¯Fthe algebraic closure of F .Let f (x,y )and g (x,y )be polynomials in F [x,y ]such that g .c .d .(f,g )=1in F [x,y ].Show that there are only finitely many (a,b )∈¯F×2such that f (a,b )=g (a,b )=0.Can you generalize this to the cases of more than two-variables?2.Let D be a PID,and D n the free module of rank n over D .Then any submodule of D n is a free module of rank m ≤n .3.Identify pairs of integers n =m ∈Z +such that the quotient rings Z [x,y ]/(x 2−y n )∼=Z [x,y ]/(x 2−y m );and identify pairs of integers n =m ∈Z +such that Z [x,y ]/(x 2−y n )∼=Z [x,y ]/(x 2−y m ).4.Is it possible to find an integer n >1such that the sum1+12+13+14+ (1)is an integer?5.Recall that F 7is the finite field with 7elements,and GL 3(F 7)is the group of all invertible 3×3matrices with entries in F 7.(a)Find a 7-Sylow subgroup P 7of GL 3(F 7).(b)Determine the normalizer subgroup N of the 7-Sylow subgroupyou found in (a).(c)Find a 2-Sylow subgroup of GL 3(F 7).6.For a ring R ,let SL 2(R )denote the group of invertible 2×2matrices.Show that SL 2(Z )is generated by T = 1101 and S = 01−10 .What about SL 2(R )?1。
大学生数学知识竞赛试题及答案【最新】

趣味数学知识竞赛复习题一、填空题1、(苏步青)是国际公认的几何学权威,我国微分几何派的创始人。
2、(华罗庚)是一个传奇式的人物,是一个自学成才的数学家。
3、编有《三角学》,被称为“李蕃三角”且自称为“三书子”的是(李锐夫)。
4、世界上攻克“哥德巴赫猜想”的第一个人是(陈景润)。
5、(姜立夫)是现代数学在中国最早而又最富成效的播种人”,这是《中国大百科全书》和《中国现代数学家传》对他的共同评价。
6. 设有n个实数,满足|xi|<1(I=1,2,3,…,n), |x1|+|x2|+…+|xn|=19+|x1+x2+…+xn| ,则n的最小值207. 三角形的一个顶点引出的角平分线,高线及中线恰将这个顶点的角四等分,则这个顶角的度数为___90° ___8. 某旅馆有2003个空房间,房间钥匙互不相同,来了2010们旅客,要分发钥匙,使得其中任何2003个人都能住进这2003个房间,而且每人一间(假定每间分出的钥匙数及每人分到的钥匙数都不限),最少得发出_16024______把钥匙.9. 在凸1900边形内取103个点,以这2003个点为顶点,可将原凸1900边形分割成小三角形的个数为______2104 _____.10. 若实数x满足x4+36<13x2,则f(x)=x3-3x的最大值为______18_____11 ."我买鸡蛋时,付给杂货店老板12美分,"一位厨师说道,"但是由于嫌它们太小,我又叫他无偿添加了2只鸡蛋给我。
这样一来,每打(12只)鸡蛋的价钱就比当初的要价降低了1美分。
" 厨师买了_18只鸡蛋?12.已知f(x)∈[0,1],则y=f(x)+1的取值范围为 ___[7/9,7/8]____13. 已知函数f(x)与g(x)的定义域均为非负实数集,对任意的x≥0,规定f(x)*g(x)=min{f(x),g(x)}.若f(x)=3-x,g(x)=,则f(x)*g(x)的最大值为____(2√3-1) _____ 14.已知a,b,cd∈N,且满足342(abcd+ab+ad+cd+1)=379(bcd+b+d),设M=a×103+b×102+c×10+d,则M的值为______ 1949 ___.15. 用E(n)表示可使5k是乘积112233…nn的约数为最大的整数k,则E(150)=__ 2975_________16. 从1到100的自然数中,每次取出不同的两个数,使它们的和大于100,则可有_2500________种不同的取法.17. 从正整数序列1,2,3,4,…中依次划去3的倍数和4的倍数,但是其中是5的倍数均保留,划完后剩下的数依次构成一个新的序列:A1=1,A2=2,A3=5,A4=7,…,则A2003的值为____3338 _____.18. .连接凸五边形的每两个顶点总共可得到十条线段(包括边在内),现将其中的几条线段着上着颜色,为了使得该五边形中任意三个顶点所构成的三角形都至少有一条边是有颜色的则n的最小值是_419. 已知x0=2003,xn=xn-1+ (n>1,n∈N),则x2003的整数部分为_______2003___21. 已知ak≥0,k=1,2,…,2003,且a1+a2+…+a2003=1,则S=max{a1+a2+a3,a2+a3+a4,…, a2001+a2002+a2003}的最小值为________3/2007 _.22. 对于每一对实数x,y,函数f满足f(x)+f(y)=f(x+y)-xy-1,若f(1)=1,那么使f(n)=n(n≠1)的整数n共有_1个.23.在棱长为a的正方体内容纳9个等球,八个角各放一个,则这些等球最大半径是____. (√3-3/2)a ___24.已知a,b,c都不为0,并且有sinx=asin(y-z),siny=bsin(z-x),sinz=csin(x-y).则有ab+bc+ca=__-1 _____.二、选择题1、被誉为中国现代数学祖师的是(1、C )。
2024年湖北武汉华中师大一附中丘成桐少年班选拔数学试题

2024年湖北省武汉市华中师范大学第一附属中学丘成桐少年班选拔真题1、解方程:[x]= 3x-2,x=____________。
2、15转化成2进制是_______,10位的2进制数有___________。
3、Φ(x)表示不超过x的非0自然数中与x互质的自然数的个数,Ф(25)=______;Ф(77)=_____.4、甲、乙、丙3人从A地出发去270km 远的B地,每天走10km,每人最多带36 天的食物。
可以把食物分给他人。
(1)、若甲、乙两人出发,甲距离A地100km 时返回,乙可以走_____千米就要返回;(2)、若甲、乙两人出发,乙最多可以走_______千米就要返回;(3)、请设计一种方案,可以互相补给,不可以存放,如何让乙走到B再返回。
5、一批衣服进价40 元,打算以40%的利润卖出,卖了80%后打折,卖完后的总利润只有原来的86%,则打了_________折。
6、对于一些数a1,a2,…, a n,若a i>a j,且i<j则成a i,a j为逆序,a i到a n的逆序数量叫逆序数(1)、1,2,3,4的任意排列中,逆序数恰好为2的排列有_______个;(2)、设a1,a2,…, a10的逆序数为k,则a10,a9,…, a1的逆序数为_______(用k表示)。
7、计算=_____________.8、小明一步跨2级或3级台阶,一共20级台阶有种___________不同的走法。
9、有一堆石子,甲乙两人轮流取,甲先取,乙后取,每次1-5颗,如果一共12颗,______必胜;如果一共2024颗,________必胜。
10、有一个水池,单开甲管12个小时可以注满,单开乙管20个小时可以注满,现在甲开1个小时,乙开1个小时,轮流下去,注满水池要_______小时。
11、把圆柱体切成2个小的圆柱体,它的表面积增加了25.12平方分米,把它切成两个半圆柱体,它的表面积增加80平方分米,这个圆柱体的体积是_________立方分米。
不考钻牛角尖的题记第2届丘成桐大学生数学竞赛

不考钻牛角尖的题记第2届丘成桐大学生数学竞赛“什么是你最喜欢的定理?”“你把这个原理再解释一下。
”“几何、拓扑和代数内容,你在大学都学了什么?”“你的几何课,是哪位老师教的?”8月19日至20日,丘成桐先生在第二届丘成桐大学生数学竞赛口试考场——中国科学院数学与系统科学研究院七层教室里,不时地向前来参加面试的学生提出上述问题。
这次竞赛考察学生在“分析与微分方程”,“几何与拓扑”,“代数、组合与数论”,“计算、统计与应用数学”四个方面的能力。
竞赛的方式和题目,都是经过世界一流数学大师精心研究的。
所以,对学生们来说,能参加这样的竞赛活动,确实是一件荣幸的事情。
个人全能冠军获得者——北京大学张瑞祥说:“过去的两天里非常愉快,要感谢丘成桐先生,他的想法非常好,把大家聚集到一起。
老师们的提问,也给了我很多的启发。
”来自北京大学的叶立早,是温州乐清人,曾是2019年第一届丘成桐中学生数学竞赛和2009年全国数学奥赛的金牌获得者,前年被保送到北京大学数学系学习,在今年的竞赛中获得个人全能银牌。
在19日上午面试之后,叶立早谈了自己的感想:“为了这次考试,自学了不少课外的基础知识。
因为这次考试,是向西方一流大学的博士生资格考试看齐。
这样的竞赛很有意义,考的都是基础知识,可以检验一下自己到底学得怎么样!”他说自己代数笔试考得不大理想,分析的口试部分回答得还比较好。
今年要上大四的北京大学数学系学生杨奔,以前是人大附中数学班的尖子生,在2019年也曾获得全国奥赛冠军。
他说:“这次北大有20多人报名,考题虽然不算难,但内容很多。
”很明显,这次考试内容完全不同于奥数竞赛的那种偏题、怪题,而是考大学数学教材中的基本知识内容。
“我们不考钻牛角尖的题,而是要考最基本、最实用的题目。
”丘成桐说。
丘成桐认为,国内数学教育启发性、创造性不够。
所以,竞赛从一开始就设置了口试部分。
他说:“这是很重要的,可以从另一个角度考察学生的能力,比如应变和阐释能力。
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。
第二届中国大学生数学竞赛决赛试卷参考答案

第二届中国大学生数学竞赛决赛试卷参考答案一.计算下列各题(本题共3小题,每小题各5分,共15分,要求写出重要步骤。
)(1)解:方法一(用两个重要极限):()()20003221sin 1cos sin 1cos 001sin cos 12limlimlim sin 11331cos 3222sin sin lim lim 1lim x x x x x xxx x x x x x x x x x x x x x x x x x x x x x x ee eee→→→-∙---→→------→-⎛⎫⎛⎫=+ ⎪ ⎪⎝⎭⎝⎭=====方法二(取对数):11cos 0002sin sin ln 1sin lim exp lim exp lim 11cos 2xx x x x x x x x x x x -→→→⎡⎤⎛⎫⎡⎤- ⎪⎢⎥⎢⎥⎛⎫⎝⎭⎢⎥==⎢⎥ ⎪-⎝⎭⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦20003221sin cos 12limlimlim 11333222x x x x x x x x x x eee e→→→----====(2).解:方法一(用欧拉公式)令111...12n x n n n n=++++++ 111ln =C+o 12n n +++-由欧拉公式得(),11111ln 2=C+o 1212n n n n++++++-+则(),其中,()1o表示n →∞时的无穷小量,-ln2o 1n x ∴=两式相减,得:(),lim ln 2.n n x →∞∴= 方法二(用定积分的定义)111lim lim lim()12n n n n x n n n→∞→∞→∞=++++111lim ()111n n n nn→∞=++++101ln 21dx x==+⎰(3)解:222222221211,121121tt t t t t t t t tte dx e dy e dy e e e e dt e dt e dx e e --++==-∴==+++ ()()222222412121224ttt tt tte e d y d dy e e dx dx dt dx e e edt+--+⎛⎫∴=∙==⎪⎝⎭二.(本题10分)解:设24,1P x y Q x y =+-=+-,则0P d x Q d y +=1,P Qy x∂∂==∴∂∂0Pdx Qdy +=是一个全微分方程,设dz Pdx Qdy =+方法一:由24zP x y x∂==+-∂得 ()()2244z x y dx x xy x C y =+-=+-+⎰由()'1zx C y Q x y y∂=+==+-∂得()()'211,2C y y C y y y c =-∴=-+22142z x xy x y y c ∴=+-+-+方法二:()()()(),0,024x y z dz Pdx Qdy x y dx x y dy==+=+-++-⎰⎰⎰,P Qy x∂∂=∴∂∂该曲线积分与路径无关 ()()2200124142xyz x dx x y dy x x xy y y ∴=-++-=-++-⎰⎰三.(本题15分)证明:由极限的存在性:()()()()1230lim 2300h k fh k f h k f h f →++-=⎡⎤⎣⎦即[]()123100k k k f ++-=,又()00f ≠,1231k k k ∴++=①由洛比达法则得()()()()()()()1232'''1230230lim2233lim 02h h k f h k f h k f h f h k f h k f h k f h h →→++-++==由极限的存在性得()()()'''1230lim 22330h k f h k f h k f h →⎡⎤++=⎣⎦即()()'1232300k k k f ++=,又()'00f ≠,123230k k k ∴++=② 再次使用洛比达法则得()()()()()()()()()'''1230"""1230""1232233lim24293lim02490000h h k f h k f h k f h hk f h k f h k f h k k k f f →→++++==∴++=≠123490k k k ∴++=③由①②③得123,,k k k 是齐次线性方程组1231231231230490k k k k k k k k k ++=⎧⎪++=⎨⎪++=⎩的解设1231111123,,01490k A x k b k ⎛⎫⎛⎫⎛⎫⎪ ⎪ ⎪=== ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭,则Ax b =, 增广矩阵*11111031230010314900011A ⎛⎫⎛⎫ ⎪ ⎪=- ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭,则()(),3R A b R A ==所以,方程Axb =有唯一解,即存在唯一一组实数123,,k k k 满足题意,且1233,3,1k k k ==-=。
丘成桐班初试题

18. 2023 GDFZ丘成桐班初试时间:80分钟满分:120分一、填空题(每小题6分,共60分)1.(量率对应)甲、乙、丙、丁四人去买电视机,甲带的钱是另外三人所带钱总数的一半,乙带的钱是另外三人所带的钱总数的13,丙带的钱是另外三人所带总钱数的14,丁带910元,四人所带的总钱数是___________元。
2.(方程的应用)两人从甲地到乙地同时出发,一人用匀速3小时走完全程,另一人用匀速4小时走完全程,经过___________小时,其中一人所剩路程的长是另一人所剩路程的长的2倍。
3.(割补法求面积)如图,直角梯形ABCD中四边形AEGF、MBKN都是正方形,且AE=MB,EP=KC=9,DF=PM= 4,则△DPC的面积为__________.4.(分数的应用)今有桃95个,分给甲、乙两班学生吃,甲班分到的桃有29坏的,其他是好的,乙班分到的桃有316是坏的,其他是好的,甲、乙两班分到的好桃共有________个。
5.(和差法求面积)如图,ABCD是平行四边形,AD=8cm,AB=10cm,∠DAB=30°,高CH=4m,弧BE,DF分别以AB,CD为半径,弧D M,BN分别以AD,CB为半径,阴影部分的面积为__________.6.(钟表问题)如图,假设某星球的一天只有6小时,每小时36分钟,那么3点18分时,时针和分针所形成的锐角是__________度。
7.(接送问题)甲、乙、丙三人同时从A地出发到距A地100千米的B地,甲与丙以25千米/时的速度驾车行进,而乙却以5千米/时的速度步行,过了一段时间后,丙下车改以5千米/时的速度步行,而甲驾车以原速折回,将乙载上后前往B地,这样甲、乙、丙三人同时到达B地,此旅程共用时为__________小时。
8.(最值问题)已知A、B、C、D、E、F、G、H、I、J、K代表十个互不相同的大于0的自然数,要使下列等式成立。
A最小是__________。
女子丘成桐数学竞赛试题

女子丘成桐数学竞赛试题一、选择题(每题3分,共15分)1. 下列哪个数是无理数?A. 3.14159B. πC. √2D. 0.333...2. 如果一个函数f(x)在区间[0, 1]上连续,且∫(0, 1) f(x)dx = 1,那么f(x)在区间[0, 1]上的最大值至少为:A. 1B. 2C. 3D. 无法确定3. 对于一个二次方程ax² + bx + c = 0,当a < 0时,方程的根:A. 都是实数B. 都是正数C. 都是负数D. 无法确定4. 一个圆的周长为2π,那么这个圆的面积是:A. πB. 2πC. 4πD. 8π5. 欧拉公式V - E + F = 2适用于:A. 所有多面体B. 所有平面图形C. 所有凸多面体D. 所有凸多边形二、填空题(每题2分,共10分)6. 一个直角三角形的两条直角边分别为3和4,其斜边的长度是________。
7. 函数y = sin(x)在区间[0, π]上的值域是________。
8. 如果一个数列是等差数列,且a_3 = 7,a_5 = 13,那么这个数列的公差d是________。
9. 将一个正六边形分成6个等边三角形,每个等边三角形的面积是正六边形面积的________。
10. 根据勾股定理,如果直角三角形的两条直角边分别为5和12,那么斜边的长度是________。
三、解答题(每题15分,共30分)11. 证明:对于任意正整数n,2^n - 1总是一个奇数。
12. 解方程组:\[\begin{cases}x + y = 5 \\2x - y = 1\end{cases}\]四、证明题(每题20分,共20分)13. 证明:对于任意实数a和b,不等式(a + b)² ≤ 2(a² + b²)总是成立。
五、综合题(每题25分,共25分)14. 一个圆的半径为r,圆心为O。
在圆上任取两点A和B,连接OA和OB,并延长交圆于点C和D。
丘成桐大学生数学竞赛试卷

丘成桐大学生数学竞赛试卷一、选择题1. 下列哪个数是一个有理数?A) √2B) πC) eD) 1/32. 若一条线段的两个端点分别是(-1,-2)和(3,4),该线段的长度是多少?A) √36B) √50C) 5D) 73. 已知函数f(x) = 2x^3 + 3x^2 - 4x + 5,求f(-2)的值是多少?A) -3B) -5C) -27D) 94. 如果(a + b)^2 = 25,且a - b = 6,那么a的值是多少?A) 3B) 2C) 4D) 55. 若正方形ABCD的边长为5 cm,以BC为直径的圆的面积是多少?A) 2.5π cm^2B) 12.5π cm^2C) 25π cm^2D) 10π cm^2二、填空题6. 在方程3x - 7 = 2x + 5中,x的解是____。
7. 若函数f(x) = 2x^2 - 3x + k,若f(1) = 5,则k的值为____。
8. 若正方形ABCD的边长为a,以AC为直径的圆的周长为____。
9. 若三角形的两个内角为60°和90°,则第三个内角的度数为____。
10. 已知a是一个整数,满足a^2 = 16,那么a的值是____。
三、解答题11. 如果有一个数字,将其4次开方后得到的结果为2,求该数字。
(请写下完整的解题过程)12. 设函数f(x) = 2x^2 + 3x - 5,求f(x)在x = 2处的导数。
(请写下完整的解题过程)13. 已知直径为10 cm的圆,求其周长和面积。
(请写下完整的解题过程)14. 计算:2 + 4 + 6 + 8 + ... + 100 的和。
(请写下完整的解题过程)15. 在一个矩形的长和宽之比为3:2的情况下,如果矩形的长为12 cm,求其宽是多少?(请写下完整的解题过程)四、解答题16. 已知倒数第二届丘成桐大学生数学竞赛中,两位学生A和B的得分为90分和85分。
如果A的得分是B的得分的a倍,求a的值。
丘成桐大学数学竞赛试题及解答

丘成桐大学数学竞赛试题及解答一、选择题1. 某数列的前5项为1, 4, 7, 10, 13,求第100项的值。
解答:根据题意可知,该数列的公差为3,第100项的值可以由以下公式得出:a_n = a_1 + (n-1)d其中,a_n 表示第n项的值,a_1 表示首项的值,d 表示公差。
代入已知条件,即可求出第100项的值:a_100 = 1 + (100-1)3= 1 + 99*3= 298所以,第100项的值为298。
2. 设函数 f(x) = x^2 + 2x + 1,求 f(3) 的值。
解答:将 x = 3 代入函数 f(x) 中,即可求得 f(3) 的值:f(3) = 3^2 + 2*3 + 1= 9 + 6 + 1= 16所以,f(3) 的值为16。
二、填空题1. 已知 a + b = 5,且 a - b = 3,求 a 和 b 的值。
解答:将两个等式相加得到:(a + b) + (a - b) = 5 + 32a = 8a = 4将 a 的值代入第一个等式可求得 b 的值:4 + b = 5b = 1所以,a 的值为4,b 的值为1。
2. 若 a:b = 2:3,且 a + b = 25,求 a 和 b 的值。
解答:设 a 的倍数为 2x,b 的倍数为 3x,则有:2x + 3x = 255x = 25x = 5将 x 的值代入 a:b 的比例中,可求得 a 和 b 的值:a = 2 * 5 = 10b = 3 * 5 = 15所以,a 的值为10,b 的值为15。
三、解答题1. 求解下列方程组:2x + y = 53x + 2y = 8解答:可以通过消元法解决这个方程组。
首先将第一个方程乘以2,得到:4x + 2y = 10然后将第二个方程减去第一个方程,消去 y 的系数:4x + 2y - (3x + 2y) = 10 - 84x + 2y - 3x - 2y = 2x = 2将 x 的值代入第一个方程,可求得 y 的值:2*2 + y = 54 + y = 5y = 1所以,该方程组的解为 x = 2,y = 1。
清华大学新领军2023年8月学科综合能力测试(TACA)丘成桐数学零试试题及详细解析

2023年8月学科能力综合测试(TACA)丘成桐数学零试试题题1.在如图所示的方格网中,每步只能从一个节点向右或向上走到相邻的节点,则从A 到Z 的路径数为.题2.对一个复方阵,定义cos A =I +∞∑n =1(−1)n (2n )!A 2n .记矩阵cos (−5π3π−10π6π)的第二行第一列元素为x ,则[|x |]=.题3.I =lim n →∞∫20231cos 4(nx +n !)d x ,则[100I ]=.题4.设g (x )=∫πx (sin t )5t d t ,记S =∫π0g (x )d x ,则[100S ]=.题5.n ×n 矩阵A n 的主对角元素为n ,其余元素为1.已知多项式f (x )满足f (A n )=O 对任意1 n 100均成立,则deg f 的最小值为.题6.矩阵10−9−9−910−9910991099109−910−9−910−9−9−910的特征值为λ1 λ2 ··· λ8,则[λ6]=.题7.在透明的球袋中有1个黑球和2个白球.接下来从袋中摸球,每次摸出一个球,然后放回袋中,并继续往袋中再放入2个与摸出球同色的球.记S n为前n次中摸出黑球的次数.已知第2次摸出的球是黑球,记S100的期望为E,则[E]=.题8.对矩阵M(t),定义其微分dd t M(t)=B(t),其中dd tM ij(t)=B ij(t).矩阵微分方程dd tM(t)=AM(t),其中A=213132321,M(0)=321213132,令a=ln|det M(2)|,则[|a|]=.题9.5级方阵A满足:第一行元素均为1,第二行元素从左至右依次为1,−1,1,−1,1,且trace(AA′)=28.记det A的最大值为M,则[M]=.题10.对n=5,令L n=1n!d nd x n(x2−1)n,S=∫1−1|L n|2d x,则[|S|]=.题11.正方体旋转群元素最多的两个共轭类的元素个数之和为.题12.A(x)=∞∑m=1∞∑n=0m!·x m·n!(m+n+1)!,则[100A(23)]=.题13.将方程(1+2+···+k)−3(1+2+···+t)=1的全体正整数解(k,t)按照从小到大的方式排列,得到序列{(k n,t n)},则k6=.题14.I=∫π/4tan101x d x,则[13I].题15.M n={A|A是n级实对称矩阵,且元素取自0,±1,±2},记a n为所有trace(A6)(A∈M n)的平均值,a=limk→∞sup n k a nn4,则[a]=.2023年8月学科能力综合测试(TACA)丘成桐数学零试试题及解析1题1.在如图所示的方格网中,每步只能从一个节点向右或向上走到相邻的节点,则从A 到Z 的路径数为.解析:54如图所示,每个节点处的红色数字表示从A 到该点的路径数.题2.对一个复方阵,定义cos A=I+∞∑n=1(−1)n(2n)!A2n.记矩阵cos(−5π3π−10π6π)的第二行第一列元素为x,则[|x|]=.解析:20令M=(−5π3π−10π6π),易知M的特征多项式为f(λ)=λ(λ−π),故其特征值为0,π.解出对应特征向量后,易知M=(3152)(π)(3152)−1于是cos M=(3152)(1−1)(3152)−1=(11−620−11)故所求答案为20.题3.I=limn→∞∫20231cos4(nx+n!)d x,则[100I]=.解析:758I=limn→∞∫20231cos4(nx+n!)d x,则[100I]=我们令I n=∫20231cos4(nx+n!)d x。
第二届全国大学生数学竞赛预赛试卷参考答案(非数学类)

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丘成桐少年班题目

丘成桐少年班题目篇一:丘成桐少年班是丘成桐教授创立的一个专门针对 13 岁到 18 岁之间的年轻数学天才的培养计划。
这个计划旨在发掘和培养有潜力的年轻数学天才,帮助他们进一步发展和成就自己的数学事业。
丘成桐少年班的题目非常困难,要求考生具备扎实的数学基础和深厚的数学思维能力。
这些题目往往需要考生进行深入的思考和推导,同时也需要考生具备良好的数学方法和技巧。
下面是一些丘成桐少年班的经典题目:1. 如果一个三角形的两边长分别为 3 和 4,那么这个三角形的第三边长可能为多少?2. 如果一个圆的半径为 3,那么这个圆的直径可能为多少?3. 如果一个正多边形的每一个顶点都沿着一条边移动,那么这个正多边形的面积将如何变化?4. 如果一个方程的解为 x=3,那么这个方程的另一个解为多少?这些题目都非常具有挑战性,需要考生具备扎实的数学基础和强大的数学思维能力才能解决。
同时,这些题目也可以帮助考生更好地理解数学的本质和美感。
除了丘成桐少年班的题目,还有很多其他有趣的数学问题和挑战,例如: 1. 一个巨大的冰块被切成了若干块,每块冰块的表面积都是原来的 2 倍,那么这个冰块最终需要多少时间才能融化完毕?2. 一个小学生想要测量一条河流的长度,他使用了一根长为 10 米的绳子,但是这条河流非常弯曲,请问这个小学生应该如何测量河流的长度?3. 如果一个足球的表面积为 25 平方米,那么这个足球的表面积和体积之间有什么关系?这些问题都充满了数学的魅力和趣味,可以帮助考生更好地理解数学的本质和应用。
篇二:丘成桐少年班是中国著名的数学培训班,旨在培养青少年数学人才。
这个培训班提出了许多有趣的数学问题,吸引了许多对数学感兴趣的年轻人。
以下是一些问题:1. 一只羊是另一只羊的几倍?在丘成桐少年班中,一位学生提出了一个问题:如果一只羊是另一只羊的 2 倍,那么两只羊的数量是否相等?这个问题看似简单,但实际上具有很高的难度。
2. 有多少个骰子?在这个问题中,我们需要在一个 6 面的骰子上放置多少个骰子,才能使它们的总和为偶数?这个问题的解法需要一定的技巧和思考能力。
丘成桐高考数学试卷

考试时间:150分钟满分:150分一、选择题(每小题5分,共50分)1. 下列函数中,在x=0处连续的是:A. f(x) = x²B. f(x) = |x|C. f(x) = x² - xD. f(x) = (x² - 1) / (x - 1)2. 设a > 0,b > 0,且a + b = 1,则下列不等式中恒成立的是:A. a² + b² ≥ 2B. a² + b² ≤ 2C. a³ + b³ ≥ 2D. a³ + b³ ≤ 23. 函数y = log₂x的图像上,斜率为-1的切线方程是:A. y = -xB. y = xC. y = 2xD. y = -2x4. 已知等差数列{an}的前n项和为Sn,若S10 = 55,S20 = 145,则第10项a10的值为:A. 5B. 6C. 7D. 85. 在平面直角坐标系中,点P的坐标为(2, -3),点Q在直线y = x上,且PQ的中点坐标为(1, 1),则点Q的坐标为:A. (0, 0)B. (1, 1)C. (2, 2)D. (3, 3)6. 若复数z满足|z - 1| = |z + 1|,则复数z的实部为:A. 0B. 1C. -1D. 27. 设函数f(x) = x³ - 3x² + 4x - 1,则f'(1)的值为:A. 1B. 2C. 3D. 48. 在△ABC中,角A、B、C的对边分别为a、b、c,若a² + b² - c² = ab,则角C的大小为:A. 30°B. 45°C. 60°D. 90°9. 已知函数f(x) = ax² + bx + c(a ≠ 0),若f(-1) = 0,f(1) = 2,且f(x)的图像开口向上,则a的取值范围是:A. a > 0B. a < 0C. a = 0D. a ≠ 010. 设集合A = {x | x² - 4x + 3 < 0},集合B = {x | x² + 4x + 3 > 0},则集合A ∩ B是:A. {x | -1 < x < 3}B. {x | x < -1 或 x > 3}C. {x | -1 ≤ x ≤ 3}D. {x | x ≤ -1 或x ≥ 3}二、填空题(每小题5分,共25分)11. 函数y = sin(x + π/6)的最小正周期为______。
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S.-T.Yau College Student Mathematics Contests 2011Analysis and Differential EquationsIndividual2:30–5:00pm,July 9,2011(Please select 5problems to solve)1.a)Compute the integral: ∞−∞x cos xdx (x 2+1)(x 2+2),b)Show that there is a continuous function f :[0,+∞)→(−∞,+∞)such that f ≡0and f (4x )=f (2x )+f (x ).2.Solve the following problem: d 2u dx 2−u (x )=4e −x ,x ∈(0,1),u (0)=0,dudx(0)=0.3.Find an explicit conformal transformation of an open set U ={|z |>1}\(−∞,−1]to the unit disc.4.Assume f ∈C 2[a,b ]satisfying |f (x )|≤A,|f(x )|≤B for each x ∈[a,b ]and there exists x 0∈[a,b ]such that |f (x 0)|≤D ,then |f (x )|≤2√AB +D,∀x ∈[a,b ].5.Let C ([0,1])denote the Banach space of real valued continuous functions on [0,1]with the sup norm,and suppose that X ⊂C ([0,1])is a dense linear subspace.Suppose l :X →R is a linear map (not assumed to be continuous in any sense)such that l (f )≥0if f ∈X and f ≥0.Show that there is a unique Borel measure µon [0,1]such that l (f )= fdµfor all f ∈X .6.For s ≥0,let H s (T )be the space of L 2functions f on the circle T =R /(2πZ )whose Fourier coefficients ˆf n = 2π0e−inx f (x )dx satisfy Σ(1+n 2)s ||ˆf n |2<∞,with norm ||f ||2s =(2π)−1Σ(1+n 2)s |ˆf n |2.a.Show that for r >s ≥0,the inclusion map i :H r (T )→H s (T )is compact.b.Show that if s >1/2,then H s (T )includes continuously into C (T ),the space of continuous functions on T ,and the inclusion map is compact.1S.-T.Yau College Student Mathematics Contests2011Geometry and TopologyIndividual9:30–12:00am,July10,2011(Please select5problems to solve)1.Suppose M is a closed smooth n-manifold.a)Does there always exist a smooth map f:M→S n from M into the n-sphere,such that f is essential(i.e.f is not homotopic to a constant map)?Justify your answer.b)Same question,replacing S n by the n-torus T n.2.Suppose(X,d)is a compact metric space and f:X→X is a map so that d(f(x),f(y))=d(x,y)for all x,y in X.Show that f is an onto map.3.Let C1,C2be two linked circles in R3.Show that C1cannot be homotopic to a point in R3\C2.4.Let M=R2/Z2be the two dimensional torus,L the line3x=7y in R2,and S=π(L)⊂M whereπ:R2→M is the projection map. Find a differential form on M which represents the Poincar´e dual of S.5.A regular curve C in R3is called a Bertrand Curve,if there existsa diffeomorphism f:C→D from C onto a different regular curve D in R3such that N x C=N f(x)D for any x∈C.Here N x C denotes the principal normal line of the curve C passing through x,and T x C will denote the tangent line of C at x.Prove that:a)The distance|x−f(x)|is constant for x∈C;and the angle made between the directions of the two tangent lines T x C and T f(x)D is also constant.b)If the curvature k and torsionτof C are nowhere zero,then there must be constantsλandµsuch thatλk+µτ=16.Let M be the closed surface generated by carrying a small circle with radius r around a closed curve C embedded in R3such that the center moves along C and the circle is in the normal plane to C at each point.Prove thatMH2dσ≥2π2,and the equality holds if and only if C is a circle with radius √2r.HereH is the mean curvature of M and dσis the area element of M.1S.-T.Yau College Student Mathematics Contests 2011Algebra,Number Theory andCombinatoricsIndividual2:30–5:00pm,July 10,2011(Please select 5problems to solve)For the following problems,every example and statement must be backed up by proof.Examples and statements without proof will re-ceive no-credit.1.Let K =Q (√−3),an imaginary quadratic field.(a)Does there exists a finite Galois extension L/Q which containsK such that Gal(L/Q )∼=S 3?(Here S 3is the symmetric group in 3letters.)(b)Does there exists a finite Galois extension L/Q which containsK such that Gal(L/Q )∼=Z /4Z ?(c)Does there exists a finite Galois extension L/Q which containsK such that Gal(L/Q )∼=Q ?Here Q is the quaternion group with 8elements {±1,±i,±j,±k },a finite subgroup of the group of units H ×of the ring H of all Hamiltonian quaternions.2.Let f be a two-dimensional (complex)representation of a finite group G such that 1is an eigenvalue of f (σ)for every σ∈G .Prove that f is a direct sum of two one-dimensional representations of G3.Let F ⊂R be the subset of all real numbers that are roots of monic polynomials f (X )∈Q [X ].(1)Show that F is a field.(2)Show that the only field automorphisms of F are the identityautomorphism α(x )=x for all x ∈F .4.Let V be a finite-dimensional vector space over R and T :V →V be a linear transformation such that(1)the minimal polynomial of T is irreducible;(2)there exists a vector v ∈V such that {T i v |i ≥0}spans V .Show that V contains no non-trivial proper T -invariant subspace.5.Given a commutative diagramA →B →C →D →E↓↓↓↓↓A →B →C →D →E1Algebra,Number Theory and Combinatorics,2011-Individual2 of Abelian groups,such that(i)both rows are exact sequences and(ii) every vertical map,except the middle one,is an isomorphism.Show that the middle map C→C is also an isomorphism.6.Prove that a group of order150is not simple.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?S.-T.Yau College Student Mathematics Contests 2011Analysis and Differential EquationsTeam9:00–12:00am,July 9,2011(Please select 5problems to solve)1.Let H 2(∆)be the space of holomorphic functions in the unit disk ∆={|z |<1}such that ∆|f |2|dz |2<∞.Prove that H 2(∆)is a Hilbert space and that for any r <1,the map T :H 2(∆)→H 2(∆)given by T f (z ):=f (rz )is a compact operator.2.For any continuous function f (z )of period 1,show that the equation dϕdt=2πϕ+f (t )has a unique solution of period 1.3.Let h (x )be a C ∞function on the real line R .Find a C ∞function u (x,y )on an open subset of R containing the x -axis such that u x +2u y =u 2and u (x,0)=h (x ).4.Let S ={x ∈R ||x −p |≤c/q 3,for all p,q ∈Z ,q >0,c >0},show that S is uncountable and its measure is zero.5.Let sl (n )denote the set of all n ×n real matrices with trace equal to zero and let SL (n )be the set of all n ×n real matrices with deter-minant equal to one.Let ϕ(z )be a real analytic function defined in a neighborhood of z =0of the complex plane C satisfying the conditions ϕ(0)=1and ϕ (0)=1.(a)If ϕmaps any near zero matrix in sl (n )into SL (n )for some n ≥3,show that ϕ(z )=exp(z ).(b)Is the conclusion of (a)still true in the case n =2?If it is true,prove it.If not,give a counterexample.e mathematical analysis to show that:(a)e and πare irrational numbers;(b)e and πare also transcendental numbers.1S.-T.Yau College Student Mathematics Contests2011Applied Math.,Computational Math.,Probability and StatisticsTeam9:00–12:00am,July9,2011(Please select5problems to solve)1.Let A be an N-by-N symmetric positive definite matrix.The con-jugate gradient method can be described as follows:r0=b−A x0,p0=r0,x0=0FOR n=0,1,...αn= r n 22/(p TnA p n)x n+1=x n+αn p n r n+1=r n−αn A p nβn=−r Tk+1A p k/p TkA p kp n+1=r n+1+βn p nEND FORShow(a)αn minimizes f(x n+αp n)for allα∈R wheref(x)≡12x T A x−b T x.(b)p Ti r n=0for i<n and p TiA p j=0if i=j.(c)Span{p0,p1,...,p n−1}=Span{r0,r1,...,r n−1}≡K n.(d)r n is orthogonal to K n.2.We use the following scheme to solve the PDE u t+u x=0:u n+1 j =au nj−2+bu nj−1+cu njwhere a,b,c are constants which may depend on the CFL numberλ=∆t ∆x .Here x j=j∆x,t n=n∆t and u njis the numerical approximationto the exact solution u(x j,t n),with periodic boundary conditions.(i)Find a,b,c so that the scheme is second order accurate.(ii)Verify that the scheme you derived in Part(i)is exact(i.e.u nj =u(x j,t n))ifλ=1orλ=2.Does this imply that the scheme is stable forλ≤2?If not,findλ0such that the scheme is stable forλ≤λ0. Recall that a scheme is stable if there exist constants M and C,which are independent of the mesh sizes∆x and∆t,such thatu n ≤Me CT u0for all∆x,∆t and n such that t n≤T.You can use either the L∞norm or the L2norm to prove stability.1Applied Math.Prob.Stat.,2011-Team2 3.Let X and Y be independent random variables,identically dis-tributed according to the Normal distribution with mean0and variance 1,N(0,1).(a)Find the joint probability density function of(R,),whereR=(X2+Y2)1/2andθ=arctan(Y/X).(b)Are R andθindependent?Why,or why not?(c)Find a function U of R which has the uniform distribution on(0,1),Unif(0,1).(d)Find a function V ofθwhich is distributed as Unif(0,1).(e)Show how to transform two independent observations U and Vfrom Unif(0,1)into two independent observations X,Y fromN(0,1).4.Let X be a random variable such that E[|X|]<∞.Show thatE[|X−a|]=infE[|X−x|],x∈Rif and only if a is a median of X.5.Let Y1,...,Y n be iid observations from the distribution f(x−θ), whereθis unknown and f()is probability density function symmetric about zero.Suppose a priori thatθhas the improper priorθ∼Lebesgue(flat) on(−∞,∞).Write down the posterior distribution ofθ.Provides some arguments to show that thisflat prior is noninforma-tive.Show that with the posterior distribution in(a),a95%probability interval is also a95%confidence interval.6.Suppose we have two independent random samples{Y1,i=1,...,n} from Poisson with(unknown)meanλ1and{Y i,i=n+1,...,2n}from Poisson with(unknown)meanλ2Letθ=λ1/(λ1+λ2).(a)Find an unbiased estimator ofθ(b)Does your estimator have the minimum variance among all un-biased estimators?If yes,prove it.If not,find one that has theminimum variance(and prove it).(c)Does the unbiased minimum variance estimator you found at-tain the Fisher information bound?If yes,show it.If no,whynot?S.-T.Yau College Student Mathematics Contests2011Geometry and TopologyTeam9:00–12:00am,July9,2011(Please select5problems to solve)1.Suppose K is afinite connected simplicial complex.True or false:a)Ifπ1(K)isfinite,then the universal cover of K is compact.b)If the universal cover of K is compact thenπ1(K)isfinite.pute all homology groups of the the m-skeleton of an n-simplex, 0≤m≤n.3.Let M be an n-dimensional compact oriented Riemannian manifold with boundary and X a smooth vectorfield on M.If n is the inward unit normal vector of the boundary,show thatM div(X)dV M=∂MX·n dV∂M.4.Let F k(M)be the space of all C∞k-forms on a differentiable man-ifold M.Suppose U and V are open subsets of M.a)Explain carefully how the usual exact sequence0−→F(U∪V)−→F(U)⊕F V)−→F(U∩V)−→0 arises.b)Write down the“long exact sequence”in de Rham cohomology as-sociated to the short exact sequence in part(a)and describe explicitly how the mapH kdeR (U∩V)−→H k+1deR(U∪V)arises.5.Let M be a Riemannian n-manifold.Show that the scalar curvature R(p)at p∈M is given byR(p)=1vol(S n−1)S n−1Ric p(x)dS n−1,where Ric p(x)is the Ricci curvature in direction x∈S n−1⊂T p M, vol(S n−1)is the volume of S n−1and dS n−1is the volume element of S n−1.1Geometry and Topology,2011-Team2 6.Prove the Schur’s Lemma:If on a Riemannian manifold of dimension at least three,the Ricci curvature depends only on the base point but not on the tangent direction,then the Ricci curvature must be constant everywhere,i.e.,the manifold is Einstein.S.-T.Yau College Student Mathematics Contests 2011Algebra,Number Theory andCombinatoricsTeam9:00–12:00pm,July 9,2011(Please select 5problems to solve)For the following problems,every example and statement must be backed up by proof.Examples and statements without proof will re-ceive no-credit.1.Let F be a field and ¯Fthe algebraic closure of F .Let f (x,y )and g (x,y )be polynomials in F [x,y ]such that g .c .d .(f,g )=1in F [x,y ].Show that there are only finitely many (a,b )∈¯F×2such that f (a,b )=g (a,b )=0.Can you generalize this to the cases of more than two-variables?2.Let D be a PID,and D n the free module of rank n over D .Then any submodule of D n is a free module of rank m ≤n .3.Identify pairs of integers n =m ∈Z +such that the quotient rings Z [x,y ]/(x 2−y n )∼=Z [x,y ]/(x 2−y m );and identify pairs of integers n =m ∈Z +such that Z [x,y ]/(x 2−y n )∼=Z [x,y ]/(x 2−y m ).4.Is it possible to find an integer n >1such that the sum1+12+13+14+ (1)is an integer?5.Recall that F 7is the finite field with 7elements,and GL 3(F 7)is the group of all invertible 3×3matrices with entries in F 7.(a)Find a 7-Sylow subgroup P 7of GL 3(F 7).(b)Determine the normalizer subgroup N of the 7-Sylow subgroupyou found in (a).(c)Find a 2-Sylow subgroup of GL 3(F 7).6.For a ring R ,let SL 2(R )denote the group of invertible 2×2matrices.Show that SL 2(Z )is generated by T = 1101 and S = 01−10 .What about SL 2(R )?1。