一道第50届IMO试题的探究

2009年第50届IMO的第6题是一个组合问题： 设α，α2，…，an是互不相同的正整数．M是有n-1个元素的正整数集，且不含数s=a1＋α2＋…＋an．     

对一道联赛题的多方位探究

1999年全国高中联赛试题的第五大题是：给定正整数n和正数M，对于满足条件a1^2＋an＋1^2≤M的所有等差数列：a1，a2，a3，…，试求S=an＋1＋an＋2＋…＋a2n＋1的最大值．     

第55届IMO试题解答

1．设a0〈a1〈…为无穷正整数数列．证明：存在唯一的整数n（n≥1），使得 an〈a0＋a1＋…＋an/n≤an＋1.     

第54届IMO预选题（二）

1．已知n为正整数．求满足下述性质的最小正整数k：给定任意实数a1,a2,…,ad,且a1＋a2＋…＋ad=n（0≤ni≤1,i=1,2,…,d）,总能将这些数拆分为k组（某些组可以是空的）,使得每组中的数的和最大为1．     

第46届IMO两道试题另解

题1 设a1，a2，…是整数序列，其中有无穷多项为正整数，也有无穷多项为负整数．假设对每个正整数n，数a1，a2，…，an被n除的余数都各不相同．证明：在数列a1，a2，…中，每个整数都刚好出现一次．     

江苏卷第20题

题目 设M为部分正整数组成的集合,数列{an）的首项a1=1,前n项的和为Sn,已知对任意整数k∈M,当n〉k时,Sn＋k＋Sn-k=2（Sn＋Sk）都成立．     

1999年全国高中数学联赛第五题简解

给定正整数n和正数M，对于满足条件的所有等差数列a1，a2，a3，…，试求S＝an＋1＋an＋2＋…＋a2n＋1的最大值．简月一（三角换元法）设山一／页rsln6，a。＋1一／页。os9（0＜r＜l，0＜6＜Zx）．6等号当r。1，0＋922。时取得，其中税。＿＿3．1角9满足cospory，sinpory．，·。r·r’。—一。he”－－”he一简解二（Cauchy不等式）1999年全国高中数学联赛第五题简解@徐民江$浙江省绍兴县越崎中学!312050 @樊宏标$浙江省绍兴县越崎中学!312050…     

浅析二项式定理的应用

二项式定理： 对于任意两个数a和b以及正整数n，总有（a＋b）n=Cn0an＋Cn2an-1b＋Cn2an-2b2＋…＋Cnran-rbr＋…＋Cnnbn,式中Cnm为组合数．公式右边的多项式称为二项展开式，又称牛顿二项展开式．     

未知量为正整数的二次不等式

问题1 在数列{an}中,an=n^2-kn．若对任意的正整数n,an≥a3都成立,求实数k的取值范围． 问题2 不等式n^2＋pn＋q≥0对任意正整数n恒成立的充要条件是什么？（常数p,q∈R）     

关于最大公因数和最小公倍数的一个方程

证明了：方程gcd（a1,a2,…，an）＋lcm（a1,a2,…，an）=a1,a2,…，an。仅当n=2时有正整数解（a1,a2）=（2，2）．     

Burning a Candle in a Vessel, a Simple Experiment with a Long History

The experiment in which a candle is burned inside an inverted vessel partially immersed in water has a history of more than 2,200 years, but even nowadays it is common that students and teachers relate the change in volume of the enclosed air to its oxygen content. Contrary to what many people think, Lavoisier concluded that any change in volume in this experiment is negligible; moreover, the explanation relating oxygen consumption in the air with its change in volume is known to be wrong. In this work we briefly review the history behind the candle experiment and its relationship with some typical erroneous explanations. One of the key factors behind Lavoisier’s success was the use of experiments carefully designed to test different hypotheses. Following these steps, we performed several closed volume experiments where the candle wick was replaced by a capillary stainless steel cylinder supported and heated by a nichrome filament connected to an external power supply. Our recorded experiments are displayed as web pages, designed with the purpose that the reader can easily visualize and analyze modern versions of Lavoisier’s experiments. These experiments clearly show an initial phase of complete combustion, followed by a phase of incomplete combustion with elemental carbon or soot rising to the top of the vessel, and a final phase where the hot artificial wick only evaporates a white steam of wax that cannot ignite because no oxygen is left in the closed atmosphere. After either a complete or incomplete combustion of the oxygen, our experiments show that the final gas volume is nearly equal to the initial air volume.     

天保房地产开发公司发展战略研究

天保房地产开发公司根植于天津滨海新区。依托控股公司带来的资源优势，基于滨海新区列入国家发展战略部署的背景，在近年来取得了迅速的发展。但是随着国家一系列政策的出台，中国的房地产行业面临着一次大的重新洗牌过程，同时企业内部仍然存在着制度缺陷、管理不规范、流程不清晰、人员激励不足等各方面的问题，所以，天保房地产公司在面临着滨海新区良好发展机遇的同时，也面临着很大的挑战。如何在机遇与挑战并存的环境中，认真分析和评估企业的外部环境及内部资源与能力的优劣势，对企业内部的资源与能力进行积累与培养，制定相应的竞争与发展战略，引导企业的健康长远发展具有一定的必要性与迫切性。 本文从竞争战略的角度，综合分析天保房地产开发公司所面临的外部环境和内部条件，研究探讨天保房地产开发公司的发展战略框架，包括战略的指导思想、战略定位、战略目标及相应的实施措施，以期实现天保房地产开发公司的发展愿景和核心价值。     

指鹿为马

秦二世时,丞相赵高野心勃勃,日夜盘算着要篡夺皇位。可朝中大臣有多少人能听他摆布,有多少人反对他,他心中没底。于是,他想了一个办法,准备试一试自己的威信,同时也可以摸清敢于反对他的人。     

Short a and Long a

Easy Phonics带你学习自然拼读,让你学会字母的发音规律,正确地拼读单词。本期让我们继续来看看、读读、写写重读音节中包含字母a的单词,区分它们读短音short a-/e/,还是长音long a-/ei/。     

Short a and Long a

Easy Phonics栏目带你轻松地学习自然拼读!在读读、听听、拼拼和玩游戏中,你将了解字母的发音规律,并正确地拼读单词。这一期,我们来学习元音字母a的短音(short a)和长音(long a).     

做校长·做学问·做人

从前,有一句话,叫做"吃得苦中苦,方为人上人".这句话被认为动机不纯,怎么可以做"人上人"呢?其实,对于"人上人"可以作不同的理解,一种理解是"骑在人民头上,作威作福",这种"人上人"当然不好;但也可以解释为比别人做更多的贡献,吃更多的苦,经过更多的挫折与困难而比一般人更成熟、更坚强,更知道怎么做人.实际上,对这句话应是作后一种理解的,因为这种"人上人"是与"苦中苦"联系在一起的.     

毕飞宇:一镜、一钟、一影、一胡琴而已

本文从毕飞宇小说中一再出现的镜、钟、影和胡琴的意向入手,考察毕飞宇小说的叙述人称、叙述速度、叙述结构和叙述意图。深入阐释＂我＂与＂他＂的分离与揉合,＂快＂与＂慢＂的交替与演变,＂是＂与＂非＂的分明与模糊,＂语言＂与＂沉默＂的相反而相成。     

Paving a clear path in a thick forest: a conceptual analysis of a metacognitive component

The concept of metacognition refers to one’s knowledge and control of one’s own cognitive system. However, despite being widely used, this concept is confusing because of several reasons. First, sometimes it is not at all clear what is cognitive and what is metacognitive. Second, researchers often use the same term, namely, “metacognition” even when they refer to very different aspects of this complex concept. Alternatively, researchers may use different terms to indicate the same metacognitive elements. Another foggy matter is the interrelationships among the various components of metacognition discussed in the literature. This conceptual confusion regarding the concept of metacognition and its sub-components calls for in-depth theoretical and conceptual clarifications. The goal of this article is to portray a detailed example of a conceptual analysis of meta-strategic knowledge (MSK) which is one specific component of metacognition. This specific example is used to draw a general model for conceptual analyses of additional metacognitive components. The approach suggested here is to begin with a clear definition of the target sub component of metacognition, followed by a systematic examination of this sub component according to several dimensions that are relevant to metacognition in general and to that sub component in particular. The examination should include an analysis of how the details of the definition of the target sub-component refer to: (a) general theoretical metacognitive issues raised by prominent scholars; (b) definitions formulated and issues raised by other researchers who have investigated the same (or a similar) sub-component and, (c) empirical findings pertaining to that sub-component. Finally, it should be noted that since metacognition is a relational rather than a definite concept it is important to situate the context within which the conceptual analysis takes place.