综合题新编选登

题155设f(n,p)=C2pn(n,p∈N,p≤2n).数列{a(n,p)}满足a(1,p) a(2,p) … a(n,p)=f(n,p).1)求证:{a(n,2)}是等差数列;2)求证:f(n,1) f(n,2) … f(n,n)=22n-1 21C2nn-1;3)设函数H(x)=f(n,1)x f(n,2)x2 … f(n,2n)·x2n,试比较H(x)-H(a)与2n(1 a)2n-1(x-a)的大小.解1)由a(1,2) a(2,     

（a＋b＋c）^n展开式的系数表

(a b)n展开式的系数表是杨辉三角.本文试对(a b c)n展开式的系数具有怎样的结构和性质进行探讨.1　(a b c)n展开式的一个系数表先看n=1,2,3,4,5,6时(a b c)n展开式的系数(表1).表1n(a b c)n展开式各项的系数11,1,121,1,1,2,2,231,1,1,3,3,3,3,3,3,641,1,1,4,4,4,4,4,4,6,6,6,12     

与二项式系数有关的求和问题的解题策略

1赋值求和例1设(2x-3)10=a10(x-1)10 a9(x-1)9 … a2(x-1)2 a1(x-1) a0,求a1 a2 a3 … a10的值.解令x=2,得a0 a1 a2 a3 … a10=1;令x=1,得a0=(-1)10=1,所以a1 a2 a3 … a10=1-1=0.例2设(1 x x2)n=a0 a1x a2x2 … a2nx2n,求a1 a3 a5 … a2n-1的值.解令x=1,得a0 a1 a2 … a2n=3n;令x=-1,得a0-a1 a2-…-a2n-1 a2n=1.两式相减得a1 a3 a5 … a2n-1=3n-12.2逆用定理例3已知等比数列{an}的首项为a1,公比为q,求和:a1C0n a2C1n a3C2n … an 1Cnn.解a1C0n a2C1n a3C2n … an 1Cnn=a1C0n a1qC1n a1q2C2n … a1qnCnn=a1(C0n qC1n q2C2n … qnCnn)…     

数论问题

本讲通过数学竞赛中的一些数论问题,简要地介绍初等数论中较为基本的思考方法.对于问题所涉及的数论基础知识,我们将直接引用而不作讨论(可以参看,例如,《奥数教程》,高三年级,华东师范大学出版社) .例1　设a ,b是给定的正整数,证明,仅有有限多个正整数n ,使得(a + 12 ) n+ (b + 12 ) n为整数.证　问题等价于证明,仅有有限多个n ,使得2 n整除(2a + 1) n+ (2b + 1) n.我们希望分解被除数(2a + 1) n+ (2b + 1) n.这在n为奇数时易于实现:我们有(2a + 1) n + (2b + 1) n =(2a + 2b + 2 ) (2a +1) n -1- (2a + 1) n -2 (2b + 1) +…- (2a + 1) (2…     

一个包含平方补数的方程

对任意正整数n,我们定义a(n)为n的平方补数,即a(n)表示能够使na(n)为完全平方数的最小正整数.本文的主要目的是利用初等方法研究方程a(n1)+a(n2)+…+a(nk)=m·a(n1+n2+…+nk)的可解性,并证明对某些特殊的正整数m及任意正整数k>1,该方程有无穷多组正整数解(n1,n2,…,nk).     

Cauchy不等式的几个应用

若ai,bi∈R ,i=1 ,2 ,… ,n(n≥ 2 )则 (a21 +a22 +… +a2 n) (b21 +b22 +… +b2 n)≥ (a1 b1 +a2 b2 +… +anbn) 2证明 :若ai=0 ,命题显然成立若ai 不全为零 ,则设f(x) =(a21 +a22 +… +a2 n)x2 +2 (a1 b1 +a2 b2+… +anbn)x+(b1 +b2 +… +bn)=(a1 x+b1 ) 2 +(a2 x+b2 ) 2 +…+(anx+bn) 2 ≥ 0由于二次项系数a21 +a22 +… +a2 n>0所以Δ≤ 0即 4(a1 b1 +a2 b2 +… +anbn) 2 - 4 (a21 +a22 +…b2 n)(b21 +b22 +… +b2 n)≤0故 (a21 +a22 +… +a2 n) (b21 +b22 +… +b2 n)≥ (a1 b1 +a2 b2 +… +anbn) 2这是著名的柯西 (Cauchy)不等式 .下面…     

让解题思路来得更自然一些——一道高中数学联赛试题引起的思考

1999年全国高中数学联赛的第五大题为:给定正整数n和正数M,对于满足条件　　　　　a21+a2n+1≤M(1)的所有等差数列a1,a2,…,试求S=an+1+an+2+…+a2n+1的最大值.这是一个关于数列、不等式和极值等知识的综合性题,着重考查学生综合应用知识的能力.下面是命题组提供的解答:解法1(配方法)　设公差为d,an+1=a,则　　S=an+1+an+2+…+a2n+1=(n+1)a+n(n+1)2d,故有　　　a+nd2=Sn+1.于是　M≥a21+a2n+1=(a-nd)2+a2=410(a+nd2)2+110(4a-3nd)2≥410(Sn+1)2.(2)因此　|S|≤102(n+1)M,且当a=310M,d=4101nM时,S=(n+1)[310M+n24101nM]=(n+1)510M=102(n+1)M,且由于此时4a=3nd,故a21+a2n+1=410(Sn+1)2=410.104M=M.所以　S的最大值为102(n+1)M.显然,解法1不失为一种“优美”的解答,它所用到的凑配技巧确实构思精巧,解法独特,充分体现了(凑)配方技术的魅力和解题技巧性的高明.可以说,将(1)式凑配为(...     

一个不等式的再推广及应用

2003年第64届普特兰数学竞赛A2题:设a1,a2,…,an和b1,b2,…,bn都是非负实数,证明:(a1a2…an)1n (b1b2…bn)1n≤[(a1 b1)(a2 b2)…(an bn)]1n.文[1]给出该不等式的如下推广:如果xij(i=1,2,…,m;j=1,2,…,n)为非负实数,则(x11x12…x1n)1n (x21x22…x2n)1n … (xm1xm2…xmn)1n≤[(     

试题研讨(14)

试题 ( 2 0 0 3年省际重点中学大联考 )设数列 {an}的前 n项和 Sn =n2 ( an + 1 ) ,n∈N+ ,a2 =a.( 1 )求证 :数列 {an}为等差数列 ;( 2 )若 a =3,Tn =a1a2 - a2 a3 + a3 a4-a4a5+… + ( - 1 ) n-1anan+ 1,求 Tn.命题溯源　此题是在 1 993年上海市高考试题 ( 2 5)的基础上 ,根据 1 994年全国高考试题 ( 2 5)改编的 .主要考查等差数列的基础知识、数学归纳法及推理论证能力 .原解思路　由　 Sn =n2 ( an + 1 )　得a1=1 ,又 a2 =a,则可猜想an =1 + ( n - 1 ) ( a - 1 ) ( * )下面用数学归纳法加以验证 .1 n =1、n =2时 ( * )式都成立 ;2假设…     

一个不等式的推广及应用

第 6 4届普特南数学竞赛 ( 2 0 0 3年 ) A2题为 [1 ] :设 a1 ,a2 ,… ,an 和 b1 ,b2 ,… ,bn 都是非负实数 ,则　 ( a1 a2 … an) 1 n ( b1 b2 … bn) 1 n≤ [( a1 b1 ) ( a2 b2 )… ( an bn) ]1 n ( 1 )此不等式显然等价于　 ( a1 b1 ) ( a2 b2 )… ( an bn)≥ [( a1 a2 … an) 1 n ( b1 b2 … bn) 1 n]n ( 2 )当且仅当 a1 b1=a2b2=… =anbn或 b1 ,b2 ,… ,bn 全为 0时取等号 .最近文 [2 ]给出了此不等式的一些应用 .本文首先给出 ( 2 )的一个推广 ,然后给出推广结果的一些应用 .定理　设 aij>0 ( i=1 ,2 ,… ,n;j=1 ,2 ,… ,…     

Half Space and Quarter Space Dirichlet Problems for the Partial Differential Equation

A customary, heuristic, method, by which the Poisson integral formula for the Dirichlet problem, for the half space, for Laplace's equation is obtained, involves Green's function, and Kelvin's method of images. Although this heuristic method leads one to guess the correct result, this Poisson formula still has to be verified directly, independently of the method by which it was arrived at, in order to be absolutely certain that a solution of the Dirichlet problem for the half space, for Laplace's equation, has been actually obtained. A similar heuristic method, as seems to be generally known, could be followed in solving the Dirichlet problem, for the half space, for the equation where is a real constant. However, in Part 1, a different, labor-saving, method is used to study Dirichlet problems for the equation. This method is essentially based on what Hadamard called the method of descent. Indeed, it is shown that he who has solved the half space Dirichlet problem for Laplace's equation has already solved the half space Dirichlet problem for the equation In Part 2, the solution formula for the quarter space Dirichlet problem for Laplace's equation is obtained from the Poisson integral formula for the half space Dirichlet problem for Laplace's equation. A representation theorem for harmonic functions in the quarter space is deduced. The method of descent is used, in Part 3, to obtain the solution formula for the quarter space Dirichlet problem for the equation by means of the solution formula for the quarter space Dirichlet problem for Laplace's equation. So that, indeed, it is also shown that he who has solved the quarter space Dirichlet problem for Laplace's equation has already solved the quarter space Dirichlet problem for the " equation" For the sake of completeness and clarity, and for the convenience of the reader, the appendix, at the end of Part 3, contains a detailed proof that the Poisson integral formula solves the half space Dirichlet problem for Laplace's equation. The Bibliography for Parts 1,2, 3 is to be found at the end of Part 1.     

Asymptotic for the cumulative distribution function of the degrees and homomorphism densities for random graphs sampled from a graphon

We give asymptotics for the cumulative distribution function (CDF) for degrees of large dense random graphs sampled from a graphon. The proof is based on precise asymptotics for binomial random variables. This result is a first step for giving a nonparametric test for identifying the degree function of a large random graph. Replacing the indicator function in the empirical CDF by a smoother function, we get general asymptotic results for functionals of homomorphism densities for partially labeled graphs. This general setting allows to recover recent results on asymptotics for homomorphism densities of sampled graphon.     

Various motivations for managerial (mis)reporting – an experimental study

In the experiment we model all possible consequences from misreporting for both the shareholder and for the manager, since we are interested in patterns in reporting behaviour resulting from different motivations for potential misrepresentation. This allows for examining the stability of the (mis)reporting behaviour in different treatments. Agents are primarily driven by the consequences for themselves rather than by the consequences for the principal, while deciding on misreporting. Participants are willing to sacrifice a small gain for themselves in order to prevent a greater loss for the principal. If agents misreport, they do it in order to generate positive rather than negative consequences for themselves. Reports in favour of the principal, but fruitless or even costly for the agent are very rare. The experiment indicates also that pro-social agents report more truthfully than pro-self agents.     

Trigonometric polynomial or exponential fitting approach?

The application of a trigonometric polynomial and an exponential fitting approach is compared for a three-point formula for second-order derivatives, for Simpson’s quadrature rule and for Numerov’s scheme for second-order differential equations. The expressions for the occurring parameters are constructed in both the approaches and the behaviour of these parameters with respect to the introduced frequency is studied. The errors for specific problems obtained in both the approaches as a function of the frequency are compared.     

非强制混合向量变分不等式解的存在性研究

本文利用例外簇方法研究非强制混合向量变分不等式的弱有效解的存在性:首先证明若混合向量变分不等式问题不存在例外簇,则混合向量变分不等式问题的弱有效解集为非空集合:利用向量值映射的渐近映射给出自反Banach空间中非强制混合向量变分不等式的弱有效解集不存在例外簇的充分条件,从而得到混合向量变分不等式问题的弱有效解的存在性结...     

‘Strong’-‘weak’ precedence in scheduling: Extensions to series-parallel orders

We examine computational complexity implications for scheduling problems with job precedence relations with respect to strong precedence versus weak precedence. We propose a consistent definition of strong precedence for chains, trees, and series-parallel orders. Using modular decomposition for partially ordered sets (posets), we restate and extend past complexity results for chains and trees as summarized in Dror (1997) [5]. Moreover, for series-parallel posets we establish new computational complexity results for strong precedence constraints for single- and multi-machine problems.     

Non-compact generalized variational inequalities for quasi-monotone and hemi-continuous operators with applications

Some results are obtained for non-compact cases in topological vector spaces for the existence problem of solutions for some set-valued variational inequalities with quasi-monotone and lower hemi-continuous operators, and with quasi-semi-monotone and upper hemi-continuous operators. Some applications are given in non-reflexive Banach spaces for these existence problems of solutions and for perturbation problems for these set-valued variational inequalities with quasi-monotone and quasi-semi-monotone operators. This revised version was published online in June 2006 with corrections to the Cover Date.     

微分、差分方程的机械化方法

介绍了微分与差分方程机械化方法研究若干最新进展.主要结果包括: 微分、差分方程的特征列理论与算法,微分、差分方程系统的分解算法以及微分、差分方程解析解求解算法.     

Geometric control of hybrid systems

In this paper, we present a geometric approach for computing controlled invariant sets for hybrid control systems. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support functions.