第2届大学生数学竞赛决赛试题
大学生高等数学竞赛试题汇总及答案
前三届高数竞赛预赛试题（非数学类）（参加高等数学竞赛的同学最重要的是好好复习高等数学知识，适当看一些辅导书及相关题目，主要是一些各大高校的试题。
）2009-2010年 第一届全国大学生数学竞赛预赛试卷一、填空题（每小题5分）1．计算=--++⎰⎰y x yx x yy x Dd d 1)1ln()(16/15，其中区域D 由直线1=+y x 与两坐标轴所围成三角形区域.解: 令v x u y x ==+,，则v u y v x -==,，v u v u y x d d d d 1110det d d =⎪⎪⎭⎫⎝⎛-=，⎰-=12d 1u uu （*） 令u t -=1，则21t u -=dt 2d t u -=，42221t t u +-=，)1)(1()1(2t t t u u +-=-，2．设)(x f 是连续函数，且满足⎰--=222d )(3)(x x f x x f , 则=)(x f ____________.解: 令⎰=20d )(x x f A ，则23)(2--=A x x f ，A A x A x A 24)2(28d )23(202-=+-=--=⎰,解得34=A 。
因此3103)(2-=x x f 。
3．曲面2222-+=y x z 平行平面022=-+z y x 的切平面方程是__________. 解: 因平面022=-+z y x 的法向量为)1,2,2(-，而曲面2222-+=y x z 在),(00y x 处的法向量为)1),,(),,((0000-y x z y x z y x ，故)1),,(),,((0000-y x z y x z y x 与)1,2,2(-平行，因此，由x z x =，y z y 2=知0000002),(2,),(2y y x z x y x z y x ====， 即1,200==y x ，又5)1,2(),(00==z y x z ，于是曲面022=-+z y x 在)),(,,(0000y x z y x 处的切平面方程是0)5()1(2)2(2=---+-z y x ，即曲面2222-+=y x z 平行平面022=-+z y x 的切平面方程是0122=--+z y x 。
历届大学生高等数学竞赛真题及答案非数学类14页
前三届高数竞赛预赛试题（非数学类）（参加高等数学竞赛的同学最重要的是好好复习高等数学知识，适当看一些辅导书及相关题目，主要是一些各大高校的试题。
）2009年 第一届全国大学生数学竞赛预赛试卷一、填空题（每小题5分，共20分）1．计算=--++⎰⎰y x y x x yy x Dd d 1)1ln()(____________，其中区域D 由直线1=+y x 与两坐标轴所围成三角形区域.解: 令v x u y x ==+,，则v u y v x -==,，v u v u y x d d d d 1110det d d =⎪⎪⎭⎫⎝⎛-=，令u t -=1，则21t u -=2．设)(x f 是连续函数，且满足⎰--=2022d )(3)(x x f x x f , 则=)(x f ____________.解: 令⎰=20d )(x x f A ，则23)(2--=A x x f ，解得34=A 。
因此3103)(2-=x x f 。
3．曲面2222-+=y x z 平行平面022=-+z y x 的切平面方程是__________.解: 因平面022=-+z y x 的法向量为)1,2,2(-，而曲面2222-+=y x z 在),(00y x 处的法向量为)1),,(),,((0000-y x z y x z y x ，故)1),,(),,((0000-y x z y x z y x 与)1,2,2(-平行，因此，由x z x =，y z y 2=知0000002),(2,),(2y y x z x y x z y x ====，即1,200==y x ，又5)1,2(),(00==z y x z ，于是曲面022=-+z y x 在)),(,,(0000y x z y x 处的切平面方程是0)5()1(2)2(2=---+-z y x ，即曲面 2222-+=y x z 平行平面022=-+z y x 的切平面方程是0122=--+z y x 。
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。
第二届全国大学生数学竞赛预赛试卷（非数学类）
22,x y +x x 2t te2111))[n n s s s s s14解：（简要过程）（简要过程）二阶导数为正，则一阶导数单增，f(x)先减后增，因为f(x)有小于0的值，所以只需在两边找两大于0的值。
的值。
将f(x)二阶泰勒展开二阶泰勒展开'''2()()(0)(0)2f f x f f x x x =++因为二阶倒数大于0，所以，所以lim ()x f x ®+¥=+¥，lim ()x f x ®-¥=-¥证明完成。
证明完成。
三、（15分）设函数()y f x =由参数方程22(1)()x t t t y t y ì=+>-í=î所确定，其中()t y 具有二阶导数，曲线()y t y =与22132t u y e du e -=+ò在1t =出相切，求函数()t y 。
解：（这儿少了一个条件22d y dx = ）由()y t y =与22132t u y e du e-=+ò在1t =出相切得出相切得3(1)2ey =，'2(1)e y ='//()22dy dy dt dx dx dt t ty ==+ 22d y dx ='3''()(2(/)(/)//(22)2)2()d dy dx d dy dx dt dx dx d t t t t t y y ==++-=。
上式可以得到一个微分方程，求解即可。
上式可以得到一个微分方程，求解即可。
四、（15分）设10,,nn n k k a S a =>=å证明：证明：（1）当1a >时，级数1nn na S a +¥=å收敛；收敛； （2）当1a £且()ns n ®¥®¥时，级数1nn na S a +¥=å发散。
大学生高等数学竞赛试题汇总与答案
原式=
(ln(1t)t)1/(1t)111
2
2(1t)
t2t2
limelimelimee
t0t0t0
（3）
11
sxnnsxnsxsxn
Iexdx()xde()[xe|edx]
n0
000
ss
nnn(n1)n!n!
sxn1
exdxIII
n12n2n0n1
sssss
0
二、（15分）设函数f(x)在(,)上具有二阶导数，并且
''()(2'
t2t)2(t)''()(2'
3
dxdx/dt(22t)
=。。。
上式可以得到一个微分方程，求解即可。
四、（15分）设
n
a0,Sa,证明：
nnk
k1
（1）当1时，级数
a
n
S
nn
1
收敛；
（2）当1且()
sn时，级数
n
a
n
S
nn
1
发散。
解：
（1）
a>0,
n
s单调递增
n
当
n1
a收敛时，
n
aa
nn
一、（25分，每小题5分）
（1）设
n
22
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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 ）。
第二届全国大学生数学竞赛决赛试题及详细解答
第二届全国大学生数学竞赛决赛试题及答案（非数学类，2011）一．计算下列各题（本题共3小题，每小题各5分，共15分。
）（1）.求11cos 0sin lim xx x x -→⎛⎫ ⎪⎝⎭；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）：0202000322sin 1sin 1ln lim11cos lim1cos 201sin cos 12limlimlim 11333222sin lim x x x x x xx x x xx xx x x xx x x x x eex ee e e→→→→→-⎛⎫ ⎪⎝⎭--→----⎛⎫== ⎪⎝⎭====（2）.求111lim ...12n n n n n →∞⎛⎫+++ ⎪+++⎝⎭； 解：方法一（用欧拉公式）令111...12n x n n n n=++++++ 111ln =C+o 1211111ln 2=C+o 1212n nn n n n+++-++++++-+由欧拉公式得（），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）已知()2ln 1arctan tt x e y t e ⎧=+⎪⎨=-⎪⎩，求22d y dx 。
历届全国大学生高等数学竞赛真题及答案非数学类
前三届高数竞赛预赛试题（非数学类）（参加高等数学竞赛的同学最重要的是好好复习高等数学知识，适当看一些辅导书及相关题目，主要是一些各大高校的试题。
）2009年 第一届全国大学生数学竞赛预赛试卷一、填空题（每小题5分，共20分）1．计算=--++⎰⎰y x yx x yy x Dd d 1)1ln()(____________，其中区域D 由直线1=+y x 与两坐标轴所围成三角形区域.解: 令v x u y x ==+,，则v u y v x -==,，v u v u y x d d d d 1110det d d =⎪⎪⎭⎫ ⎝⎛-=， v u u v u u u y x y x x yy x D D d d 1ln ln d d 1)1ln()(⎰⎰⎰⎰--=--++⎰⎰⎰⎰----=---=1021000d 1)ln (1ln d )d ln 1d 1ln (u uu u u u u u u u v v uuv u u u u u ⎰-=12d 1u uu （*） 令u t -=1，则21t u -=dt 2d t u -=，42221t t u +-=，)1)(1()1(2t t t u u +-=-，⎰+--=0142d )21(2(*)tt t⎰+-=1042d )21(2t t t 1516513221053=⎥⎦⎤⎢⎣⎡+-=t t t2．设)(x f 是连续函数，且满足⎰--=2022d )(3)(x x f x x f , 则=)(x f ____________.解: 令⎰=20d )(x x f A ，则23)(2--=A x x f ，A A x A x A 24)2(28d )23(202-=+-=--=⎰,解得34=A 。
因此3103)(2-=x x f 。
3．曲面2222-+=y x z 平行平面022=-+z y x 的切平面方程是__________. 解: 因平面022=-+z y x 的法向量为)1,2,2(-，而曲面2222-+=y x z 在),(00y x 处的法向量为)1),,(),,((0000-y x z y x z y x ，故)1),,(),,((0000-y x z y x z y x 与)1,2,2(-平行，因此，由x z x =，y z y 2=知0000002),(2,),(2y y x z x y x z y x ====，即1,200==y x ，又5)1,2(),(00==z y x z ，于是曲面022=-+z y x 在)),(,,(0000y x z y x 处的切平面方程是0)5()1(2)2(2=---+-z y x ，即曲面2222-+=y x z 平行平面022=-+z y x 的切平面方程是0122=--+z y x 。
第二届全国大学生数学竞赛决赛试题及解答
第二届全国大学生数学竞赛决赛试题及解答一、(15分)求出过原点且和椭球面2224561x y z ++=的交线为一个圆周的所有平面.【解】 所述圆周过原点，则一定以原点为圆心，且在球面2222x y z R ++= ①上.因此，该球面与椭球面2224561x y z ++= ②的交线即为圆周.由①、②确定的平面也必包含此圆周.联立此二式，得2222221114560x y z R R R ⎛⎞⎛⎞⎛⎞−+−+−=⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠ 显然，当215R =时，有220x z −=，这是两相交平面x z =，0x z +=，即为所求.二、(15分)设()01f x <<，无穷积分()0d f x x +∞∫和()0d xf x x +∞∫都收敛.求证：()()()21d d 2xf x x f x x +∞+∞>∫∫.【证】令()0d f x x a +∞=∫，则()0,a ∈+∞.据题设条件()01f x <<，得()()()0d d d aaxf x x xf x x xf x x +∞+∞=+∫∫∫()()0d d a axf x x a f x x +∞>+∫∫()()()d d aaxf x x a a f x x =+−∫∫()()()0d 1d a axf x x a f x x =+−∫∫()()()0d 1d a axf x x x f x x >+−∫∫201d 2a x x a ==∫， 因此，得()()()21d d 2xf x x f x x +∞+∞>∫∫.三、(15分)设1nn na+∞=∑收敛，122n n n n k t a a ka +++=++++"".证明：lim 0n n t →∞=.【证】 首先，注意到1n n k k t ka +∞+==∑()1n k k kn k a n k+∞+==++∑,据题设条件1n n na +∞=∑收敛，可知()1n kk n k a +∞+=+∑收敛，而k n k ⎧⎫⎨⎬+⎩⎭关于k 单调，且01k n k <<+即有界，故由Abel 判别法知()1n k k kn k a n k+∞+=++∑收敛，即n t 有意义. 因为1nn na+∞=∑收敛，所以0ε∀>，存在N +∈]，使得当n N >时，+n kk nR ka ∞==∑(),εε∈−.此时，对任何n N >以及1m >，有()111mmn kk n k n k k k kaR R n k ++++===−+∑∑11211m m k n k n k k k k R R n k n k +++==−=−++−∑∑ 1121111m n m n k n k m k k R R R n n m n k n k ++++=−⎛⎞=−+−⎜⎟++++−⎝⎠∑，于是，有1mn kk ka+=∑21111mk m kk n n m n kn k εε=−⎛⎞⎛⎞≤++−⎜⎟⎜⎟++++−⎝⎠⎝⎠∑22m n m εε=<+. 所以，2n t ε≤，()n N >，即lim 0n n t →∞=.四、(15分)设()n A M ∈^，定义线性变换:()()A n n M M σ→^^，()A X AX XA σ=−.证明：当A 可对角化时，A σ也可对角化.这里()n M ^是复数域^上n 阶方阵组成的线性空间.【证】取()n M ^的自然基{}:,1,2,ij E i j n ="，其中ij E 是(,)i j 元等于1，其它元均为0的n 阶矩阵.因为A 可对角化，所以存在可逆矩阵()n P M ∈^，使得112diag(,,,)n P AP λλλ−=Λ=".显然，{}1:,1,2,ij PE P i j n −="也是()n M ^的一组基，并且有11111()()()()()A ij ij ij ij ij i j ij PE P A PE P PE P A P E E P PE P σλλ−−−−−=−=Λ−Λ=−，所以A σ在基11111111,,,,,,n n nn PE P PE P PE P PE P −−−−"""下的矩阵为对角矩阵12111diag(0,,,,,,,,0)n n n n λλλλλλλλ−−−−−"""，这就是说，A σ可对角化.五、(20分)设连续函数:f →\\，满足()()(),sup x y f x y f x f y ∈+−−<+∞\.证明：存在实常数a 满足()sup x f x ax ∈−<+∞\.【证】 令()()(),sup x y M f x y f x f y ∈=+−−\，则+,,x m n ∀∈∈\`，有()()()f x y f x f y M +−−≤， ①()((1))()f nx f n x f x M −−−≤.于是，有()()()2()((1))()1nk f nx nf x f kx f k x f x n M nM =−≤−−−≤−≤∑. ②因此()()()()()()()nf mx mf nx nf mx f mnx f mnx mf nx n m M −≤−+−≤+，()()11f mx f nx M m n n m ⎛⎞−≤+⎜⎟⎝⎠. 这表明函数列()f nx n ⎧⎫⎨⎬⎩⎭在(,)−∞+∞上一致收敛，设其极限为()g x ，则()g x 是连续函数. 进一步，由不等式①，有()()()()f n x y f nx f ny M nn n n+−−≤，,;x y n +∀∈∈\`. 取极限，得()()()g x y g x g y +=+，,x y ∀∈\.由此可解得()()1g x g x ax ==.另一方面，再由②式，得()()f nx f x M n−≤. 令n →∞，得()()g x f x M −≤，x ∀∈\.从而()()sup x g x f x M ∈−≤<+∞\.故存在实常数a ，使得()sup x f x ax M ∈−≤<+∞\.六、(20分) 设:()n M ϕ→\\是非零线性映射，满足()()XY YX ϕϕ=，,()n X Y M ∀∈\，这里()n M \是实数域\上n 阶方阵组成的线性空间.在()n M \上定义双线性型(-，-)：()()n n M M ×→\\\为(,)()X Y XY ϕ=.(1)证明(-，-)是非退化的，即若(,)0X Y =，()n Y M ∀∈\，则X O =； (2)设212,,,n A A A "是()n M \的一组基，212,,,n B B B "是相应的对偶基，即0,(,)1,.i j ij i j A B i j δ≠⎧==⎨=⎩当,当 证明21n i ii A B =∑是数量矩阵.【证】(1)先确定ϕ的结构.取()n M \的自然基{}:,1,2,ij E i j n ="，其中ij E 是(,)i j 元等于1，其它元均为0的n 阶矩阵.令()ji ij c E ϕ=，则()()ij n C c M =∈\.()n A M ∀∈\，有1111()()tr()n n n nij ij ij ji i j i j A a E a c AC ϕϕ=======∑∑∑∑.根据题设，()()XY YX ϕϕ=，,()n X Y M ∀∈\，所以tr()tr()tr()YCX XYC YXC ==.因此XC CX =.由于X 的任意性，知C E λ=为数量矩阵.于是有()tr()A A ϕλ=，()n A M ∀∈\.因为0ϕ≠，所以0λ≠.现在，如果(,)tr()0X Y XY λ==，()n Y M ∀∈\，取TY X =，那么X O =. (2)令()ii pqA a =，()i i stB b =.设21n pq pq ii i E B ε==∑，利用{}i A 与{}j B 的对偶性，有()()21,,n pq pq jpqijij i A E A B εε===∑.另一方面，由(1)的结果，有(),tr()j j pq j pq qpA E A E a λλ==，所以21n i pq qpi i E aB λ==∑.比较等式两边的(,)s t 元，得211n i i qp st ps qt i a b δδλ==∑.注意到，pq st qs pt E E E δ=，因此，有22211,1, 1,1, 11,1,11n n n n n n n n n i i i ii i pq pq st st pq st qs pt pt qs pti i p q s t p q s t i s t p q n A B a E b E a b E E E δδδλλ=========⎛⎞⎛⎞====⎜⎟⎜⎟⎝⎠⎝⎠∑∑∑∑∑∑∑∑∑.。
第一届和第二届大学生全国数学竞赛试题
证法二： （1）根据 Green 公式，将曲线积分化为区域 D 上的二重积分
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d2y 且 f ′ ≠ 1 ，则 =____________________. dx 2