C3-2

Functions  函数 1、①函数基本概念:A mapping(映射) transforms one set of numbers into a different set of numbers. 注:区别映射与函数:映射(mapping)包括函数(function),特别的,当映射的两个sets都为数集时,称为函数。我们可以理解为两个sets of number之间的映射关系,其中,set A我们称之为domain(定义……

C3-1

Algebraic fraction 分式 1、simplify fraction 化简: Algebraic fractions can be simplified by cancelling down(互消). To do this the numerators and denominators must be fully factorised first.(首先要将分子分母因式分解) 注:这里的化简包括分子分母是数和含有未知数、字母的式子。For example: 应当注意到……

C1-6、7、8

本篇文章围绕“sequence and series”“differentiation”和“integration”即数列、微粉和积分来展开,(会涉及多种英文名称): 1、常用数列表达: 通项公式;general term   or  the nth term 递推关系:recurrence relation 等差数列:arithmetic sequence 公差:common difference 2、导数(derived function): the gradi……

C1-2

这里是数学天堂的第二篇文章,希望解题愉快 1、首先,何为函数?一个集合(自变量)映射到另一个集合(因变量)的过程即为函数。(映射的知识点暂时不在这里赘述) 拓展:生活中利用某些特殊值来设未知数建立函数,从而来预测不特殊的情况。这就可以看作是数学建模的过程,并能解决实际的问题,比如最常见的线性规划:通过……

C1-1

1、Collecting like terms(合并同类项): 3x+2xy+7-x+3xy-9=2x+5xy-2 拓展:对于m个字母(x、y…)组成的齐次式(n次)中,若有大于等于m^n+1项,则有至少两项属于同类项 2、 Rules of indices(指数): The rules also apply to all irrational number  √a (≠a/b). 补充:Surd: “√” 拓展:证明√2为无理数:若√2为有理……

CCC竞赛 2011S5: Switches

Problem Description You are walking by a row of K (4 ≤ K ≤ 25) lights, some of which are on and some of which are off. In this initial configuration, there is no consecutive sequence of four lights that are on. Whenever four or more consecutive lights are on, the lights in that consecutive block w……

CCC竞赛 2014S5: Lazy Fox

Problem Description You have a pet Fox who loves treats. You have N neighbours at distinct locations (described as points on the Cartesian plane) which hand out treats to your pet Fox, and each neighbour has an unlimited number of treats to give out. The origin (which is where the Fox starts) will……

C1-3

本篇文章围绕“equations and inequalities”即等式与不等式来展开:   常见的linear equations拥有两个solution: By elimination 消元法  and  by substitution代入法   拓展:对于联立的one quadratic equation and one linear equation ,可以参考图像的交点数来得到解的个数,并通过substitution来得到具……

C1-4

本篇文章围绕“sketch”即画图来展开,(会涉及多种英文名称): cubic curve(三次曲线):general form y=ax^3+bx^2+cx+d(a≠0)   拓展:对于n次函数,因式分解后y=(x+a)(x+b)(x+c)…,利用规律“奇穿偶不穿”即可sketch图像。 补充:Reciprocal functions(反比例函数): 有asymptote(渐近线) transformations(图像……