Research shows that the residual amount of a certain drug in a user's body decreases exponentially over time after taking the drug. It is known that 2 hours after taking the drug, half of the drug dose remains in the body. Which of the following options is correct?
(1) After 3 hours, the body still retains $\frac{1}{3}$ of the drug dose
(2) After 4 hours, the body still retains $\frac{1}{4}$ of the drug dose
(3) After 6 hours, the body still retains $\frac{1}{6}$ of the drug dose
(4) After 8 hours, the body still retains $\frac{1}{8}$ of the drug dose
(5) After 10 hours, the body still retains $\frac{1}{10}$ of the drug dose

As shown in the figure, $OABC-DEFG$ is a cube. Which of the following vectors is parallel to the cross product $\overrightarrow{AD} \times \overrightarrow{AG}$?
(1) $\overrightarrow{AE}$
(2) $\overrightarrow{BE}$
(3) $\overrightarrow{CE}$
(4) $\overrightarrow{DE}$
(5) $\overrightarrow{OE}$

Let $a \in \{-6, -4, -2, 2, 4, 6\}$ be the leading coefficient of a real-coefficient cubic polynomial $f(x)$. If the graph of the function $y = f(x)$ intersects the $x$-axis at three points whose $x$-coordinates form an arithmetic sequence with first term $-7$ and common difference $a$, how many values of $a$ satisfy $f(0) > 0$?
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

How many real numbers $x$ satisfy $\sin\left(x + \frac{\pi}{6}\right) = \sin x + \sin\frac{\pi}{6}$ and $0 \leq x < 2\pi$?
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5 or more

Divide the 50 positive integers from 1 to 50 equally into groups A and B, with 25 numbers in each group, such that the median of group A is 1 less than the median of group B. How many ways are there to divide them?
(1) $C_{25}^{50}$
(2) $C_{24}^{48}$
(3) $C_{12}^{24}$
(4) $\left(C_{12}^{24}\right)^{2}$
(5) $C_{24}^{48} \cdot C_{12}^{24}$

On the same plane, two artillery batteries $A$ and $B$ are 7 kilometers apart, with $A$ directly east of $B$. During an exercise, $A$ fires a projectile west-northwest at angle $\theta$, and $B$ fires a projectile east-northwest at angle $\theta$, where $\theta$ is an acute angle. Both projectiles hit the same target $P$ 9 kilometers away. Then $A$ fires another projectile west-northwest at angle $\frac{\theta}{2}$, landing at point $Q$ 9 kilometers away. What is the distance $\overline{BQ}$ between artillery battery $B$ and landing point $Q$?
(1) 4 kilometers
(2) 4.5 kilometers
(3) 5 kilometers
(4) 5.5 kilometers
(5) 6 kilometers

Let $\Gamma$ be the graph formed by points $(x, y)$ satisfying $y = \log x$ on the coordinate plane. Which of the following relationships produce graphs that are completely identical to $\Gamma$?
(1) $y + \frac{1}{2} = \log(5x)$
(2) $2y = \log\left(x^{2}\right)$
(3) $3y = \log\left(x^{3}\right)$
(4) $x = 10^{y}$
(5) $x^{3} = 10^{\left(y^{3}\right)}$

For any positive integer $n \geq 2$, let $T_{n}$ denote a triangle with side lengths $n, n+1, n+2$. Select the correct options. (Note: If a triangle has side lengths $a, b, c$ respectively, let $s = \frac{a+b+c}{2}$, then the area of the triangle is $\sqrt{s(s-a)(s-b)(s-c)}$)
(1) $T_{n}$ is always an acute triangle
(2) The perimeters of $T_{2}, T_{3}, T_{4}, \cdots, T_{10}$ form an arithmetic sequence
(3) The area of $T_{n}$ increases as $n$ increases
(4) The three altitudes of $T_{5}$ form an arithmetic sequence in order
(5) The largest angle of $T_{3}$ is greater than the largest angle of $T_{2}$

A laboratory collected a large number of two similar species $A$ and $B$, recording their body length $x$ (in centimeters) and body weight $y$ (in grams). The average body lengths of species $A$ and $B$ are $\overline{x_{A}} = 5.2$ and $\overline{x_{B}} = 6$ respectively, with standard deviations 0.3 and 0.1 respectively. Let the average body weights of species $A$ and $B$ be $\overline{y_{A}}$ and $\overline{y_{B}}$ respectively. If the regression lines of body weight $y$ on body length $x$ for species $A$ and $B$ are $L_{A}: y = 2x - 0.6$ and $L_{B}: y = 1.5x + 0.4$ respectively, with correlation coefficients 0.6 and 0.3 respectively. An individual $P$ with body length 5.6 centimeters and body weight 8.6 grams is discovered. Select the correct options.
(1) $\overline{y_{A}} < \overline{y_{B}}$
(2) The standard deviation of body weight for species $A$ is less than that for species $B$
(3) For species $A$, the absolute difference between individual $P$'s body weight and the average body weight $\overline{y_{A}}$ is greater than one standard deviation
(4) The distance from point $(5.6, 8.6)$ to line $L_{A}$ is less than its distance to line $L_{B}$
(5) The distance from point $(5.6, 8.6)$ to point $(\overline{x_{A}}, \overline{y_{A}})$ is less than its distance to point $(\overline{x_{B}}, \overline{y_{B}})$

On the coordinate plane, there is a square and a regular hexagon, with the square to the right of the hexagon. Both regular polygons have one side on the $x$-axis, and the center $A$ of the square and the center $B$ of the hexagon are both above the $x$-axis. The two polygons have exactly one intersection point $P$. The side length of the square is 6, and the distance from point $P$ to the $x$-axis is $2\sqrt{3}$. Select the correct options.
(1) The distance from point $A$ to the $x$-axis is greater than the distance from point $B$ to the $x$-axis
(2) The side length of the regular hexagon is 6
(3) $\overrightarrow{BA} = (7, 3 - 2\sqrt{3})$
(4) $\overline{AP} > \sqrt{10}$
(5) The slope of line $AP$ is greater than $-\frac{1}{\sqrt{3}}$

Consider the system of linear equations in two variables $\left\{\begin{array}{c} ax + 6y = 6 \\ x + by = 1 \end{array}\right.$, where the coefficients $a, b$ are determined by rolling a fair die and flipping a fair coin respectively. Let $a$ be the number of points shown on the die; if the coin shows heads, $b = 1$; if the coin shows tails, $b = 2$. Select the correct options.
(1) The probability of rolling $a = b$ is $\frac{1}{3}$
(2) The probability that the system has no solution is $\frac{1}{12}$
(3) The probability that the system has a unique solution is $\frac{5}{6}$
(4) The probability that the coin shows tails and the system has a solution is $\frac{1}{2}$
(5) Given that the coin shows tails and the system has a solution, the probability that $x$ is positive is $\frac{2}{5}$

Three points $A(1,0), B(0,1), C(-1,0)$ are given on the coordinate plane. Let $\Gamma$ be the graph obtained by transforming $\triangle ABC$ by the matrix $T = \left[\begin{array}{ll} 3 & 0 \\ a & 1 \end{array}\right]$, where $a$ is a real number. Select the correct options.
(1) If $a = 0$, then $\Gamma$ is an isosceles right triangle
(2) At least two points on the sides of $\triangle ABC$ have unchanged coordinates after transformation by $T$
(3) $\Gamma$ must have part of it in the fourth quadrant
(4) There exists a figure $\Omega$ on the plane such that after transformation by $T$ it becomes $\triangle ABC$
(5) The area of $\Gamma$ is a constant value

A sales station sells three types of mobile phones: A, B, and C. The profit per unit is 100 yuan for type A, 400 yuan for type B, and 240 yuan for type C. Last year, $A, B, C$ units of types A, B, C were sold respectively, with an average profit of 260 yuan per unit. It is also known that the average profit for selling types A and B together ($A + B$ units) is 280 yuan per unit. The ratio of the quantities of the three types of mobile phones sold last year is $A : B : C =$ (13－1)：(13－2)：(13－3) (expressed as a ratio of integers in lowest terms)

It is known that $f(x), g(x), h(x)$ are all real-coefficient cubic polynomials, and their remainders when divided by $x^{2} - 2x + 3$ are $x + 1$, $x - 3$, and $-2$ respectively. If $xf(x) + ag(x) + bh(x)$ is divisible by $x^{2} - 2x + 3$, where $a, b$ are real numbers, then $a =$ (14－1)(14－2), $b =$ (14－3).

A shopping mall holds a raffle drawing activity with on-site registration. After registration closes, the host places the same number of raffle balls as the number of registrants, of which 10 balls are marked as lucky prizes: 5 balls for a 5000 yuan gift voucher and 5 balls for an 8000 yuan gift voucher. Each ball has an equal probability of being drawn, and balls are not replaced after drawing. Before the drawing, the organizers announce a winning probability of 0.4\% based on the number of prizes and registrants. After the drawing begins, each person draws a ball in order, and each person has only one chance to draw. If among the first 100 participants, exactly 1 person draws a 5000 yuan voucher and no one draws an 8000 yuan voucher, then the expected value of the prize amount that the 101st person can receive is (15－1)(15－2) yuan.

On the coordinate plane, it is known that the orthogonal projection length of vector $\vec{v}$ in the direction of vector $(2, -3)$ is 1 less than its original length, and the orthogonal projection length in the direction of vector $(3, 2)$ is 2 less than its original length. If $\vec{v}$ makes acute angles with both vectors $(2, -3)$ and $(3, 2)$, then the orthogonal projection length of $\vec{v}$ in the direction of vector $(4, 7)$ is

On the coordinate plane, within the square (including boundary) with vertices $O(0,0), A(0,1), B(1,1), C(1,0)$, let $R$ be the region formed by points $P(x, y)$ satisfying the following condition: the set of all points at distance $|x - y|$ from point $P(x, y)$ is completely contained within the square $OABC$ (including boundary). The area of region $R$ is \hspace{2cm}. (expressed as a fraction in lowest terms)

In coordinate space, let $O$ be the origin and $E$ be the plane $x - z = 4$.
If the projection of the origin $O$ onto plane $E$ is point $Q$, and the angle between vector $\overrightarrow{OQ}$ and vector $(1, 0, 0)$ is $\alpha$, what is the value of $\cos\alpha$? (Single choice question, 3 points)
(1) $-\frac{\sqrt{2}}{2}$
(2) $-\frac{1}{2}$
(3) $\frac{1}{2}$
(4) $\frac{\sqrt{2}}{2}$
(5) $\frac{\sqrt{3}}{2}$

In coordinate space, let $O$ be the origin and $E$ be the plane $x - z = 4$.
It is known that there is a point $P(a, b, c)$ in space such that the angle $\theta$ between vector $\overrightarrow{OP}$ and vector $(1, 0, 0)$ satisfies $\theta \leq \frac{\pi}{6}$. Show that the real numbers $a, b, c$ satisfy the inequality $a^{2} \geq 3\left(b^{2} + c^{2}\right)$. (Non-multiple choice question, 4 points)

In coordinate space, let $O$ be the origin and $E$ be the plane $x - z = 4$.
Continuing from question 19, it is known that point $P$ is on plane $E$ and $b = 0$. Find the maximum possible range of $c$ and the minimum possible length of line segment $\overline{OP}$. (Non-multiple choice question, 8 points)