【2010-2018丘成桐大学生数学竞赛笔试真题】4.Algebra-Individual-2011
2010年全国大学生高等数学竞赛总决赛(答案)
2010年全国大学生数学竞赛决赛答 tian27546这是献给博士论坛一个礼物 转载时请勿注明是博士论坛一、（20分）计算下列各题：1．求极限 211sin )1(lim n k n k n k n π∑-=→∞+解法1因211sin )1(n k n k n k π∑-=+211222sin sin 21(2sin 21n n k n k nn k πππ∑-=+=） )22cos 22(cos 1(2sin 2122112n k n k n k nn k πππππ+--+=∑-=） )22cos 22(cos 1(22112nk n k n k n n k πππππ+--+≈∑-=） 2112211222cos 1(22cos 1(n k nk n n k n k n n k n k ππππππ++--+=∑∑-=-=）） 222211222cos 11(22cos 1(n k n k n n k n k n nk n k ππππππ--+--+=∑∑=-=））2122222222cos 12)12(cos 11(2cos )11(n k n n n n n n n n n n n k πππππππ-+--+-+=∑-=） 21222222)12(cos 2)12(cos 12(2cos )11(nk n n n n n n n n n k ππππππ-+---+=∑-=）（*） 而2122)12(cos n k n k π-∑-=212222sin 2)12(cos22sin 21n n k nn k πππ∑-=-=])1(sin [sin2sin2121222n k n k nn k πππ--=∑-= 2222sin 2sin )1(sinn n n n πππ--=222sin2)2(sin 2cos n n n n πππ-=（**） 将（**）代入（*），然后取极限，得原式]2sin2)2(sin2cos2)12(cos 12(2cos )11([lim 222222n n n nn n n n n n n n n ππππππππ-+---+=→∞）]2)2(sin 2cos 2)8)12(1(12()11([lim 22342222n n n n n n n n n n n ππππππ-+----+=∞→） ]2)2(sin 2cos 2)21(12()11([lim 2232222n n n n n n n n n n ππππππ-+---+=∞→） )]48)2(2)2()(81(2)21(12()11([lim 633222232222nn n n n n n n n n n n πππππππ----+---+=∞→）)]482)(81(2)21(12()11([lim 33222232222n n n n n n n n n n n ππππππππ---+---+=∞→） 65π=上式中含2n 的项的系数为0121=+-πππ，含n 的项的系数为0)2(111=-++πππ，常数项系数为656824ππππππ=-=--解法2 Step 1因∑-=112sin n k n k π211222sinsin 22sin 21n nk nn k πππ∑-==)22cos 22(cos2sin2122112n k n k nn k πππππ+--=∑-=)2)12(cos2(cos2sin21222n n n n πππ--=故)2)12(cos 2(cos 2sin 21lim sinlim 222112n n n nn k n n k n ππππ--=→∞-=→∞∑)2)12(cos2(cos1lim222n n n n n πππ--=→∞nn n n n 2sin 2)1(sin2lim22πππ-=→∞n n n n n 22)1(2lim22πππ-=∞→2π= Step 2因222)12(cosn k nk π-∑=22222sin 2)12(cos22sin21n n k nnk πππ∑=-=])1(sin [sin2sin212222nk n k nnk πππ--=∑= 2222sin 2sinsin n n n n πππ-=2222sin 2)1(sin 2)1(cos nn n n n πππ-+=因此∑-=112sin n k n k nk π211222sin sin 22sin 21n n k n k n n k πππ∑-== ]2)12(cos 2)12(cos [2sin 212112112n k n k n k n k nn k n k πππ+--=∑∑-=-= ]2)12(cos 12)12(cos [2sin 21222112n k n k n k n k nnk n k πππ----=∑∑=-=⎥⎦⎤⎢⎣⎡-+---=∑-=2122222)12(cos 12)12(cos 12cos 12sin 21n k n n n n n n n nn k ππππ ⎥⎦⎤⎢⎣⎡-+--=∑=222222)12(cos 12)12(cos 2cos 12sin 21n k n n n n nnnk ππππ(*) ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-++--=2222222sin 2)1(sin 2)1(cos 2)12(cos 2cos 12sin 21nn n n n n n n n n n ππππππ 于是∑-=→∞112sin lim n k n n k nk π⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-++--=→∞2222222sin 2)1(sin 2)1(cos 2)12(cos 2cos 12sin 21lim nn n n n n n n n n n n ππππππ ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡-++---=→∞n n n n n n n n n n 22)1(sin2)1(cos 8)12(11lim 224222πππππ）（ ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡---+-++-=∞→n n n n n n n n n n n 2)48)1(2)1()(8)1(1211lim 6332422222ππππππ（⎥⎦⎤⎢⎣⎡----++-=∞→)24)1(1)(81211lim 52322222n n n n n n n n n ππππ（ ⎥⎦⎤⎢⎣⎡---++-=∞→)241()(81211lim 2222222n n n n n n n n ππππ（ ⎥⎦⎤⎢⎣⎡---++-=∞→)2411)(81211lim 2222222n n n n n n n ππππ（ ）（222222282411211lim n n n n n n n ππππ---++-=→∞ ）（22222228242lim n n n n n ππππ--=∞→62ππ-=3π=原式6532πππ=+=2．计算⎰⎰∑++++2222)(zy x dxdya z axdydz ，其中 ∑为下半球面222y x a z ---= 的上侧, 0>a .解 记1∑为平面 222,0a y x z ≤+= 的上侧，2∑为下半球面 222y x a z ---= 的下侧，Ω是由1∑和2∑所围成的立体，则422222211)(adxdy a dxdy a dxdy a z axdydz ay x ⎰⎰⎰⎰⎰⎰≤+∑∑===++π，设,sin ,cos θθr y r x ==则⎰⎰∑+∑++212)(dxdy a z axdydz ⎰⎰⎰Ω+++=dxdydz a z a )220(⎰⎰⎰Ω+=dxdydz a z )32(⎰⎰⎰≤+---+=2222220)32(a y x y x a dz a z dxdy⎰⎰≤+---+=22222202]3[a y x y x a dxdy az z⎰⎰≤+--+++-=222)3(222222a y x dxdy y x a a y x a ⎰⎰≤≤≤≤-++-=πθθ2002222d d )3(ar r r r a a r a⎰-++-=a r r r a a r a 02222d )3(2π ⎰-++-=ar r a a r a 022222)d()3(π⎰-++-=22122d ))(3(a u u a a u a π223222)(42a u a a uu a ⎥⎦⎤⎢⎣⎡--+-=π274a π=⎰⎰∑++++2222)(zy x dxdya z axdydz⎰⎰⎰⎰∑∑+∑+++++-=12122)(1)(1dxdy a z axdydz a dxdy a z axdydz a 227333a a a πππ-=+-=3．现 设计一个容积为V 的圆柱体容器. 已知上下两底的材料费为单位面积a元，而侧面的材料费为单位面积b 元. 试给出最节省的设计方案；即高与的上下底直径之比为何值时所需费用最少？解 设圆柱体的底半径为r ，高为h ，则h r V 2π=，2rVh π=总造价为222r a rh b P ππ+=222r a rbVπ+=， 则2322242r r a bV r a r bV P ππ--=+-='，由0='P 知，解得312⎪⎭⎫⎝⎛=πa bV r ，312⎪⎭⎫ ⎝⎛=ππa bV V h ， 因为是惟一的驻点，所以当3122323131222222:2⎪⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛=Vab a bV V a bV a bV V h r ππππππ 时，所需费用最少.4．已知 x x x f 33cos sin 1)(+='，)21,41(∈x ，求)(x f 解 因x x x f 33cos sin 1)(+='，)21,41(∈x ，故 ⎰+=x xx x f d cos sin 1)(33⎰+-+=x x x x x x x d )cos )(sin cos sin cos (sin 122⎰+-=x x x x x d )cos )(sin cos sin 1(1⎰+-=x x x d )4sin()2sin 211(21π⎰+⎪⎭⎫ ⎝⎛++=x x x d )4sin()22cos(211121ππ⎰+⎪⎭⎫ ⎝⎛++=x x x d )4sin()4(2cos 211121ππ 令)4(21π+=x t ，则⎰+=t tt x f d 2sin )4cos 211(2)(⎰+=t tt t d cos sin )4cos 2(2⎰-+=t t t t t d cos sin )2sin 2cos 2(222⎰+=t t t t t d cos sin )2sin 2cos 3(222 ⎰+-=t tt t t t t d cos sin )cos sin 4)sin (cos 3(222222⎰-++=t t t t t t t t t d cos sin )cos sin 2sin 3cos 3()cos (sin 22244222 ⎰-+++=t t t t t t t tt t t d cos sin )cos sin 2sin 3cos 3(cos sin 2sin cos 222442244⎰-+++=t t t t tt tan d tan )tan 2tan 33(tan 2tan 122424 令t u tan =，2u v =，则⎰-+++=u u u u u u x f d )233(212)(2424⎰-+++=224224d )233(2122u u u u u u ⎰-+++=v v v v v v d )233(212222⎰+-++=v v v v v v d )323(122222 令)()323(1222v R vAv v v v v +=+-++，则31=A ，)323(332336331)323(12)(22222+--+-++=-+-++=v v v v v v v v v v v v v v R )323(382+-=v v 因此⎰⎰+-+=323d 324d 62)(2v v vv v x f ⎰+-+=323d 324ln 622v v vv ⎰+-+=98)31(d 924ln 622v v v C v v +-+=32231arctan 3221924ln 62C v v +-+=2213arctan 32ln 62 C t t +-+=221tan 3arctan 32tan ln 6222C t t +-+=221tan 3arctan 32tan ln 6222C x x +-+++=221)82(tan 3arctan 32)82(tan ln 6222ππ 二、（10分）求下列极限1．⎪⎭⎫ ⎝⎛-+∞→e n n n n )11(lim解 设xx x f 1)1()(+=, 则))1ln()1(1()1()(21xx x x x x f x+-++=')1()1ln()1()(2x x x x x x f +++-= 原式=)(lim )1(lim010x f x e x x xx '=-+→→)()(lim )(lim 00x f x f x f x x '=→→)1()1ln()1(lim)(lim 20x x x x x x f x x +++-=→→20)1ln()1(limx x x x e x ++-=→22)1ln(lim 0e x x e x -=+-=→2．nnn n n c b a ⎪⎪⎪⎭⎫⎝⎛++∞→3lim 111，其中0>a ，0>b ，0>c 解 因300ln 3ln ln ln 3ln ln ln lim 33lim abc c b a c c b b a a x c b a x x x x x x x x =++=++=-++→→ 故 原式=333lim)13(1lim 10003lim abc ee c b a x c b a c b axxxx x x x x x x xx xx ===⎪⎪⎭⎫⎝⎛++-++-++→→→三、（10分）设)(x f 在1=x 处可导，0)1(=f ，2)1(='f ，求xx x x x f x tan )cos (sin lim 220++→ 解 设)(x f 在1=x 处可导，0)1(=f ，2)1(='f ，则xx x f x x f x x x x x f x x tan )1()cos (sin lim tan )cos (sin lim 220220+-+=++→→ 1cos sin )1()cos (sin lim 1cos sin lim tan lim 220220220-+-+-++=→→→x x f x x f x x x x x x x x x x 1cos sin )1()cos (sin lim 2sin cos sin 2lim cos 111lim220020-+-+-+=→→→x x f x x f x x x x xx x x 1cos sin )1()cos (sin lim 2sin cos sin 2lim 212200-+-+-=→→x x f x x f x x x x x x 1cos sin )1()cos (sin lim 21cos 2lim sin lim 2122000-+-+-=→→→x x f x x f x x x x x x1cos sin )1()cos (sin lim 41220-+-+=→x x f x x f x 1)1()(lim 411--=→t f t f t )1(41f '=21= 四、（10分）设)(x f 在),0[+∞上连续，⎰+∞0d )(x x f 收敛，求⎰+∞→yy x x xf y 0d )(1lim.解 令⎰=xt t f x G 0d )()(，则因⎰+∞0d )(x x f 收敛，故)(lim y G y +∞→，不妨设R A y G y ∈=+∞→)(lim ，则[]}d )()(1{lim )(d 1lim d )(1lim0000⎰⎰⎰-==+∞→+∞→+∞→y yy y y y y x x G x xG yx G x y x x xf y)d )(1)((lim 0⎰-=+∞→yy x x G yy G ⎰+∞→-=yy x x G y A 0d )(1lim 0)(lim =-=-=+∞→A A y G A y五、（12分）设函数)(x f 在]1,0[上连续，在)1,0(内可微，且0)1()0(==f f ，1)21(=f ，证明：（1）存在⎪⎭⎫⎝⎛∈1,21ξ使得ξξ=)(f ；（2）存在()ξη,0∈使得1)()(+-='ηηηf f .证 （1）记x x f x F -=)()(，则函数)(x F 在]1,21[上连续，且1)1(-=F ，21)21(=F ，故由零点存在性定理知存在⎪⎭⎫⎝⎛∈1,21ξ使得0)(=ξF ，即ξξ=)(f . （2）因x x x f x f e x d )1)()((⎰+-'--x e x xe x x f e x x f e x x x x d d d )(d )(⎰⎰⎰⎰----+-'-= x e e x x f e x x f e x x x x d d )(d d )(⎰⎰⎰⎰----++-=x x xe x f e --+-=)(故令x e x x f x F --=))(()(, 则函数)(x F 在],0[ξ上连续，在()ξ,0内可微，0)0(=F ,0)(=ξF ,x x e x x f e x f x F -----'='))(()1)(()(, 故由罗尔定理知,存在()ξη,0∈使得0)(='ηF , 1)()(+-='ηηηf f .六、设)(x f 在),(+∞-∞上有定义，在0=x 的某邻域内有一阶连续导数，且0)(lim 0>=→a x x f x ，证明级数∑∞=-1)1()1(n n n f 条件收敛. 证 因 0)(lim>=→a xx f x ，故存在一个正数δ，使得当δ<-<00x 时，有 2)(aa x x f <-因此x x f a )(2<（δ<-<00x ），于是，当δ1>n 时， δ<-<010n ，nn f a 1)1(2<，n a n f 2)1(>，这表明级数∑∞=1)1(n n f 发散，即级数∑∞=-1)1()1(n n n f 发散.下证原级数收敛：由0)(lim0>=→a xx f x 知，0)(lim lim )(lim )0(000====→→→a x x f x x f f x x x ，0)(lim )0()(lim )0(00>==-='→→a xx f x f x f f x x由)(x f 在0=x 的某邻域内有一阶连续导数知，)(lim )0(0x f f a x '='=→，因此存在一个正数η，使得当η<-0x 时，有2)(aa x f <-' 因此)(20x f a '<<（),(ηη-∈x ）. 特别地，)(x f 在),0(η上单调增，于是当η1>n 时，)1()11(n f n f <+，且0)0()1(lim ==∞→f nf .最后由Leibniz 判别法知，原级数收敛.综上可知，原级数条件收敛.六、（14分）设1>n 为整数，⎰⎪⎪⎭⎫ ⎝⎛++++=-x n tt n t t t e x F 02d !!2!11)( ，证明：方程 2)(n x F =在⎪⎭⎫⎝⎛n n ,2内至少有一个根. 证 记!!2!11)(2n t t t t p nn ++++= ,)!!2!11()(2n t t t e t r ntn ++++-= ,则)()(t r e t p n t n -=,且当0>t 时,0)(>t p n , 0)(>t r n ,0)(>-t r e n t .记2)()(n x F x -=ψ，则⎰--=n n t t t r e nx 0d )(2)(ψ,因⎰⎪⎪⎭⎫⎝⎛++++=-x n tt n t t t e x F 02d !!2!11)( ，故函数)(x ψ在],2[n n 上连续，在⎪⎭⎫⎝⎛n n ,2内可微，且2)2()2(n n F n -=ψ⎰⎰<-=--=--20200d )(2d ))(1(nn t n n tt t r e n t t r e ,2d )()(0nt t p e n nn t -=⎰-ψ⎰⎰⎰⎰----+-=+--=202220d )(d )(d )(2d ))(1(n nn n t n t n n n t n n t tt p e t t r e tt p e nt t r e⎰⎰++-=---20202d )2(d )(n n n n t n tt n t p et t r e⎰⎰+++-=---20202d )2(d )!1(1nnn nt t t n t p e t e e n ξ ⎰⎰+-++-=+---202022d ))2((d )!1(1nnn nt nt t t n t r e e t e e n ξ ⎰⎰+---+-+-=202022d )!1(1d )!1(121nnnnt t t e e n t e e n n ξξ ⎰⎰--+-+-=2020d )!1(1d )!1(121n nt t t e e n t e e n n ξξ ⎰-+->202d )!1(22n nt t e e n n []202)!1(22nt ne e n n -++= )1()!1(222-+-=ne n n )!1(2)!1(222+++-=n n e n n )!1(22)!1(2222+-=+->n en n e n n n012>->n（若2>n ，则左边的两个不等式都成立） ()()⎰⎰-+-=-+=-=--101021d 121d 121)1()1(t te t t t e F ψ()[]⎰-++-=--101021d 1t e e t t t 032321)1(2111>-=--+-=--ee e 031)2(>->eψ01223!4223)3(1223144144314923232333>-=->⇒>⇒>>>e e e e ψ 01232452!522)4(2>->->->e e e ψ,0122212e e 12)(>->++->n n n n n e n n ψ 故由零点存在性定理知, 存在),2(n n ∈ξ使得0)(=ξψ, 即2)(nF =ξ.七、（12分）是否存在R 中的可微函数)(x f 使得53421))((x x x x x f f --++=? 若存在，请给出一个例子；若不存在，请给出证明.解 不存在假如存在R 中的可微函数)(x f 使得54321))((x x x x x f f -+-+=，则4325432)))((x x x x x f x f f -+-=''（， 若1)1(=f ，则025432)1))1(()]1[2<-=-+-=''='（（f f f f 矛盾。
【2010-2018丘成桐大学生数学竞赛笔试真题】algebra2018-individual
(c) (8 points) Let v1, . . . , vn, w1, . . . , wn ∈ V . Assume that for every K-linear map f : V → K, (f (v1), . . . , f (vn)) and (f (w1), . . . , f (wn)) coincide up to permutation of the indices. Deduce that (v1, . . . , vn) and (w1, . . . , wn) coincide up to permutation of the indices. Here we make no assumptions on K.
Problem 5 (20 points). Let p be a prime number and let vp(·) denote the p-adic valuation on Qp. Let A = (aij)1≤i,j≤n ∈ Mn(Qp) be an n × n matrix with entries in Qp. Assume that we know the following:
(c) (8 points) Prove the following theorem of Goldbach: for integers i, j ≥ 0 with i = j, the integers 22i + 1 and 22j + 1 are coprime. √
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。
丘成桐数学竞赛2020年笔试真题algebra_and_numbertheory_20
S.-T.Yau College Student Mathematics Contests2020Algebra and Number TheorySolve every problem.Problem1.Let F be afield of characteristic zero.Consider the polynomial ring F[x1,...,x n].(a)Prove Newton’s identity over thefield Fp k−p k−1e1+···+(−1)k−1p1e k−1+(−1)k ke k=0, wheree k(x1,...,x n)=1≤i1<···<ßk≤n x i1···x ikfor1≤k≤n,e0=1,e k=0when k>n,andp k(x1,...,x n)=x k1+···+x k n.(b)Prove that over thefield of F of characteristic zero,an n×n matrix A is nilpotent if and onlyif the trace of A k is equal to zero for all k=1,2...Hint:use Part(a).(c)Prove that over thefield of F of characteristic zero,two n×n matrix A and B have thesame characteristic polynomial if and only if the trace of A k and trace of B k are equal for all k=1,2...Hint:use Part(a).Problem2.(a)Let M be afinitely generated R-module and a⊂R an ideal.Supposeφ:M→M is anR-module map such thatφ(M)⊆a M.Prove that there is a monic polynomial p(t)⊂R[t] with coefficients from a such that p(φ)=0.Hint:p(t)is basically just the characteristic polynomial.(b)If M is afinitely generated R-module such that a M=M for some ideal a⊂R,then there exitsx∈R such that1−x∈a and xM=0.Problem3.Let R=F[x,y]/(y2−x2−x3)for somefield F.(a)Prove that R is an integral domain.(b)Compute the normalization of R(i.e.,the integral closure of R in itsfield of fraction). Problem4.Let p and be two prime numbers and[ x]denote the -th cyclotomic polynomial 1+x+···+x −1.(a)Prove that[ x]is an irreducible element of Q[x].(b)Show that[ x]is divisible by x−1in F p[x]if p= .Here F p is thefinitefield Z/p Z.(c)Suppose p .let a be the order of p in F .Show that a is thefirst value of m for whichthe group GL m(F p)of invertible m×m matrices with entries from F p contains an element of order .Hint:Derive and use the formula for the number of elements in GL m(F p).Problem5.Let p≥3be a prime number and let Z p be the ring of p-adic integers.(a)Show that an element in1+p Z p is a p-th power in Z p if and only if it lives in1+p2Z p.(b)Let Z×p denote the group of units in Z p.Show that there exist a,b,c∈Z×p such that a p+b p=c pif and only ifp−1i=1i p−2t i≡0(mod p)for some integer t∈{2,3,...,p−1}.(In particular,this condition holds for p=7by taking t=3.Therefore,Fermat’s Last Theorem does not hold for Z7.)Problem6.Recall that a metric space is called spherically complete if any decreasing sequence of closed balls has nonempty intersection.Let p be a prime number and let Q p be thefield of p-adic numbers.For every integer n≥1, consider thefinite extension Q p(µp n)of Q p generated by all p n-th roots of unity.Let Q p(µp∞)=∪n≥1Q p(µp n).All of these algebraic extensions of Q p are equipped with the unique norm|·| extending the usual p-adic norm on Q p.Question:Which of the following are spherically complete?Explain why.(a)Q p;(b)Q p(µp n);(c)Q p(µp∞);(d)Q p(µp∞),the completion of Q p(µp∞).Hint:Show that there exists a sequence a1,a2,...∈Q p(µp∞)such that|a1|>|a2|>···and lim|a i|>0,and such that the closed ballsB i:=x∈Q p(µp∞):|x−a1−a2−···−a i|≤|a i|have empty intersection.。
清华丘成桐数学英才班2018-2020届考试真题
2020年试题2020年“丘成桐数学英才班”招生考试形式内容英才班考试内容包括综合考核、学术能力测试（两场笔试、一场面试）和心理测评，其中对学术能力的考查是关键。
针对考试形式和往届真题解析如下，供大家参考。
笔试第一场考查中学数学常见内容和微积分及线性代数。
主要考查数学基础知识是否扎实，以及对微积分、线性代数等未来研究数学真正的基础方法和工具的掌握程度。
笔试第二场采取现学现考的形式，考试当天上午两位教授讲授两个专题，着重讲解中学数学未涉及的概念或者定理，下午考核上午所学内容。
重点考查学生对新知识的接受程度，学习理解能力和分析解决问题能力，而非解题技巧的运用。
两场笔试的优胜者获得面试资格，面试采取考生和评委面对面的方式进行，评委由三至五位教授组成，进一步了解学生的数学知识体系以及现场沟通、临场反应能力。
考生要在短时间内抓住重点、讲清要点、流畅沟通、有效反馈。
2019年试题2018年试题（一）初试综合测试（语数英一起考210分钟理化一起考120分钟）（二）复试复试一：一个数论、一个不等式、个求不定方程解个数、一个积分一个组合几何一个证明欧拉定理（边点面关系那个）都是代数数论真题还原：积分那道是证明pi的无理性，分两个小题(相当于提示)(证明方法和数论书(闵嗣鹤）里基本一样)；最后一道是类似欧拉定理在空间中的一些推广。
复试二：群论和简单随机游走，讲完测评。
往年试题（官方版）针对英才班考试特点，以下管中窥豹，对往届试题略加分析，进一步帮助考生了解如何应考。
笔试第一场真题包括以下三个题目。
例1是不等式题目，例2是立体几何与组合相结合的问题，都是中学数学的常见内容。
例3是微积分题目。
例1．例2.例3.这道题要求考生能真正掌握一元Riemann积分的概念以及分部积分等重要技术手段。
笔试第二场现学现考真题包括以下两个题目。
例4是关于平面整点上随机游动的专题，例5是关于群的作用的专题。
例4.这道试题的还有一个不同于传统考试的特点：它由一系列的小问题组成，每一小问都可以利用前面已经得到的结论作答。
1集合-2010-2018年全国高中数学联赛真题分类汇编
全国高中数学联赛一试试题分类汇编1、集合2018A1、设集合{}99,,3,2,1 =A ，集合{}A x x B ∈=|2，集合{}A x x C ∈=2|，则集合C B 的元素个数为2018B1、设集合{}8,1,0,2=A ，集合{}A a a B ∈=|2，则集合B A 的所有元素之和是2013A1、设集合{}3,1,0,2=A ,集合{}A x A x xB ∉-∈-=22,|，则集合B 中所有元素的和为2014A 2、设集合⎭⎬⎫⎩⎨⎧≤≤≤+21|3b a b a 中最大元素与最小元素分别为N M ,，则N M -的值为2014B 3、对于实数R 的任意子集U ，我们在R 上定义函数U x Ux x f U ∉∈⎩⎨⎧=,,01)(，如果B A ,是实数R 的两个子集，则1)()(≡+x f x f B A ，的充分必要条件是。
呕心沥血，不畏辛苦，历时6个多月，我和多位数学教练尽微薄之力，为广大数学竞赛学生、老师、大学生、爱好者等能更有效的备考每年9月份的全国高中数学联赛考试，重新对全国高中数学联赛历年真题编辑，按难易度、年份、一试二试分开分类汇编，针对性强化训练，考试一样的试卷（答案都是详解，全部在最后）训练更到位。
已经联系印刷厂，按要求最少出版600本，除了我和几位教练的学生需要100多本，其余以成本价出售，按付款顺序发货，不加印。
详情请您联系：QQ ：850756813微信：lovechenyilove(添加时，请您备注信息)PS ：各省的历年预赛真题、竞赛视频、竞赛电子书、解题策略、有效备考经验等等全部免费赠送。
全国高中数学联赛一试试题分类汇编1、集合答案2018A1、答案：24解析：由条件知，{}48,,6,4,2 =C B ，故C B 的元素个数为24。
2018B1、答案：31解析:易知{}16,2,0,4=B ，所以{}16,8,4,2,1,0=B A ，元素之和为31.。
【2010-2018丘成桐大学生数学竞赛笔试真题】Algebra2012Individual
INDIVIDUAL TEST Algebra and Number Theory 1.Prove that the polynomial x 6+30x 5−15x 3+6x −120cannot be written as a product of two polynomials of rational coefficients and positive degrees.2.Let F p be the field of p -elements and GL n (F p )the group of invertible n by n matrices.(1)Compute the order of GL n (F p ).(2)Find a Sylow p-subgroup of GL n (F p ).(3)Compute the number of Sylow p-subgroups.3.Let V be a finite dimensional vector space over complex field C with a nondegenerate symmetric bilinear form (,).LetO (V )={g ∈GL (V )|(gu,gv )=(u,v ),u,v ∈V }be the orthogonal group.Prove that fixed point subspace (V ⊗C V )O(n )is 1-dimensional.4.Let D be the ring consisting of all linear differential operators of finite order on R with polynomial coefficients,of the formD =n i =0a i (x )d idx i for some natural number n ∈N and polynomials a 0(x ),···,a n (x )∈R [x ].This ring R operates naturally on M :=R [x ],making M a left D -module.(1)(to warm up)Suppose that b (x )∈R [x ]is a non-zero polynomialin M ,and let c (x )be any element in M .Show that there is an element D ∈D such that D (b (x ))=c (x ).(2)Suppose that m is a positive integer,b 1(x ),···,b m (x )are mpolynomials in M linearly independent over R and c 1(x ),···,c m (x )are m polynomials in M .Prove that there exists an element D ∈D such that D (b i (x ))=c i (x )for i =1,···,m .5.Let Λbe a lattice of C ,i.e.,a subgroup generated by two R -linear independent elements.Let R be the subring of C consists of elements αsuch that αΛ⊂Λ.Let R ×denote the group of invertible elements in R .(1)Show that either R =Z or have rank 2over Z .1S.-T YAU COLLEGE MATH CONTESTS 2012Please solve 5 out of the following 6 problems,or highest scores of 5 problems will be counted.O(V)2INDIVIDUAL TEST(2)Let n≥3be a positive integer and(R/nR)×the group of invert-ible elements in the quotient R/nR.Show that the canonicalgroup homomorphismR×→(R/nR)×is injective.(3)Find maximal size of R×.6.Let V be a(possible)infinite dimensional vector space over R witha positive definite quadratic norm · .Let A be an additive subgroup of V with following properties:(1)A/2A isfinite;(2)for any real number c the set{a∈A: a <c}isfinite.Prove that A is offinite rank over Z.。
【2010-2018丘成桐大学生数学竞赛笔试真题】2018-team
a) Let N = 2m − 1, and Y = Xm, please find the A and B such that Y −A NB
has nontrivial distributionቤተ መጻሕፍቲ ባይዱ and please find this distribution. b) Let N = 2m, and Y = Xm − Xm−1, please find the A and B such that Y −A NB
1
S.-T. Yau College Student Mathematics Contests 2018
Probability and Statistics
Team (5 problems)
Problem 1. Let Xi, 1 ≤ i ≤ N be i.i.d. random variables. Here X1 is uniformly distributed on [0, 1]. We reorder them as
has nontrivial distribution, and please find this distribution.
Problem 2. Let X = (Z2)N, i.e., X = (X1, X2 · · · , XN · · · ), Xi ∈ (0, 1). It can be considered as countable lightbulbs. 0 means off, 1 means on. We start with X0 = 0. Keep generating independent geometric random variables, whose distribution are
丘成桐大学生数学竞赛,应用数学与概率统计,考纲
Computational Mathematics,Applied Mathematics,Probability and Statistics(the second draft)Computational MathematicsInterpolation and approximationPolynomial interpolation and least square approximation;trigonometric interpolation and approximation,fast Fourier transform;approximations by rational functions;splines.Nonlinear equation solversConvergence of iterative methods(bisection,secant method,Newton method, other iterative methods)for both scalar equations and systems;finding roots of polynomials.Linear systems and eigenvalue problemsDirect solvers(Gauss elimination,LU decomposition,pivoting,operation count,banded matrices,round-off error accumulation);iterative solvers (Jacobi,Gauss-Seidel,successive over-relaxation,conjugate gradient method, multi-grid method,Krylov methods);numerical solutions for eigenvalues and eigenvectorsNumerical solutions of ordinary differential equationsOne step methods(Taylor series method and Runge-Kutta method);stability, accuracy and convergence;absolute stability,long time behavior;multi-step methodsNumerical solutions of partial differential equationsFinite difference method;stability,accuracy and convergence,Lax equivalence theorem;finite element method,boundary value problems References:[1] C.de Boor and S.D.Conte,Elementary Numerical Analysis,an algorithmic approach,McGraw-Hill,2000.[2]G.H.Golub and C.F.van Loan,Matrix Computations,third edition,Johns Hopkins University Press,1996.[3] E.Hairer,P.Syvert and G.Wanner,Solving Ordinary Differential Equations,Springer,1993.[4] B.Gustafsson,H.-O.Kreiss and J.Oliger,Time Dependent Problems and Difference Methods,John Wiley Sons,1995.[5]G.Strang and G.Fix,An Analysis of the Finite Element Method,second edition,Wellesley-Cambridge Press,2008.Applied MathematicsODE with constant coefficients;Nonlinear ODE:critical points,phase space&stability analysis; Hamiltonian,gradient,conservative ODE’s.Calculus of Variations:Euler-Lagrange Equations;Boundary Conditions,parametric formulation; optimal control and Hamiltonian,Pontryagin maximum principle.First order partial differential equations(PDE)and method of characteristics;Heat,wave,and Laplace’s equation;Separation of variables and eigen-function expansions;Stationary phase method;Homogenization method for elliptic and linear hyperbolic PDEs;Homogenization and front propagation of Hamilton-Jacobi equations;Geometric optics for dispersive wave equations.References:W.D.Boyce and R.C.DiPrima,Elementary Differential Equations,Wiley,2009F.Y.M.Wan,Introduction to Calculus of Variations and Its Applications,Chapman&Hall,1995G.Whitham,“Linear and Nonlinear Waves”,John-Wiley and Sons,1974.J.Keener,“Principles of Applied Mathematics”,Addison-Wesley,1988.A.Benssousan,P-L Lions,G.Papanicolaou,“Asymptotic Analysis for Periodic Structures”,North-Holland Publishing Co,1978.V.Jikov,S.Kozlov,O.Oleinik,“Homogenization of differential operators and integral functions”, Springer,1994.J.Xin,“An Introduction to Fronts in Random Media”,Surveys and Tutorials in Applied Math Sciences,No.5,Springer,2009.ProbabilityRandom Variables;Conditional Probability and Conditional Expectation;Markov Chains;The Exponential Distribution and the Poisson Process;Continuous-Time Markov Chains;Renewal Theory and Its Applications;Queueing Theory;Reliability Theory;Brownian Motion and Stationary Processes;Simulation.Reference:Sheldon M.Ross,Introduction to Probability ModelsStatisticsDistribution Theory and Basic StatisticsFamilies of continuous distributions:Chi-sq,t,F,gamma,beta;Families of discrete distributions: Multinomial,Poisson,negative binomial;Basic statistics:Mean,median,quantiles,order statisticsLikelihood principleLikelihood function,parametric inference,sufficiency,factorization theorem,ancillary statistic, conditional likelihood,marginal likelihood.TestingNeyman-Pearson paradigm,null and alternative hypotheses,simple and composite hypotheses, type I and type II errors,power,most powerful test,likelihood ratio test,Neyman-Pearson Theorem,monotone likelihood ratio,uniformly most powerful test,generalized likelihood ratio test.EstimationParameter estimation,method of moments,maximum likelihood estimation,unbiasedness, quadratic and other criterion functions,Rao-Blackwell Theorem,Fisher information,Cramer-Rao bound,confidence interval,duality between confidence interval and hypothesis testing.Bayesian StatisticsPrior,posterior,conjugate priors,Bayesian lossNonparametric statisticsPermutation test,permutation distribution,rank statistics,Wilcoxon-Mann-Whitney test,log-rank test,Kolmogorov-Smirnov-type tests.RegressionLinear regression,least squares method,Gauss-Markov Theorem,logistic regression,maximum likelihoodLarge sample theoryConsistency,asymptotic normality,chi-sq approximation to likelihood ratio statistic,large-sample based confidence interval,asymptotic properties of empirical distribution.ReferencesCasella,G.and Berger,R.L.(2002).Statistical Inference(2nd Ed.)Duxbury Press.茆诗松，程依明，濮晓龙，概率论与数理统计教程（第二版），高等教育出版社，2008.陈家鼎，孙山泽，李东风，刘力平，数理统计学讲义，高等教育出版社，2006.郑明，陈子毅，汪嘉冈，数理统计讲义，复旦大学出版社，2006.陈希孺，倪国熙，数理统计学教程，中国科学技术大学出版社，2009.。
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。
丘成桐大学生数学竞赛数学专业大纲(英汉对照版)
S．T．YAU College Student Mathematics Contests
Algebra, Number Theory and Combinatorics (second draft)
代数、数论和组合（修改稿）
Linear Algebra Abstract vector spaces; subspaces; dimension; matrices and linear transformations; matrix algebras and groups; determinants and traces; eigenvectors and eigenvalues, characteristic and minimal polynomials; diagonalization and triangularization of operators; invariant subspaces and canonical forms; inner products and orthogonal bases; reduction of quadratic forms; hermitian and unitary operators, bilinear forms; dual spaces; adjoints. tensor products and tensor algebras; 线性代数 抽象向 量空间 ；子 空间； 维度； 矩阵 和线性 变换 ；矩阵 代数 和群； 行列式 和迹 ；特 征向量 和特征 值； 特征多 项式和 最小 多项式 ；对 角化和 三角 化；不 变子空 间和 典范 形式； 内积和 正交 基；二 次型的 化简 ；埃尔 米特 算子和 酉算 子；双 线性型 ；对 偶空 间；伴随矩阵，张量积和张量代数； Integers and polynomials Integers, Euclidean algorithm, unique decomposition; congruence and the Chinese Remainder theorem; Quadratic reciprocity ; Indeterminate Equations. Polynomials, Euclidean algorithm, uniqueness decomposition, zeros; The fundamental theorem of algebra; P olynomials of integer coefficients, the Gauss lemma and the Eisenstein criterion; Polynomials of several variables, homogenous and symmetric polynomials, the fundamental theorem of symmetric polynomials. 整数和多项式 整数 ， 欧 几里 得 算法 ， 唯 一分 解 ； 同余 和 中国 剩 余 定理 ； 二次 互 反 性； 不 定方 程. 多项式 ，欧几 里得 算法， 唯一分 解， 零点； 代数 基本定 理； 整系数 多项式 ，高 斯引 理和爱 森斯坦 判别 法；多 元多项 式， 齐次多 项式 和对称 多项 式；对 称多项 式基 本定 理. Group Groups and homomorphisms, Sylow theorem, finitely generated abelian groups. Examples: permutation groups, cyclic groups, dihedral groups, matrix groups, simple groups, Jordan-Holder theorem, linear groups (GL(n, F) and its subgroups), p-groups, solvable and nilpotent groups, group extensions, semi-direct products, free groups, amalgamated products and group presentations. 群 群和同态， 西罗定 理，有限 生成阿 贝尔群. 例子：置 换群、循 环群、二 面体群、 矩阵 群、单群、若尔当-赫尔德定理、线性群（GL（n，F）及其子群）、p群、可解群和 幂零群，群扩张，半直积，自由群，融合积和群表示. Ring Basic properties of rings, units, ideals, homomorphisms, quotient rings, prime and maximal ideals,
【2010-2018丘成桐大学生数学竞赛笔试真题】applied2017-individual
where M is symmetric and positive definite, ω1 > 1, ω2, ..., ωk > 0 and c1, ..., ck−1 ∈ Rn. Deduce from (a) that the iterative scheme converges if and only if all eigenvalues of M (denote it as λ(M )) satisfies:
3. We consider a piecewise smooth function
f (x) =
f1(x), f2(x),
x ≤ 0, x>0
where f1(x) is a C∞ function on (−∞, 0] and f2(x) is a C∞ function on [0, ∞), but f1(0) = f2(0). Suppose p(x) is a k-th degree polynomial (k ≥ 1) interpolating f (x) at k + 1 equally-spaced grid points xj, j = 0, 1, 2, · · · , k with xi < 0 < xi+1 for some i between 0 and k − 1. Prove that, when the grid size h = xj+1 − xj is small enough, p (x) = 0 for xi ≤ 0 ≤ xi+1, that is, p(x) is monotone in the interval [xi, xi+1]. (Hint: first prove the case when f1(x) = c1, f2(x) = c2 and c1 = c2 are two constants.)
丘成桐中国大学生数学竞赛大纲
2010年中国大学生数学竞赛（丘成桐教授发起）竞赛大纲一．Syllabuses for Geometry and TopologyGeometry:Curves and surfaces1) Plane curves and space curves2) The fundamental theorem of curves3) Concept and examples of surfaces4) The first and second fundamental forms5) Normal curvature, principal curvature and the Gauss curvature6) Orthogonal moving frames and structure equations of surfaces7) Existence and uniqueness of surfaces8) Isometric transformation of surfaces9) Covariant derivatives on surfaces10) Geodesic curvatures and geodesics, Geodesic coordinates11) The Gauss-Bonnet formula12) Laplacian operator on surfacesGeometry on manifolds1) Manifolds2) Vector fields and differentials3) Tensors and differential forms4) Stokes formula5) De Rham theorem6) Lie derivatives7) Lie algebras8) Maurer-Cartan equations9) Vector bundles10)Connection and curvatures11) Structure equations12) Riemannian metrics13) The Hodge star operator and Laplacian operator14) The Hodge theoremReferences:M. Do Carmo, Differential geometry of curves and surfaces.S S Chern and Chen Weihuan, Lectures on differential geometry Q. Chen and CK Peng, Differential geometryT. Frenkel: Geometry from physicsJ. Milnor, Morse theoryTopologyPoint Set Topology1) Open set and closed set2) Continuous maps3) Haudorff space, seperability and countable axioms4) Compactness and Heine-Borel theorem5) Connectivity and path connectivity6) Quotient space and quotient topologyFundamental groups1) Definition of fundamental groups, homotopic maps2) Computation of fundamental groups: Van Kampen theorem3) Covering maps and covering spaces4) Applications: Brouwer fixed point theorem, Lefschetz fixed point theoremComplexes and homology groups1) Simplex, complexes and polyhedron2) Barycentric subdivision and simplex approximation3) Computation of fundamental groups of complexes4) Classification of surfaces5) Simplex homology groups6) Application: Lefschetz fixed point theoremDifferential topology1) Smooth manifolds and smooth maps2) Sard’s theorem3) Transversality and intersection4) Vector fileds and Poincare-Hopf theorem5) Differential forms and de Rham complexes6) Orientation and integration7) Poincare Lemma8) Poincare duality9) Meyer-Vietoris sequences10)de Rham theorem11)Vector bundle and Euler classesReferences:Armstrong, Basic topologyJ. Milnor, Topology from the differentiable viewpointV. Guillemin and A. Pollack, Differential topologyBott and Tu, Differential forms in algebraic topology (first chapter)二．Syllabuses on algebra, combinatorics, number theory and representation theoryAlgebra群论（31）：集合论预备知识；对称和群；子群和陪集分解；生成元集和循环群；正规子群、商群和同态定理；置换群和线性群；群在集合上的作用；Sylow定理和单群；自由群和群的表现；有限生成Abel群的结构；小阶群的结构；幂零群和可解群。
第三届丘成桐大学生数学竞赛试题-纯数学
(1) Show that either R = Z or have rank 2 over Z.
1
2
INDIVIDUAL TEST
(2) Let n ≥ 3 be a positive integer and (R/nR)× the group of invertible elements in the quotient R/nR. Show that the canonical group homomorphism
1
INDIVIDUAL TEST S.-T YAU COLLEGE MATH CONTESTS 2012
Geometry and Topology
Please solve 5 out of the following 6 problems, or highest scores of 5 problems will be counted.
C is a simple (that is, without self-intersection), smooth, closed curve
in the Euclidean plane, with length L. Show that the area enclosed by
C
is
less
than
U ⊂ M satisfying that [Xi, Xj] = 0 for 1 ≤ i, j ≤ k. Prove that for any point p ∈ U there is a coordinate chart (V, yi) with p ∈ V ⊆ U and
coordinates {y1, · · · , yn} such that Xi =
第三届丘成桐大学生数学竞赛试题-概率统计及应用数学
(c) Find the density of (U1 · · · Un)−X.
INDIVIDUAL TEST S.-T YAU COLLEGE MATH CONTESTS 2012
Applied Math. and Computational Math.
Please solve 4 out of the following 5 problems.
if γ = 0.5.
Assume that there are no tied data values.
(a)
Show
that
W
=
U
+
1 2
n(n
+
1),
which
shows
that
the
two
test
statistics differ only by a constant and yield exactly the same
(a) prove that there is a unique solution P x(x) to (2) (b) denote h = maxk |xk − x|, prove
|ci
−
1 i!
f
(i)(x)|
=
C (f ,
i)hm+1−i,
i = 0, 1, . . . , m,
where f (i)(·) is the i-th derivative of f and C(f, i) denote some constant
1. In the numerical integration formula
1
(1)
f (x)dx ≈ af (−1) + bf (c),
2010年大学生数学竞赛试题参考解答
6. 已知 u f ( x, y) 满足 r
u u u u ， 。 x, y ，这里 ( , r ) 为极坐标。求 y r x
u u u u u u cos sin r x y y y x r cos r x r x 解： ， 。 y r sin u u r ( sin ) u r cos u u ( y ) u x y y x x u u x x y y x u 2 xy u x 2 y 2 2 2 由条件知 ，解之得： ， . 2 y x y 2 x x y u ( y ) u x y y x
1 令 ( x) e x e x x 2 2 ，则 ( x) 为偶函数，且 (0) 0 。 2
1 6
（ ' x） e x e- x x ， ''( x) e x e x 1 。
显然， ''( x) 0 ， '( x) 单调增加。 x 0 时 '( x) '(0) 0 。
t2 1 4 dt t 3 3 。 2 6 0 3 1 t
2
4 故 S S1 S2 3 。 3
另解：利用曲线积分。由曲线积分可知 S
1 1 ydx xdy [ ] ， 2L 2 L1 L2
其中 L1 为红色曲线， L2 为线段 x 1 (2 y 0) 。
解：由条件知 F ( x) F ( x) x(1 x)e2 x 。 两边积分 F ( x) F '( x)dx x(1 x)e2 x dx 得
第十届（2018）全国大学生数学竞赛（非数学类）预赛试题
第十届全国大学生数学竞赛（非数学类）预赛试题及一、填空题（本题满分24分, 共4小题, 每小题6分)（1）设(0,1),则lim (1)n n n=＿＿＿＿＿＿＿．（2）若曲线()yy x 由+cos +sin 1yx t t ety t确定，则此曲线在0t 对应点处的切线方程为（3）223/2ln(1)(1)xx dx x =（4）321cos cos2cos3limxx x xx=＿＿＿＿＿＿＿.f t ()0t(1)0f 二 (本题满分8分) 设函数在时一阶连续可导，且，求函数f xy 22()，使得曲线积分2222Ly(2f (x y ))dx xf (xy )dy 与路径无关，其中L 为任一不与直y x 线相交的分段光滑闭曲线．f x ()0,11)3(f x 三 (本题满分14分) 设 在区间[]上连续，且 .证明：11141)3f (x)dxdx(f x .四 (本题满分12分）计算三重积分22xy ()dV （V ）（V ），其中是由222x y(z 2)4，222xy(z 1)9，0z 所围成的空心立体.五 (本题满分14分) 设(,)f x y 在区域D 内可微，且22f f M xy，11(,)A x y ，22(,)B x y 是D 内两点，线段AB 包含在D 内。
证明：1122|(,)(,)|||f x y f x y M AB ，其AB ||AB 中表示线段的长度.)0(f x 六(本题满分14分) 证明：对于连续函数，有11lnf (x)dxln f (x)dx .七 (本题满分14分) 已知{}k a ，{}k b 是正项数列，且10,kkb b ，为一常数.证明：若级数1k k a 收敛，则级数121211()()k k k k k kk a a a b b b b b 收敛.1,2,k。
大学生数学竞赛试题
首届全国大学生数学竞赛计算下列各题(1) 求极限121lim (1)sin n n k k k n n π-→∞=+∑. (2)计算2∑∑为下半球面z =0a >.(3) 现要设计一个容积为V 的一个圆柱体的容器. 已知上下两底的材料费为单位面积a 元，而侧面的材料费为单位面积b 元.试给出最节省的设计方案：即高与上下底的直径之比为何值时所需费用最少？(4) 已知()f x 在11,42⎛⎫ ⎪⎝⎭内满足331()sin cos f x x x '=+，求()f x . 二、（10分）求下列极限(1) 1lim 1n n n e n →∞⎛⎫⎛⎫+- ⎪ ⎪ ⎪⎝⎭⎝⎭； (2) 111lim 3nn n n n a b c →∞⎛⎫++ ⎪ ⎪ ⎪⎝⎭, 其中0,0,0a b c >>>. 三、（10分）设()f x 在1x =点附近有定义，且在1x =点可导， (1)0,(1)2f f '==. 求220(sin cos )lim tan x f x x x x x→++. 四、（10分）设()f x 在[0,)+∞上连续，无穷积分0()f x dx ∞⎰收敛. 求 01lim ()y y xf x dx y →+∞⎰.五、（12分）设函数()f x 在[0,1]上连续，在(0,1)内可微，且1(0)(1)0,12f f f ⎛⎫=== ⎪⎝⎭. 证明：(1) 存在1,12ξ⎛⎫∈ ⎪⎝⎭使得()f ξξ=；(2) 存在(0,)ηξ∈使得()()1f f ηηη'=-+.六、（14分）设1n >为整数，20()1...1!2!!n x tt t t F x e dt n -⎛⎫=++++ ⎪⎝⎭⎰.证明: 方程()2n F x =在,2n n ⎛⎫ ⎪⎝⎭内至少有一个根. 七、（12分）是否存在11R 中的可微函数()f x 使得 2435(())1f f x x x x x =++--？若存在，请给出一个例子；若不存在，请给出证明.八、(12分)设()f x 在[0,)∞上一致连续，且对于固定的[0,)x ∈∞，当自然数n →∞时()0f x n +→. 证明: 函数序列{():1,2,...}f x n n +=在[0,1]上一致收敛于0.。
丘成桐大学生数学竞赛参考书
丘成桐大学生数学竞赛参考书丘成桐大学生数学竞赛参考书Geometry and Topology (the second draft)Space curves and surfacesCurves and Parametrization, Regular Surfaces; Inverse Images of Regular Values.Gauss Map and Fundamental Properties; Isometries; Conformal Maps; Rigidity of the Sphere.Topological spaceSpace, maps, compactness and connectedness, quotients; Paths and Homotopy. The Fundamental Group of the Circle. Induced Homomorphisms. Free Products of Groups. The van Kampen Theorem. Covering Spaces and Lifting Properties; Simplex and complexes. Triangulations. Surfaces and its classification.Differential ManifoldsDifferentiable Manifolds and Submanifolds, Differentiable Functions and Mappings; The Tangent Space, Vector Field and Covector Fields. Tensors and Tensor Fields and differential forms. The Riemannian Metrics as examples, Orientation and Volume Element; Exterior Differentiation and Frobenius's Theorem; Integration on manifolds, Manifolds with Boundary and Stokes' Theorem.Homology and cohomologySimplicial and Singular Homology. Homotopy Invariance. Exact Sequences and Excision. Degree. Cellular Homology. Mayer-Vietoris Sequences. Homology with Coefficients. The Universal Coefficient Theorem. Cohomology of Spaces. The Cohomology Ring. A Kunneth Formula. Spaces with PolynomialCohomology. Orientations and Homology. Cup Product and Duality.Riemannian ManifoldsDifferentiation and connection, Constant Vector Fields and Parallel DisplacementRiemann Curvatures and the Equations of Structure Manifolds of Constant Curvature,Spaces of Positive Curvature, Spaces of Zero Curvature, Spaces of Constant Negative CurvatureReferences:M. do Carmo , Differentia geometry of curves and surfaces.Prentice- Hall, 1976 (25th printing)Chen Qing and Chia Kuai Peng, Differential GeometryM. Armstrong, Basic Topology Undergraduate texts in mathematicsW.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry Academic Press, Inc., Orlando, FL, 1986M. Spivak, A comprehensive introduction to differential geometryN. Hicks, Notes on differential geometry, Van Nostrand.T. Frenkel, Geometry of PhysicsJ. Milnor, Morse TheoryA Hatcher, Algebraic Topology(/doc/6412700287.html,/~hatcher/AT/ATpa ge.html)J. Milnor, Topology from the differentiable viewpointR. Bott and L. Tu, Differential forms in algebraic topologyV. Guillemin, A. Pollack, Differential topologyAlgebra, Number Theory and Combinatorics (second draft)Linear AlgebraAbstract vector spaces; subspaces; dimension; matrices and linear transformations; matrix algebras and groups; determinants and traces; eigenvectors and eigenvalues, characteristic and minimal polynomials; diagonalization and triangularization of operators; invariant subspaces and canonical forms; inner products and orthogonalbases; reduction of quadratic forms; hermitian and unitary operators, bilinear forms; dual spaces; adjoints. tensor products and tensor algebras;Integers and polynomialsIntegers, Euclidean algorithm, unique decomposition; congruence and the Chinese Remainder theorem; Quadratic reciprocity ; Indeterminate Equations. Polynomials, Euclidean algorithm, uniqueness decomposition, zeros; The fundamental theorem of algebra; Polynomials of integer coefficients, the Gauss lemma and the Eisenstein criterion; Polynomials of several variables, homogenous and symmetric polynomials, the fundamental theorem of symmetric polynomials.GroupGroups and homomorphisms, Sylow theorem, finitely generated abelian groups. Examples: permutation groups, cyclic groups, dihedral groups, matrix groups, simple groups, Jordan-Holder theorem, linear groups (GL(n, F) and its subgroups), p-groups, solvable and nilpotent groups, group extensions, semi-direct products, free groups, amalgamated products and group presentations.RingBasic properties of rings, units, ideals, homomorphisms, quotient rings, prime and maximal ideals, fields of fractions,Euclidean domains, principal ideal domains and unique factorization domains, polynomial and power series rings, Chinese Remainder Theorem, local rings and localization, Nakayama's lemma, chain conditions and Noetherian rings, Hilbert basis theorem, Artin rings, integral ring extensions, Nullstellensatz, Dedekind domains,algebraic sets, Spec(A).ModuleModules and algebra Free and projective; tensor products; irreducible modules and Schur’s lemma; semisimple, simple and primitive rings; density and Wederburn theorems; the structure of finitely generated modules over principal ideal domains, with application to abelian groups and canonical forms; categories and functors; complexes, injective modues, cohomology; Tor and Ext.FieldField extensions, algebraic extensions, transcendence bases; cyclic and cyclotomic extensions; solvability of polynomial equations; finite fields; separable and inseparable extensions; Galois theory, norms and traces, cyclic extensions, Galois theory of number fields, transcendence degree, function fields.Group representationIrreducible representations, Schur's lemma, characters, Schur orthogonality, character tables, semisimple group rings, induced representations, Frobenius reciprocity, tensor products, symmetric and exterior powers, complex, real, and rational representations.Lie AlgebraBasic concepts, semisimple Lie algebras, root systems, isomorphism and conjugacy theorems, representation theory.Combinatorics (TBA)References:Strang, Linear algebra, Academic Press.I.M. Gelfand, Linear Algebra《整数与多项式》冯克勤余红兵著高等教育出版社Jacobson, Nathan Basic algebra. I. Second edition. W. H. Freeman and Company, New York, 1985. xviii+499 pp.。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
(b) Does there exists a finite Galois extension L/Q which contains K such that Gal(L/Q) ∼= Z/4Z?
For the following problems, every example and statement must be backed up by proof. Examples and statements without proof will receive no-credit.
√ 1. Let K = Q( −3), an imaginary quadratic field.
automorphism 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 iv | i ≥ 0} spans V .
2. Let f be a two-dimensional (complex) representation of a finite group G such that 1 is an eigenvalue of f (σ) for every σ ∈ G. Prove that f is a direct sum of two one-dimensional representations of G
6. Prove that a group of order 150 is not simple.
(c) Does there exists a finite Galois extension L/Q which contains K such that Gal(L/Q) ∼= Q? Here Q is the quaternion group with 8 elements {±1, ±i, ±j, ±k}, a finite subgroup of the group of units H× of the ring H of all Hamiltonian quaternions.
3. 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 identity
Show that V contains no non-trivial proper T -invariant subspace.
5. Given a commutative diagram
A→B →C →D→E
↓
↓
↓
↓
↓
A →B →C →D →E
1
附件/试卷 4பைடு நூலகம்
Appendix/Contest Paper 4
附件/试卷 4
Appendix/Contest Paper 4
S.-T. Yau College Student Mathematics Contests 2011
Algebra, Number Theory and Combinatorics
Individual
2:30–5:00 pm, July 10, 2011 (Please select 5 problems to solve)
Algebra, Number Theory and Combinatorics, 2011-Individual 2
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.