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

A light show display uses color-changing flashing lights. After each activation, the flashing color changes periodically according to the following sequence: Blue–White–Red–White–Blue–White–Red–White–Blue–White–Red–White…, with one cycle every four flashes. Blue light lasts 5 seconds each time, white light lasts 2 seconds each time, and red light lasts 6 seconds each time. Assuming the time to change lights is negligible, select the light color(s) between the 99th and 101st seconds after activation.
(1) All blue lights
(2) All white lights
(3) All red lights
(4) Blue light first, then white light
(5) White light first, then red light

Eight buildings are arranged in a row, numbered 1, 2, 3, 4, 5, 6, 7, 8 from left to right. A telecommunications company wants to select three of these buildings' rooftops to install telecommunications base stations. If base stations cannot be installed on two adjacent buildings to avoid signal interference, how many ways are there to select locations for the base stations if no base station is installed on building 3?
(1) 12
(2) 13
(3) 20
(4) 30
(5) 35

On a coordinate plane, it is known that vector $\overrightarrow{PQ} = \left(\log \frac{1}{5}, -10^{-5}\right)$, where point $P$ has coordinates $\left(\log \frac{1}{2}, 2^{-5}\right)$. Select the correct option.
(1) Point $Q$ is in the first quadrant
(2) Point $Q$ is in the second quadrant
(3) Point $Q$ is in the third quadrant
(4) Point $Q$ is in the fourth quadrant
(5) Point $Q$ is on a coordinate axis

Let matrix $A = \left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]$. If $A^{7} - 3A = \left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$, then which of the following is the value of $a + b + c + d$?
(1) $-8$
(2) $-5$
(3) $5$
(4) $8$
(5) $10$

Assuming the Earth is a sphere with radius $r$, a point particle moves north from location A along the meridian passing through that location. When it reaches the North Pole, the arc length traveled is $\frac{7}{12}\pi r$. Which of the following options is most likely the location of A?
(1) East longitude $75^{\circ}$, North latitude $15^{\circ}$
(2) East longitude $30^{\circ}$, South latitude $75^{\circ}$
(3) East longitude $75^{\circ}$, South latitude $15^{\circ}$
(4) West longitude $30^{\circ}$, North latitude $75^{\circ}$
(5) West longitude $15^{\circ}$, South latitude $30^{\circ}$

When an artist uses single-point perspective to draw spatial scenes on a flat piece of paper, the following principles must be followed: I. A straight line in space must be drawn as a straight line on the paper. II. The relative positions of points on a line in space must be consistent with the relative positions of the points drawn on the paper. III. The $K$ value of any four distinct points on a line in space must be the same as the $K$ value of the four points drawn on the paper, where the $K$ value is defined as follows: For any four ordered distinct points $P_1, P_2, P_3, P_4$ on a line, the corresponding $K$ value is defined as $$K = \frac{\overline{P_1P_4} \times \overline{P_2P_3}}{\overline{P_1P_3} \times \overline{P_2P_4}}$$ An artist follows the above principles to draw a line in space and four distinct points $Q_1, Q_2, Q_3, Q_4$ on that line on paper, where $\overline{Q_1Q_2} = \overline{Q_2Q_3} = \overline{Q_3Q_4}$. If the line drawn on the paper is viewed as a number line and the points on it are represented by coordinates, which of the following sets of four coordinates is most likely to be the coordinates of these four points on the paper?
(1) $1, 2, 4, 8$
(2) $3, 4, 6, 9$
(3) $1, 5, 8, 9$
(4) $1, 2, 4, 9$
(5) $1, 7, 9, 10$

There is a shooting game with the launcher placed at the origin of a coordinate plane and three circular target disks with radius 1, centered at $(2,2)$, $(4,6)$, and $(8,1)$ respectively. A player selects a positive number $a$ and presses a button. The launcher then fires a laser beam in the direction of point $(1, a)$ (forming a ray). Assume the laser beam can penetrate through the target after hitting it and continue in the original direction (grazing the edge of the disk also counts as a hit). Select the correct options.
(1) The laser beam lies on a line passing through the origin with slope $a$
(2) If $a = \frac{3}{2}$, the laser beam will hit the disk centered at $(4,6)$
(3) The player can hit all three disks with just one laser beam
(4) The player needs to fire at least three laser beams to hit all three disks
(5) If the player fires one laser beam and hits the disk centered at $(8,1)$, then $a \leq \frac{16}{63}$

Let $f(x) = 2x^3 - 3x + 1$. Select the correct statements about the graph of the function $y = f(x)$.
(1) The graph of $y = f(x)$ passes through the point $(1, 0)$
(2) The graph of $y = f(x)$ has only one intersection point with the $x$-axis
(3) The point $(1, 0)$ is a center of symmetry of the graph of $y = f(x)$
(4) The graph of $y = f(x)$ approximates a straight line $y = 3x - 3$ near the center of symmetry
(5) The graph of $y = 3x^3 - 6x^2 + 2x$ can be obtained from the graph of $y = f(x)$ by appropriate translation

Classes A and B each have 40 students taking a mathematics exam (total score 100 points). After the exam, classes A and B adjust their scores using $y_1 = 0.8x_1 + 20$ and $y_2 = 0.75x_2 + 25$ respectively, where $x_1, x_2$ represent the original exam scores of classes A and B, and $y_1, y_2$ represent the adjusted scores of classes A and B. The average adjusted scores for both classes are 60 points, with adjusted standard deviations of 16 and 15 points respectively. Select the correct options.
(1) Every student in class A has an adjusted score not lower than their original score
(2) The average original score of class A is higher than that of class B
(3) The standard deviation of original scores in class A is higher than that in class B
(4) If student A from class A has a higher adjusted score than student B from class B, then A's original score is higher than B's original score
(5) If the number of students in class A with adjusted scores below 60 (failing) is greater than the number in class B, then the number of students in class A with original scores below 60 must be greater than in class B

Consider points $O(0,0), A, B, C, D, E, F, G$ on a coordinate plane, where points $B$, $C$ and $D$, $E$ and $F$, $G$ and $A$ are located in the first, second, third, and fourth quadrants respectively. If $\vec{v}$ is a vector on the coordinate plane satisfying $\vec{v} \cdot \overrightarrow{OA} > 0$ and $\vec{v} \cdot \overrightarrow{OB} > 0$, then the dot product of $\vec{v}$ with which of the following vectors must be negative?
(1) $\overrightarrow{OC}$
(2) $\overrightarrow{OD}$
(3) $\overrightarrow{OE}$
(4) $\overrightarrow{OF}$
(5) $\overrightarrow{OG}$

Let $a, b, c$ be nonzero real numbers, and the two roots of the quadratic equation $ax^2 + bx + c = 0$ both lie between 1 and 3. Select the equation whose two roots must lie between 4 and 5.
(1) $a(x-2)^2 + b(x-2) + c = 0$
(2) $a(x+2)^2 + b(x+2) + c = 0$
(3) $a(2x-7)^2 + b(2x-7) + c = 0$
(4) $a\left(\frac{x+7}{2}\right)^2 + b\left(\frac{x+7}{2}\right) + c = 0$
(5) $a(3x-11)^2 + b(3x-11) + c = 0$

If $x, y$ are two positive real numbers satisfying $x^{-\frac{1}{3}} y^{2} = 1$ and $2\log y = 1$, then $\frac{x - y^{2}}{10} =$ (13--1) (13--2).

On a coordinate plane, there is a circle with radius 7 and center $O$. It is known that points $A, B$ are on the circle and $\overline{AB} = 8$. Then the dot product $\overrightarrow{OA} \cdot \overrightarrow{OB} =$ (14--1) (14--2).

According to a certain country's investigation of missing light aircraft: 70\% of missing light aircraft are eventually found. Among the aircraft that are found, 60\% have emergency locator transmitters installed; among the missing aircraft that are not found, 90\% do not have emergency locator transmitters installed. Emergency locator transmitters send signals when the aircraft crashes, allowing rescue personnel to locate it. A light aircraft is now missing. If it is known that the aircraft has an emergency locator transmitter installed, the probability that it will be found is (15--1)(15--2).

A bag contains blue, green, and yellow balls totaling 10 balls. Two balls are randomly drawn from the bag (each ball has an equal probability of being drawn). The probability that both balls drawn are blue is $\frac{1}{15}$, and the probability that both are green is $\frac{2}{9}$. The probability that two randomly drawn balls are of different colors is $\frac{\text{(16--1)}}{\text{(16--3)}}$. (Express as a fraction in lowest terms)

There are six students (three female and three male) who frequently interact with a teacher at school. After graduation, the teacher invites them to a gathering. After the meal, seven people stand in a row for a commemorative photo. It is known that among the students, one female and one male had an unpleasant experience and do not want to stand adjacent during the photo, while the teacher stands in the middle and the three male students do not all stand on the same side of the teacher. The total number of possible arrangements is (17--1)(17--2)(17--3).

It is known that the world's tallest tilted skyscraper is located in Abu Dhabi with a tilt angle of $18^{\circ}$. Converting this tilt angle to radians is which of the following options? (Single choice, 5 points)
(1) $\frac{\pi}{36}$
(2) $\frac{\pi}{18}$
(3) $\frac{\pi}{20}$
(4) $\frac{\pi}{10}$
(5) $\frac{\pi}{8}$

China's Tiger Hill Tower, Pearl Tower, and Italy's Leaning Tower of Pisa are three famous leaning towers with tower heights of 48, 19, and 57 meters respectively, and offset distances of 2.3, 2.3, and 4 meters respectively. Their tilt angles are denoted as $\theta_1{}^{\circ}, \theta_2{}^{\circ}$, and $\theta_3{}^{\circ}$ respectively. Compare the size relationship of $\theta_1, \theta_2$, and $\theta_3$. (Non-multiple choice, 4 points)
Note: The tilt angle $\theta^{\circ}$ is the angle between the tower body and a vertical dashed line ($0 \leq \theta < 90$), and the offset distance is the distance from the tower top to the vertical dashed line.

Suppose there are two iron towers with equal tower heights. Their tilt angles $\alpha^{\circ}, \beta^{\circ}$ satisfy $\sin\alpha^{\circ} = \frac{1}{5}$ and $\sin\beta^{\circ} = \frac{7}{25}$ respectively. It is known that the offset distances of the two towers differ by 20 meters. Find the difference in the distance from the tower tops to the ground. (Non-multiple choice, 6 points)
Note: The tilt angle $\theta^{\circ}$ is the angle between the tower body and a vertical dashed line ($0 \leq \theta < 90$), the offset distance is the distance from the tower top to the vertical dashed line, and the distance from the tower top to the ground is the vertical height.