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# 数学代写|Ross数学夏令营2023选拔代写

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?

(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.

(a) 假设$s=1$，使路径以45度角开始。

(如果$P$是一个角点，该路径就退化为一条来回追踪的线段)。

(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？