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# AMC代考美国数学竞赛代考American Mathematics Competitions代考|Introduction

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AMC美国数学思维活动是一项面向世界中学生的数学竞赛，由美国数学协会MAA主办，目前每年全球超过6000所学校的30万名同学参赛，是全球非常有影响力的青少年数学竞赛之一。AMC的命题由美国AMC委员会全权负责，该委员会成员皆来自MIT、Harvard、Princeton等全美一流学府。

AMC活动不仅促进了数学在全球的交流与发展，而且为国际高校了解入学申请者在数学上的学习成就提供了重要依据。随着同学们对美国数学思维活动AMC的了解，未来将有更多中国学生通过AMC活动走向世界舞台，与全球学生共同探索数学问题，感受数学学习的快乐。

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## AMC代考美国数学竞赛代考American Mathematics Competitions代考|Introduction

As an appetizer, here is a typical problem of the kind Sections $1-5$ should equip you to solve. You are invited to try it as soon as you wish. You may find it hard for now, but by the end of Section 5, where its solution is given, it should not look difficult to you.

Appetizer Problem: Yeukai’s grandmother has given her $\$ 10$, and asked her to buy the maximum possible total number of mangoes and oranges, using all the money, but getting more oranges than mangoes. Mangoes cost 7 cents each and oranges cost 13 cents each. What should she buy? The term ‘Diophantine equation’ refers to any equation in one or more variables whose solution(s) must come from a restricted set of numbers. The set might be the rationals, integers, non-negatives, etc. For example, suppose we have to find all$x, y \in \mathbb{Z}$such that $$21 x+7 y=11,$$ then we have to solve a linear Diophantine equation in two variables. In fact, the equation above has no integral solutions, for reasons you shall soon find out. In our discussion we shall restrict ourselves to solutions coming from the set of integers, unless otherwise stated. Diophantine equations are named after Diophantus of Alexandria, who lived around 250 CE. Not much is known about him, but he was an Egyptian who received a Greek education. Diophantus did much work on the exact solution of equations and gave considerable impetus to the slow development of algebraic symbolism that culminated in our modern economical symbolic notation. Before we can study the methods for solution of Diophantine equations, we must introduce some necessary concepts and procedures. ## AMC代考美国数学竞赛代考American Mathematics Competitions代考|Division algorithm and greatest common divisor This algorithm formalizes the procedure of ‘division with remainders’ in the integers. Given integers$a, b$with$b>0$, there exist unique$q, r \in \mathbb{Z}: a=q b+r$, with$0 \leq r<b$; that is,$\frac{a}{b}=q+\frac{r}{b}$, where$q$is the quotient and$r$is the remainder. If$r=0$we say that$b$divides$a$, and we write$b \mid a$. The greatest common divisor of$a$and$b$, denoted by$\operatorname{gcd}(a, b)$, is the largest positive integer which divides both$a$and$b$. For this to exist, at least one of the integers$a$and$b$must be non-zero, for 0 is divisible by any number. Let’s define the gcd formally: Definition: Let$a, b \in \mathbb{Z}$not both zero; then$\operatorname{gcd}(a, b)$is the unique natural number$d$such that: (i) both$d \mid a$and$d \mid b$; (ii) if$c \mid a$and$c \mid b$, then$c \mid d$. The natural question is: How do we find$\operatorname{gcd}(a, b)$, for any given$a, b \in \mathbb{Z}$? One way is to factorize both, and then select the factors that appear in both. This can be very time-consuming for large numbers; even modern computers lack the speed to factorize very large numbers efficiently. (This is exploited in methods of safe encryption of information.) Another way is to use the following schema, given by Euclid. The basic idea is to divide larger by smaller, then divide smaller by remainder (which is smaller than it), and so on, until the division is exact. ## 美国数学竞赛代考 ## AMC代考美国数学竞赛代考American Mathematics Competitions代 考|lntroduction 作为开甶萫，这里有一个典型的问题类$1-5$应该装备你来解决。诚䢍您怸快尝试。你现在可能会觉得很难，但是到第 5 节结束 时，给出了它的解决方菲，对你来说应该不难。 开胃菜问题: Yeukai的祖母给了她$\$10$ ，并要求她用所有的钱购买层可能茤的芒果和橙子，但橙子比芒果岁。芒果一个 7 美分， 橙子—个 13 美分。她应该买什么?

$$21 x+7 y=11$$

## AMC代考美国数学竞赛代考American Mathematics Competitions代考|Division algorithm and greatest common divisor

(i) 两者 $d \mid a$ 和 $d \mid b$;
(ii) 如果 $c \mid a$ 和 $c \mid b$ ，然后 $c \mid d$.

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。