1998年全国初中数学竞赛试题及参考答案

一、选择题（每小题6分，满分30分）1已知a，b，c都是实数，并且a＞b＞c，那么下列式子中正确的是（B）（A）ah＞ie（B）a＋b＞b＋c（C）a－b＞b－c（D）H＞上co解根据不等式性质，选旧）2如果方程x2＋pH＋1＝0（户＞0）的两根之差为1，那么户等于（D）（A）2（B）4（C）月（D）后解由6一p’－4＞0及p＞2，设xl，。2为方程的两根，那么有一l＋12一一p，x112＝1又由3在OABC中，已知BD和CE分别是两边上的中线，并且BD上CE，BD＝4，CE—6，那么］ABC的面积等于（C）解如图，连ED，4已知de产0，并且在上上一旦u＝7厂一P，那么直线…     

On sums of a prime and four prime squares in short intervals

In this paper, we prove that each sufficiently large integer N ≠1（mod 3） can be written as N=p＋p1^2＋p2^2＋p3^2＋p4^2, with
｜p-N/5｜≤U,｜pj-√N/5｜≤U,j=1,2,3,4,
where U=N^2/20＋c and p,pj are primes.     

分担多项式的亚纯函数的进一步结果(英文)

In this paper,we use the theory of value distribution and study the uniqueness of meromorphic functions.We will prove the following result：Let f（z）and g（z）be two transcendental meromorphic functions,p（z）a polynomial of degree k,n≥max{11,k＋1}a positive integer.If fn（z）f（z）and gn（z）g（z）share p（z）CM,then either f（z）=c1ec p（z）dz, g（z）=c2e ？c p（z）dz ,where c1,c2 and c are three constants satisfying（c1c2） n＋1 c2=-1 or f（z）≡tg（z）for a constant t such that tn＋1=1.     

一类脉冲微分方程正周期解存在的充要条件

考虑非线性脉冲微分方程{x＇（t）=x（t）[a（t）-b（t）x^p（t）],t≠tk, △x｜t=tk=ckx（tk）,k∈N.得到了该方程存在正周期解的充要条件为m∏k=1（1＋ck）^pexp（p∫^w 0）a（σ）dσ）＞1.     

一个推广的具有最佳常数的Hardy-Hilbert不等式

本文的目的是建立新的具有最佳常数因子的Hardy-Hilbert不等式的推广式．对二重级数适当配方,利用Hlder不等式及β-函数,得到下面的推广式：∑_(m=1)~∞∑_(n=1)~∞((a_nb_n)/(m~c n~c)■)＜cλ,p(∑n~((P-1)(1-λ))a_n~p)~(1/p)(∑n~((q-1)(1-λ))b_n~q)~(1/q),这里λ＞0,c＞0,p＞1,(1/p) (1/q)=1,a_n≥0,b_n≥0,cλ,p=(1/c)B((λ/cp),(λ/cq)),通过选取两个特殊序列,证明了常数因子cλ,p是最佳的；还给出了它的等价形式,用类似方法给出了重积分形式的Hardy-Hilbert不等式的推广式及其等价形式．     

赋Day范数的c0（Г,X）的某些性质

设Г为一非空集，（X1||·||）为Banach空间，本主要结果如下：（1）U（C0（Г，X），p）为稳定的当且仅当U（X）是稳定的。（2）设Г为无限集，那么下列三条等价：（a)(c0(Г,X),p)有λ-性质，（b)(c0(Г,X),p)有一致λ-性质，（c)(X1||·||）有一致λ-性质。（3）设Г为有限集，那么（c0(Г,X),p)有λ-性质（相应地，一致λ-性质）当且仅当（X1||·||）有λ-性质（相应地，一致λ-性质）。（4）（C0（Г，X），p)有Kadets性质（相应地，Kadets-Klce性质）当且仅当（X1||·||）有Kadeta性质（相应地，Kadets-Klee性质）。（5）w∈S（Cp(Г,X),p)是U（c0(Г,X),p)的可凹点（相应地，PC）当且仅当对于任意的t∈S（w),w(t)是(x∈X:||x||≤||w(t)||）的可凹点（相应地，PC）。     

Sums of nine almost equal prime cubes

We prove that each sufficiently large odd integer N can be written as sum of the form N = p1^3 ＋p2^3 ＋... ＋p9^3 with [pj - （N/9）^1/31 ≤ N^（1/3）-θ, where pj, j = 1,2,...,9, are primes and θ = （1/51） -ε.     

具有非线性边界条件半线性热方程解的爆破估计(英)

本文考虑具有非线性边界条件-u_x（0，t）=u~q（0，t）在（0，T）半线性热方程u_t-u_(xx)=u~p在（0，∞）×（0，T）的正解．在初值的一些单调性假定下证明了c（T-t）~(-t)≤supu（x，t）≤C（T-0）~(-r)其中r=1／（p-1）如户＞2q-1且r=1／[2（q-1）]如p≤2q-1.     

Nonradial Entire Large Solutions of Semilinear Elliptic Equations

We consider the problem of whether the equation △u = p（x）f（u） on RN, N ≥ 3, has a positive solution for which lim ｜x｜→∞（x） = ∞ where f is locally Lipschitz continuous, positive, and nondecreasing on （0,oo） and satisfies ∫1∞[F （t）]^- 1/2dt = ∞ where F（t） = ∫0^tf（s）ds. The nonnegative function p is assumed to be asymptotically radial in a certain sense. We show that a sufficient condition to ensure such a solution u exists is that p satisfies ∫0∞ r min｜x｜=r P （x） dr = ∞. Conversely, we show that a necessary condition for the solution to exist is that p satisfies ∫0∞r1＋ε min ｜x｜=rp（x）dr =∞ for all ε〉0.     

一道高考试题的别解

2011年高考浙江数学理科试题第18题为：在△ABC中,角A,B,C所对的边分别为a,b,c已知sinA＋sinC=psinB（p∈R）且ac=4/1b2.（1）当p=54,b=1时,求a,c的值;（2）若B为锐角,求p的取值范围.     

Difference Equations and Divisibility Properties of Sequences

There are many different ways of defining a sequence in terms of solutions to difference equations. In fact, if a sequence satisfies one recurrence then it satisfies an infinite number of recurrences. Arithmetic properties of an integral sequence are often studied by direct methods based on the combinatorial or algebraic definition of the numbers or using their generating function. The rational generating function is the main tool in obtaining various difference equations with coefficients and initial values exhibiting divisibility patterns that can imply particular arithmetic properties of the solutions. In this process, we face the challenging task of finding difference equations that are relevant to the divisibility properties by transforming the original rational generating function. As a matter of fact, it is not necessarily the simple difference equation that helps the most in proving the properties. We illustrate this process on several examples and a sequence involving a p -sected binomial sum of the form y n = y n ( p , a )= ~ k =0 X n kp a k where p is an arbitrary prime. Let 𝜌 p ( m ) denote the exponent of the highest power of a prime p which divides m . Recently, the author obtained lower bounds for 𝜌 p ( y n ) based on recurrence relations of order p and p m 1. The cases with tight bounds have also been characterized. In this paper, we prove that 𝜌 p ( y np ( p , a ))= n for 𝜌 p ( a +1)=1, p S 3. We obtain alternative difference equations of order p 2 for y n and order p for the p -sected sequence y np by a generating function based method. We also extend general divisibility results relying on the arithmetic properties of the coefficients and initial values.     

非结合非分配的环(Ⅱ)

本文继上文(Ⅰ)的理论,共分二节,第一节继续上文的理想理论,引进两非环的半素理想及半准素理想概念,并对它们作基本性质的研究.第二节引进根及半单纯概念建立分解成单纯子环直和的有关诸定理.     

产品检验中的抽样个数

设某批产品 (总体 )的次品率为 p,对总体提出假设H0 :p≤ p0 ,　 H1:p >p1其中 0 相似文献   

一个素变数丢番图方程

证明了如果 １＜ｃ＜２５８／２３５，则素变数丢番图方程对充分大的 Ｎ是可解的，改进了 Ｋｕｍｃｈｅｖ和 Ｎｅｄｅｖａ的结果 １＜ｃ＜１２／１１．     

一个素变数丢番图不等式

本文证明了如果１＜ｃ＜２３７／２１４，则对于充分大的Ｎ和ε≥Ｎ－１／５（２３７／２１４－ｃ）＋ｖ（ｖ＞０）表达式Ｄ（Ｎ）：＝Σ｜ｐｃ１＋ｐｃ２＋ｐｃ３－Ｎ｜＜－εｌｏｇｐ１ｌｏｇｐ２ｌｏｇｐ３ 关于素变数ｐ１，ｐ２，ｐ３有渐近公式，改进了文献［１］的结果１＜Ｃ＜１．１．     

在零点的隣區內彼此相等的特徵函数

<正> §1.引言 大家知道,兩個不相恆等的特徵函數(以下简称特函)可以在零點的隣區內相等。為固定用語起見,在本文中我們說特函f(t)属於集合(U),如果存在一個特函,它与f(t)在零的隣區內相等,但並不恆等於f(t);如果f(t)不屬於(U),就說它屬於(U)。     

奇素数的三项方幂和中的平方数

研究了一类基本而又重要的指数Diophantine方程,利用广义Ramanujan-Nagell方程的性质证明了这类方程有非负整数解的充要条件,并得出这类方程的全部非负整数解.     

球面Sn+p(c)中子流形的拓扑

研究了球面Sn+p(c)子流形Mn的Pinching定理,证明了当s＜2√n-1c时,Mn(n＞3)与n维球面同胚.     

ASYMPTOTICALLY ISOMETRIC COPIES OF lp (1≤p＜∞) AND c0 IN BANACH SPACES

Let X be a Banach space. If there exists a quotient space of X which is asymptotically isometric to l1, then X contains complemented asymptotically isometric copies of l1. Every infinite dimensional closed subspace of l1 contains a complemented subspace of l1 which is asymptotically isometric to l1. Let X be a separable Banach space such that X* contains asymptotically isometric copies of lp (1 ＜ p ＜∞). Then there exists a quotient space of X which is asymptotically isometric to lq (1/p 1/q=1). Complementedasymptotically isometric copies of c0 in K(X, Y) and W(X, Y) are discussed. Let X be a Gelfand-Phillips space. If X contains asymptotically isometric copies of c0, it has to contain complemented asymptotically isometric copies of c0.     

Lehmer问题的两个推广

设素数P>2,整数C与P互素．对任意整数1≤a≤P-1,存在惟一的整数 1≤b≤P-1满足ab≡c mod P．Lehmer建议我们研究a与b的奇偶性不同的情形．本文给出了这一问题的两个推广,并获得了两个有趣的混合均值公式．