Robot Rossie moves within a square room $A B C D$. Rossie moves along straight line segments, never leaving that room.
When Rossie encounters a wall she stops, makes a right-angle turn (with direction chosen to face into the room), and continues in that new direction.
If Rossie comes to one of the room’s corners, she rotates through two right angles, and moves back along her previous path.
Suppose Rossie starts at point $P$ on $A B$ and her path begins as a line segment of slope $s$.
We hope to describe Rossie’s path.
For some values of $P$ and $s$, Rossie’s path will be a tilted rectangle with one vertex on each wall of the room. (Often, this inscribed rectangle is itself a square.) In this case, Rossie repeatedly traces that stable rectangle.
(a) Suppose $s=1$ so that the path begins at a 45 degree angle.
For every starting point $P$, show: Rossie’s path is a stable rectangle.
(If $P$ is a corner point, the path degenerates to a line segment traced back and forth.)
Now draw some examples with various $P$ and $s$.
Given $P$ and $s$, does Rossie’s path always converge to a stable rectangle?
Here are some steps that might help you answer this question:
(b) First consider the case: $01$ or when $s<0$ ? Does the argument above still apply?
Let $\mathbb{Z}$ denote the set of integers. If $m$ is a positive integer, we write $\mathbb{Z}m$ for the system of “integers modulo $m$.” Some authors write $\mathbb{Z} / m \mathbb{Z}$ for that system. For completeness, we include some definitions here. The system $\mathbb{Z}_m$ can be represented as the set ${0,1, \ldots, m-1}$ with operations $\oplus$ (addition) and $\odot$ (multiplication) defined as follows. If $a, b$ are elements of ${0,1, \ldots, m-1}$, define: $a \oplus b=$ the element $c$ of ${0,1, \ldots, m-1}$ such that $a+b-c$ is an integer multiple of $m$. $a \odot b=$ the element $d$ of ${0,1, \ldots, m-1}$ such that $a b-d$ is an integer multiple of $m$. For example, $3 \oplus 4=2$ in $\mathbb{Z}_5$, $3 \odot 3=1$ in $\mathbb{Z}_4$, and $-1=12$ in $\mathbb{Z}{13}$.
To simplify notations (at the expense of possible confusion), we abandon that new notation and write $a+b$ and $a b$ for the operations in $\mathbb{Z}_m$, rather than writing $a \oplus b$ and $a \odot b$.
Let $\mathbb{Q}$ denote the system of rational numbers.
We write $4 \mathbb{Z}$ for the set of multiples of 4 in $\mathbb{Z}$. Similarly for $4 \mathbb{Z}{12}$. Consider the following number systems: $$ \mathbb{Z}, \quad \mathbb{Q}, \quad 4 \mathbb{Z}, \quad \mathbb{Z}_3, \quad \mathbb{Z}_8, \quad \mathbb{Z}_9, \quad 4 \mathbb{Z}{12}, \quad \mathbb{Z}_{13} .
$$
One system may be viewed as similar to another in several different ways. We will measure similarity using only algebraic properties.
(a) Consider the following sample properties:
(i) If $a^2=1$, then $a=\pm 1$.
(ii) If $2 x=0$, then $x=0$.
(iii) If $c^2=0$, then $c=0$.
Which of the systems above have properties (i), (ii), and/or (iii)?
(b) Formulate another algebraic property and determine which of those systems have that property. [Note: Cardinality is not considered to be an algebraic property.]
Write down some additional algebraic properties and investigate them.
(c) In your opinion, which of the listed systems are “most similar” to each another?
Please spend extra effort to write up this problem’s solution as an exposition that can be read and understood by a beginning algebra student. That student knows function notation and standard properties of polynomials (as taught in a high school algebra course). Your solution will be graded not only on the correctness of the math but also on the clarity of exposition.
(a) Find all polynomials $f$ that satisfy the equation:
$$
f(x+2)=f(x)+2 \text { for every real number } x .
$$
(b) Find all polynomials $g$ that satisfy the equation:
$$
g(2 x)=2 g(x) \text { for every real number } x .
$$
(c) The problems above are of the following type: Given functions $H$ and $J$, find all polynomials $Q$ that satisfy the equation:
$$
J(Q(x))=Q(H(x)) \text { for every } x \text { in } S
$$
where $S$ is a subset of real numbers. In parts (a) and (b), we have $J=H$ and $S$ is all real numbers, but other scenarios are also interesting. For example, the choice $J(x)=1 /(x-1)$ and $H(x)=1 /(x+1)$, generates the question:
Find all polynomials $Q$ that satisfy the equation:
$$
\frac{1}{Q(x)-1}=Q\left(\frac{1}{x+1}\right)
$$
for every real number $x$ such that those denominators are nonzero.
Is this one straightforward to solve?
(d) Make your own choice for $J$ and $H$, formulate the problem, and find a solution. Choose $J$ and $H$ to be non-trivial, but still simple enough to allow you to make good progress toward a solution.
机器人Rossie在一个正方形房间$A B C D$内移动。罗西沿着直线段移动，从不离开这个房间。
当Rossie遇到一堵墙时，她会停下来，做一个直角转弯（方向选择为面向房间），然后继续沿着这个新方向前进。
如果Rossie走到房间的一个角落，她会旋转两个直角，然后沿着之前的路径移动回去。
假设Rossie从$A B$上的$P$点开始，她的路径是一条斜率为$s$的线段。
我们希望描述一下Rossie的路径。
对于$P$和$s$的某些值，Rossie的路径将是一个倾斜的矩形，在房间的每一面墙上都有一个顶点。 (通常，这个内嵌的矩形本身就是一个正方形。)在这种情况下，Rossie重复地追踪这个稳定的矩形。
(a) 假设$s=1$，使路径以45度角开始。
对于每一个起点$P$，表明。罗西的路径是一个稳定的矩形。
(如果$P$是一个角点，该路径就退化为一条来回追踪的线段)。
现在画一些有不同$P$和$s$的例子。
考虑到$P$和$s$，罗西的路径是否总是收敛到一个稳定的矩形？
下面是一些步骤，可能有助于你回答这个问题。
(b) 首先考虑以下情况：$01$或$s<0$时，Rossie的行为是什么？上面的论证是否仍然适用？
让$mathbb{Q}$表示有理数系统。
我们用$4\mathbb{Z}$表示$mathbb{Z}$中4的倍数的集合。类似地，4美元\mathbb{Z}{12}$。 请考虑以下数系。 $$ `mathbb{Z}, quadmathbb{Q}, quad 4mathbb{Z}, quadmathbb{Z}_3, quadmathbb{Z}_8, quadmathbb{Z}_9, quad 4mathbb{Z}{12}, quadmathbb{Z}_{13}.
$$
一个系统可以通过几种不同的方式被视为与另一个系统相似。我们将只用代数性质来衡量相似性。
(a) 考虑以下的样本属性。
(i) 如果$a^2=1$，那么$a=\pm 1$。
(ii) 如果$2 x=0$，那么$x=0$。
(iii) 如果$c^2=0$，则$c=0$。
上述系统中哪一个具有(i)、(ii)和/或(iii)的特性？
(b) 提出另一个代数性质，并确定这些系统中哪些具有该性质。[注意：Cardinality不被认为是一个代数属性。］
写下一些额外的代数性质，并对它们进行研究。
(c) 在你看来，所列的系统中哪些是 “最相似 “的？
请花更多的精力把这个问题的解决方案写成一个初学代数的学生可以阅读和理解的论述。该学生知道函数符号和标准的多项式性质（如高中代数课程中所教授的）。你的答案不仅要看数学的正确性，还要看论述的清晰性。
(a) 找到所有满足方程的多项式$f$。
$$
f(x+2)=f(x)+2\text {对于每一个实数}x 。
$$
(b) 找出所有满足方程的多项式$g$。
$$
g(2 x)=2 g(x)\text { 对于每个实数 } x .
$$
(c) 上面的问题属于以下类型: 给出函数$H$和$J$, 找出所有满足方程的多项式$Q$:
$$
J(Q(x))=Q(H(x)) \J(Q(x))=Q(H(x))。S
$$
其中$S$是实数的一个子集。在(a)和(b)部分，我们有$J=H$，$S$为所有实数，但其他情况也很有趣。例如, 选择$J(x)=1 /(x-1)$和$H(x)=1 /(x+1)$, 产生了问题:
找到所有满足方程的多项式$Q$。
$$
\frac{1}{Q(x)-1}=Q\left(\frac{1}{x+1}\right)
$$
对于每个实数$x$来说，这些分母都是非零的。
这个问题是否可以直接解决？
(d) 自己选择$J$和$H$，提出问题，并找到解决方案。选择$J$和$H$是不难的，但仍然简单到足以让你在解决问题上取得良好进展。
Ross数学夏令营2023选拔代写
斯坦福大学数学夏令营保录取Sumac代写2023 请认准avatest™. avatest™为您的留学生涯保驾护航。
