For any $x$ and $y$ satisfying $x > 0$ and $y > 0$, let $m$ be the smallest value among $\frac { y } { x } , x$ and $\frac { 8 } { y }$.
Also, let $A$ be the set of points $( x , y )$ where $m = \frac { y } { x }$, and let $B$ be the set of points $( x , y )$ where $m = \frac { 8 } { y }$.
(1) For $\mathbf { M } \sim$ S in the following sentence, choose the correct answer from among choices (0) $\sim$ (7) below. $A$ and $B$ can be expressed as follows:
$$\begin{aligned} & A = \{ ( x , y ) \mid \mathbf { M } \leqq \mathbf { N } , \quad \mathbf { O } \leqq 8\mathbf { P } \} \\ & B = \{ ( x , y ) \mid 8 \mathbf { Q } \leqq \mathbf { R } , \quad 8 \leqq \mathbf { S } \} . \end{aligned}$$
(0) $x$
(1) $y$
(2) $x + y$
(3) $x - y$
(4) $x ^ { 2 }$
(5) $x y$ (6) $y ^ { 2 }$ (7) $x ^ { 2 } + y ^ { 2 }$
(2) For $\mathbf { T }$ and $\mathbf { U }$ in the following sentence, choose the correct answer from among choices (0) $\sim$ (8).
When sets $A$ and $B$ are indicated on the $xy$-plane, $A$ is the shaded portion of graph $\mathbf{T}$ and $B$ is the shaded portion of graph $\mathbf{U}$. Note that the $x$ and $y$ axes are not included in the shaded portions.
(3) We are to find the maximum value of $m$ when a point $\mathrm { P } ( x , y )$ moves within $A \cup B$.
When $\mathrm { P } ( x , y ) \in A$, since $y = m x$, we need to find the point P which maximizes the slope of the straight line passing through the origin O and P.
Also, when $\mathrm { P } ( x , y ) \in B$, since $m = \frac { 8 } { y }$, we need to find the point P at which the $y$ coordinate of P is minimized.
From the above, at $( x , y ) = ( \mathbf { V } , \mathbf { W } ) , m$ takes the maximum value $\mathbf { X }$.

A manufacturer produces two types of electric vehicles, Type A and Type B. The costs for producing these two types involve three categories: battery, motor, and others. The costs for each category are shown in the table below (unit: 10,000 yuan):

	Battery Cost	Motor Cost	Other Cost
Type A	56	26	48
Type B	40	20	56

The selling price formula for the two types of electric vehicles is the sum of ``$x$ times the battery cost'', ``$y$ times the motor cost'', and ``$\frac { x + y } { 2 }$ times the other cost'', that is,
Selling Price $=$ Battery Cost $\times x +$ Motor Cost $\times y +$ Other Cost $\times \frac { x + y } { 2 }$ where the multipliers $x, y$ must satisfy ``$1 \leq x \leq 2, 1 \leq y \leq 2$, and the selling prices of both Type A and Type B electric vehicles do not exceed 200 (10,000 yuan)''. To differentiate its products, the manufacturer wants to maximize the price difference between Type A and Type B electric vehicles. Based on the above information, answer the following questions.
(1) Write the selling prices of Type A and Type B electric vehicles (in terms of $x$ and $y$), and explain why ``the selling price of Type A electric vehicles is always higher than that of Type B electric vehicles''. (4 points)
(2) On a coordinate plane, draw the feasible region of $(x, y)$ satisfying the conditions in the problem, and shade the region with diagonal lines. (4 points)
(3) Find the values of multipliers $x$ and $y$ that maximize the price difference between Type A and Type B electric vehicles. What is the maximum price difference in units of 10,000 yuan? (6 points)

How many integers $x$ satisfy $2|x| + x < 10$?
(1) 13
(2) 14
(3) 15
(4) 16
(5) Infinitely many

Let integer $n$ satisfy $| 5 n - 21 | \geq 7 | n |$ . Select the correct options.
(1) $| 5 n - 7 n | \geq 21$
(2) $- 1 \leq \frac { 7 n } { 5 n - 21 } \leq 1$
(3) $7 n \leq 5 n - 21$
(4) $( 5 n - 21 ) ^ { 2 } \geq 49 n ^ { 2 }$
(5) There are infinitely many integers $n$ satisfying the given inequality

A person wants to plant two types of fruits, A and B, on farmland, and sets the planting area of fruit A (area A) and the planting area of fruit B (area B) to satisfy the following three conditions: (I) Area A does not exceed 15 hectares. (II) The sum of area A and area B does not exceed 24 hectares. (III) Area A does not exceed 3 times area B, and area B does not exceed 2 times area A. Let area A be $x$ hectares and area B be $y$ hectares.
Which of the following options for the ordered pair $(x, y)$ satisfies the above three conditions? (Single choice)
(1) $(7,15)$
(2) $(12,13)$
(3) $(14,10)$
(4) $(15,4)$
(5) $(16,8)$

A person wants to plant two types of fruits, A and B, on farmland, and sets the planting area of fruit A (area A) and the planting area of fruit B (area B) to satisfy the following three conditions: (I) Area A does not exceed 15 hectares. (II) The sum of area A and area B does not exceed 24 hectares. (III) Area A does not exceed 3 times area B, and area B does not exceed 2 times area A. Let area A be $x$ hectares and area B be $y$ hectares.
Express the three conditions set by the person for area A and area B as a system of linear inequalities in $x$ and $y$.

A person wants to plant two types of fruits, A and B, on farmland, and sets the planting area of fruit A (area A) and the planting area of fruit B (area B) to satisfy the following three conditions: (I) Area A does not exceed 15 hectares. (II) The sum of area A and area B does not exceed 24 hectares. (III) Area A does not exceed 3 times area B, and area B does not exceed 2 times area A. Let area A be $x$ hectares and area B be $y$ hectares.
Given that when the farmland is harvested, fruit A yields a profit of 6 ten-thousand yuan per hectare and fruit B yields a profit of 7 ten-thousand yuan per hectare, find the maximum profit from planting both fruits in ten-thousand yuan. Show the calculation process in the solution area of the answer sheet, and draw the feasible region in the diagram area of the answer sheet, marking all vertices of the region and shading the region with diagonal lines.

$$(2x-1)\left(4x^{2}-1\right)<0$$
Which of the following open intervals is the solution set of the inequality in real numbers?
A) $\left(-\infty, \frac{-1}{2}\right)$
B) $\left(\frac{-1}{2}, 0\right)$
C) $\left(\frac{-1}{2}, \frac{1}{2}\right)$
D) $\left(\frac{1}{4}, \frac{1}{2}\right)$
E) $\left(\frac{1}{2}, \infty\right)$

For given positive real numbers $a$, $c$ and negative real number $b$, $$a^{2}b > abc + c^{2}$$ Given that the inequality is satisfied, which of the following is necessarily true?
A) $a = |b|$
B) $a = c$
C) $c > |b|$
D) $a < c$
E) $c < a$

$$\frac { - 5 } { 4 } < x < \frac { 7 } { 3 }$$
What is the sum of the integers $x$ that satisfy this inequality?
A) - 2
B) - 1
C) 0
D) 1
E) 2

For real numbers $x , y$ and $z$
$$\begin{aligned} & y > 0 \\ & x - y > z \end{aligned}$$
Given this, which of the following is always true?
A) $x > z$
B) $x > y$
C) $z > y$
D) $x > 0$
E) $z > 0$

The parabola $f(x)$ and the line $d$ are shown in the Cartesian coordinate plane above.
Accordingly, which of the following systems of inequalities has the shaded region as its solution set?
A) $\left.\begin{array}{l} y - x^{2} + 2x \leq 0 \\ y - x + 2 \geq 0 \end{array}\right\}$
B) $\left.\begin{array}{l} y - x^{2} + 2x \geq 0 \\ 2y - x + 2 \geq 0 \end{array}\right\}$
C) $\left.\begin{array}{l} y - x^{2} + 4x \leq 0 \\ 2y - x + 2 \leq 0 \end{array}\right\}$
D) $\left.\begin{array}{l} y + x^{2} - 4x \leq 0 \\ 2y - x + 4 \leq 0 \end{array}\right\}$
E) $\left.\begin{array}{l} y + x^{2} - 4x \leq 0 \\ 2y - x + 2 \geq 0 \end{array}\right\}$

$$\sqrt { 2 } < x < \sqrt { 3 }$$
Given this, which of the following can x be?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 4 } { 3 }$
D) $\frac { 7 } { 4 }$
E) $\frac { 6 } { 5 }$

For integers x and y, $x + 2y = 11$. Given that,
I. x is an odd number. II. x is greater than y. III. Both x and y are positive.
Which of the following statements are always true?
A) Only I B) Only III C) I and II D) I and III E) II and III

$$\left. \begin{array} { l } x ( 3 - x ) > 0 \\ ( 2 x + 1 ) ( x - 2 ) < 0 \end{array} \right\}$$
If the solution set of the inequality system given above is the open interval $(\mathbf { a } , \mathbf { b })$, what is the difference $\mathbf { a - b }$?
A) - 2
B) 0
C) 1
D) $\frac { 1 } { 2 }$
E) $\frac { 3 } { 2 }$

For points $(x, y)$ on the boundary of the bounded region between the parabola $y = x ^ { 2 }$ and the line $y = 2 - x$, what is the maximum value of the expression $\mathbf { x } ^ { \mathbf { 2 } } + \mathbf { y } ^ { \mathbf { 2 } }$?
A) 25
B) 20
C) 17
D) 13
E) 10

$$-2 < x < 4$$
Given that, what is the greatest integer value that the expression $1 - x$ can take?
A) $-3$
B) $-2$
C) $-1$
D) 2
E) 3

Let x and y be real numbers with $-1 < y < 0 < x$. Which of the following statements are always true?
I. $x + y > 0$ II. $x - y > 1$ III. $x \cdot ( y + 1 ) > 0$
A) Only I
B) Only III
C) I and II
D) I and III
E) II and III

Given that $x < 0 < y$, I. $y - x ^ { - 1 }$ II. $x ^ { 2 } + y ^ { - 1 }$ III. $( x \cdot y ) ^ { - 1 }$ Which of these expressions have negative values?
A) Only I
B) Only II
C) Only III
D) I and III
E) II and III

For real numbers $x , y$ and $z$
$$x + y < 0 < x < y + z$$
Given this, which of the following orderings is correct?
A) $x < y < z$
B) $x < z < y$
C) $y < x < z$
D) $y < z < x$
E) $z < y < x$

a, b are real numbers and
$$\begin{aligned} & 0 < a < 3 a ^ { 2 } \\ & b - 1 = 6 a \end{aligned}$$
Given this, what is the smallest integer value that b can take?
A) 3
B) 4
C) 5
D) 6
E) 7

For positive real numbers $x$ and $y$
$$\frac { x } { 8 } = \frac { y } { 12 } = \frac { 9 } { y - x }$$
Given this, what is the sum $x + y$?
A) 10
B) 15
C) 20
D) 25
E) 30

Given that $| a | = 2 , | b | = 5$ and $| c | = 6$,
$$\begin{aligned} & \mathrm { c } < \mathrm { a } < \mathrm { b } \\ & \mathrm { a } \cdot \mathrm {~b} \cdot \mathrm { c } > 0 \end{aligned}$$
What is the sum $a + b + c$?
A) - 9
B) - 3
C) - 1
D) 1
E) 3

Let $f : \mathbf { R } \backslash \{ 0 \} \rightarrow \mathbf { R }$ with
$$f ( x ) = \frac { 2 } { x } - x + 1$$
For this function, which of the following is the set of all $x$ points such that $f ( x ) \in ( 0 , \infty )$?
A) $( - \infty , 0 )$
B) $( - 1 , \infty )$
C) $( 0,1 ) \cup ( 2 , \infty )$
D) $( - 2,0 ) \cup ( 2 , \infty )$
E) $( - \infty , - 1 ) \cup ( 0,2 )$

For positive integers $n$, the subsets of the set $R$ of real numbers are defined as
$$A _ { n } = \left\{ x \in R : \frac { ( - 1 ) ^ { n } } { n } < x < \frac { 2 } { n } \right\}$$
Accordingly, $$A _ { 1 } \cap A _ { 2 } \cap A _ { 3 }$$
the intersection set is equal to which of the following?
A) $\left( \frac { 1 } { 2 } , \frac { 2 } { 3 } \right)$
B) $\left( \frac { 1 } { 2 } , 2 \right)$
C) $\left( \frac { - 1 } { 3 } , \frac { 2 } { 3 } \right)$
D) $\left( \frac { - 1 } { 3 } , 1 \right)$
E) $\left( - 1 , \frac { 2 } { 3 } \right)$