1. A sequence $a_{1} + 2, \cdots, a_{k} + 2k, \cdots, a_{10} + 20$ has ten terms, and their sum is 240. Then the value of $a_{1} + \cdots + a_{k} + \cdots + a_{10}$ is
(1) 31
(2) 120
(3) 130
(4) 185
(5) 218

2. Let $a = \cos(\pi^{2})$. Which of the following options is correct?
(1) $a = -1$
(2) $-1 < a \leq -\frac{1}{2}$
(3) $-\frac{1}{2} < a \leq 0$
(4) $0 < a \leq \frac{1}{2}$
(5) $\frac{1}{2} < a \leq 1$

3. Given that $f(x)$ and $g(x)$ are two real-coefficient polynomials, and the remainder when $f(x)$ is divided by $g(x)$ is $x^{4} - 1$. Which of the following options cannot be a common factor of $f(x)$ and $g(x)$?
(1) 5
(2) $x - 1$
(3) $x^{2} - 1$
(4) $x^{3} - 1$
(5) $x^{4} - 1$

4. Three high schools A, B, and C have 3, 4, and 5 classes respectively in their first year. One class is randomly selected from these 12 classes to participate in a Chinese language test, and then one class is randomly selected from the remaining 11 classes to participate in an English test. What is the probability that the two classes participating in the tests are from the same school closest to which of the following options?
(1) $21\%$
(2) $23\%$
(3) $25\%$
(4) $27\%$
(5) $29\%$

5. Assume that the distances between towns A, B, and C are all equal to 20 kilometers. Two straight roads intersect at town D, one passing through towns A and B, and the other passing through town C. On an accurately scaled map, the angle between the two roads is measured to be $45^{\circ}$. Then the distance between towns C and D is approximately
(1) 24.5 kilometers
(2) 25 kilometers
(3) 25.5 kilometers
(4) 26 kilometers
(5) 26.5 kilometers

6. How many lines on the coordinate plane are there such that the distance from point $O(0,0)$ to the line is 1, and the distance from point $A(3,0)$ to the line is 2?
(1) 1 line
(2) 2 lines
(3) 3 lines
(4) 4 lines
(5) Infinitely many lines

II. Multiple-Choice Questions (25 points)

Instructions: For questions 7 through 11, each of the five options is independent, and at least one option is correct. Select the correct options and mark them on the ``Answer Sheet''. No deductions are made for incorrect answers. Full credit (5 points) is given for all five options correct; 2.5 points are given for exactly one incorrect option; no credit is given for two or more incorrect options.

8. On the coordinate plane, four lines $L_{1}, L_{2}, L_{3}, L_{4}$ have the relative positions with respect to the $x$-axis, $y$-axis, and the line $y = x$ as shown in the figure. $L_{1}$ is perpendicular to $L_{3}$, and $L_{3}$ is parallel to $L_{4}$. The equations of $L_{1}, L_{2}, L_{3}, L_{4}$ are $y = m_{1}x$, $y = m_{2}x$, $y = m_{3}x$, and $y = m_{4}x + c$ respectively. Which of the following options are correct?
(1) $m_{3} > m_{2} > m_{1}$
(2) $m_{1} \cdot m_{4} = -1$
(3) $m_{1} < -1$
(4) $m_{2} \cdot m_{3} < -1$
(5) $c > 0$ [Figure]

9. A company commissioned a polling organization to survey the percentage of residents in locations A and B who have heard of a certain product (hereinafter referred to as ``awareness''). The results are as follows: At a 95\% confidence level, the confidence intervals for the product's awareness in locations A and B are $[0.50, 0.58]$ and $[0.08, 0.16]$ respectively. Which of the following options are correct?
(1) In location A, 54\% of the survey respondents have heard of the product
(2) The number of survey respondents in location B was less than in location A
(3) The survey results can be interpreted as: the probability that more than half of all residents in location A have heard of the product is greater than 95\%
(4) If multiple surveys are conducted in location B using the same method, the awareness has a 95\% chance of falling in the interval $[0.08, 0.16]$
(5) After intensive advertising, a follow-up survey is conducted in location B with the number of respondents increased to four times the original number. Then at a 95\% confidence level, the width of the confidence interval for the product's awareness will be reduced by half (i.e., 0.04)

10. Let $a, b, c$ be real numbers. Which of the following statements about the linear system $\left\{\begin{array}{c} x + 2y + az = 1 \\ 3x + 4y + bz = -1 \\ 2x + 10y + 7z = c \end{array}\right.$ are correct?
(1) If the linear system has a solution, then it must have exactly one solution
(2) If the linear system has a solution, then $11a - 3b \neq 7$
(3) If the linear system has a solution, then $c = 14$
(4) If the linear system has no solution, then $11a - 3b = 7$
(5) If the linear system has no solution, then $c \neq 14$

11. As shown in the figure, a rectangular prism $ABCD - EFGH$ has edge length equal to 2 (i.e., $\overline{AB} = 2$). $K$ is the center of square $ABCD$, and $M$, $N$ are the midpoints of segments $BF$ and $EF$ respectively. Which of the following options are correct?
(1) $\overrightarrow{KM} = \frac{1}{2}\overrightarrow{AB} - \frac{1}{2}\overrightarrow{AD} + \frac{1}{2}\overrightarrow{AE}$
(2) (Dot product) $\overrightarrow{KM} \cdot \overrightarrow{AB} = 1$
(3) $\overline{KM} = 3$
(4) $\triangle KMN$ is a right triangle
(5) The area of $\triangle KMN$ is $\frac{\sqrt{10}}{2}$ [Figure]

Part II: Fill-in-the-Blank Questions (45 points)

Instructions: 1. For questions A through I, mark your answers on the ``Answer Sheet'' at the row numbers indicated (12–33). 2. Each completely correct answer is worth 5 points; incorrect answers do not result in deductions; incomplete answers receive no credit.
A. From the positive integers 1 to 100, after removing all prime numbers, multiples of 2, and multiples of 3, the largest remaining number is (12)(13).
B. On the coordinate plane, there are four points $O(0,0)$, $A(-3,-5)$, $B(6,0)$, $C(x,y)$. A particle starts at point $O$ and moves in the direction of $\overrightarrow{AO}$ for a distance of $\overline{AO}$ and stops at $P$. Then it moves in the direction of $\overrightarrow{BP}$ for a distance of $2\overline{BP}$ and stops at $Q$. Suppose the particle continues to move in the direction of $\overrightarrow{CQ}$ for a distance of $3\overline{CQ}$ and returns to the origin $O$. Then $(x, y) = ($（14）（15），（16）（17）$)$.
C. In a raffle game, participants draw a ball from a box, confirm its color, and return it. Only those who draw a blue or red ball receive a shopping voucher with amounts of 2000 yuan (for blue ball) and 1000 yuan (for red ball) respectively. The box currently contains 2 blue