An ice cream shop needs to prepare at least $n$ buckets of different flavors of ice cream to satisfy the advertisement claim that ``the number of combinations of selecting two scoops of different flavors exceeds 100 types.'' How many ways can a customer select two scoops (which may be the same flavor) from $n$ buckets?
(1) 101
(2) 105
(3) 115
(4) 120
(5) 225

A certain brand of calculator computes the logarithm $\log _ { a } b$ by pressing $\log$（1）$a$（ $b$ ）. A student computing $\log _ { a } b$ (where $a > 1$ and $b > 1$ ) pressed the buttons in the wrong order, pressing $\log$（1）$b$（ $a$ ） instead, obtaining a result that is $\frac { 9 } { 4 }$ times the correct value. Select the relationship between $a$ and $b$.
(1) $a ^ { 2 } = b ^ { 3 }$
(2) $a ^ { 3 } = b ^ { 2 }$
(3) $a ^ { 4 } = b ^ { 9 }$
(4) $2 a = 3 b$
(5) $3 a = 2 b$

When processing two-dimensional data, one method is to project the data vertically onto a certain line and use that line as a number line to understand the variance of the one-dimensional data formed by the projection points. For the set of two-dimensional data shown in the figure, which line in the following options would result in the smallest variance of the one-dimensional projected data?
(1) $y = 2 x$
(2) $y = - 2 x$
(3) $y = - x$
(4) $y = \frac { x } { 2 }$
(5) $y = - \frac { x } { 2 }$

Let the arithmetic sequence $\left\langle a _ { n } \right\rangle$ have first term $a _ { 1 }$ and common difference $d$ both positive, and $\log a _ { 1 } , \log a _ { 3 } , \log a _ { 6 }$ also form an arithmetic sequence in order. Select the common difference of the sequence $\log a _ { 1 } , \log a _ { 3 } , \log a _ { 6 }$.
(1) $\log d$
(2) $\log \frac { 2 } { 3 }$
(3) $\log \frac { 3 } { 2 }$
(4) $\log 2 d$
(5) $\log 3 d$

It is known that 30\% of the population in a certain region is infected with a certain infectious disease. For a rapid screening test of the disease, there are two results: positive or negative. The test has an 80\% probability of identifying an infected person as positive and a 60\% probability of identifying an uninfected person as negative. To reduce the situation where the test incorrectly identifies an infected person as negative, experts recommend three consecutive tests. If $P$ is the probability that an infected person is among those who test negative in a single test, and $P'$ is the probability that an infected person is among those who test negative in all three consecutive tests, what is $\frac { P } { P' }$ closest to?
(1) 7
(2) 8
(3) 9
(4) 10
(5) 11

Two lines $L _ { 1 } , L _ { 2 }$ on the coordinate plane both have positive slopes, and the angle bisector of one of the angles formed by $L _ { 1 } , L _ { 2 }$ has slope $\frac { 11 } { 9 }$ . Another line $L$ passes through the point $( 2 , \frac { 1 } { 3 } )$ and forms a bounded region with $L _ { 1 } , L _ { 2 }$ that is an equilateral triangle. Which of the following options is the equation of $L$?
(1) $11 x - 9 y = 19$
(2) $9 x + 11 y = 25$
(3) $11 x + 9 y = 25$
(4) $27 x - 33 y = 43$
(5) $27 x + 33 y = 65$

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

On the coordinate plane, the three vertices of $\triangle A B C$ have coordinates $A ( 0,2 ) , B ( 1,0 ) , C ( 4,1 )$ respectively. Select the correct options.
(1) Among the three sides of $\triangle A B C$, $\overline { A C }$ is the longest
(2) $\sin A < \sin C$
(3) $\triangle A B C$ is an acute triangle
(4) $\sin B = \frac { 7 \sqrt { 2 } } { 10 }$
(5) The circumradius of $\triangle A B C$ is less than 2

Let $P$ be a point inside $\triangle A B C$, and $\overrightarrow { A P } = a \overrightarrow { A B } + b \overrightarrow { A C }$ , where $a , b$ are distinct real numbers. Let $Q , R$ be on the same plane, with $\overrightarrow { A Q } = b \overrightarrow { A B } + a \overrightarrow { A C } , ~ \overrightarrow { A R } = a \overrightarrow { A B } + ( b - 0.05 ) \overrightarrow { A C }$ . Select the correct options.
(1) $Q , R$ are also both inside $\triangle A B C$
(2) $| \overrightarrow { A P } | = | \overrightarrow { A Q } |$
(3) Area of $\triangle A B P$ = Area of $\triangle A C Q$
(4) Area of $\triangle B C P$ = Area of $\triangle B C Q$
(5) Area of $\triangle A B P$ > Area of $\triangle A B R$

Given a real-coefficient cubic polynomial function $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + 3$ . Let $g ( x ) = f ( - x ) - 3$ . It is known that the graph of $y = g ( x )$ has a center of symmetry at $( 1,0 )$ and $g ( - 1 ) < 0$ . Select the correct options.
(1) $g ( x ) = 0$ has three distinct integer roots
(2) $a < 0$
(3) The center of symmetry of the graph of $y = f ( x )$ is $( - 1 , - 3 )$
(4) $f ( 100 ) < 0$
(5) The graph of $y = f ( x )$ near the point $( - 1 , f ( - 1 ) )$ can be approximated by a line with slope $a$

Let $f ( x ) , g ( x )$ both be real-coefficient polynomials, where $g ( x )$ is a quadratic with positive leading coefficient. It is known that the remainder when $( g ( x ) ) ^ { 2 }$ is divided by $f ( x )$ is $g ( x )$ , and the graph of $y = f ( x )$ has no intersection with the $x$-axis. Select the option that cannot be the $y$-coordinate of the vertex of the graph of $y = g ( x )$.
(1) $\frac { \sqrt { 2 } } { 2 }$
(2) 1
(3) $\sqrt { 2 }$
(4) 2
(5) $\pi$

A certain online game offers a ``ten-draw'' card-pulling mechanism. ``Ten-draw'' means the system automatically performs ten card-pulling actions. If each ``ten-draw'' requires 1500 tokens, the probability of drawing a gold card is 2\% for the first nine draws and 10\% for the tenth draw. A certain student has 23000 tokens and continuously uses ``ten-draw'' until unable to draw anymore. The expected value of the number of gold cards drawn is (13-1)(13-2) cards.

Let $a , b$ be real numbers, and the system of equations $\left\{ \begin{array} { l } a x + 5 y + 12 z = 4 \\ x + a y + \frac { 8 } { 3 } z = 7 \\ 3 x + 8 y + a z = 1 \end{array} \right.$ has exactly one solution. After a series of Gaussian elimination operations, the original augmented matrix can be transformed to $\left[ \begin{array} { c c c | c } 1 & 2 & b & 7 \\ 0 & b & 5 & - 5 \\ 0 & 0 & b & 0 \end{array} \right]$ . Then $a = ( 14 - 1 ) , b = \frac { ( 14 - 2 ) } { ( 14 - 3 ) }$ . (Express as a fraction in lowest terms)

As shown in the figure, the Wang family owns a triangular piece of land $\triangle A B C$ , where $\overline { B C } = 16$ meters. The government plans to requisition the trapezoid $D B C E$ portion to develop a road with straight lines $D E , B C$ as edges, with road width $h$ meters, leaving the Wang family with only $\frac { 9 } { 16 }$ of the original land area. After negotiation, the plan is changed to develop a road with parallel lines $B E , F C$ as edges, with the same road width, where $\angle E B C = 30 ^ { \circ }$ . Only the $\triangle B C E$ region needs to be requisitioned. According to this agreement, the Wang family's remaining land $\triangle A B E$ has (15-1)(15-2)(15-3) square meters.

In coordinate space, on the plane $x - y + 2 z = 3$ there are two distinct lines $L : \frac { x } { 2 } - 1 = y + 1 = - 2 z$ and $L ^ { \prime }$ . It is known that $L$ also lies on another plane $E$ , and the projection of $L ^ { \prime }$ on $E$ coincides with $L$ . Then the equation of $E$ is $x +$ (16-1)(16-2) $y +$ (16-3)(16-4) $z =$ (16-5) .

In coordinate space, a parallelepiped has three vertices of one base at $( - 1,2,1 ) , ( - 4,1,3 ) , ( 2,0 , - 3 )$ , and one vertex of another face lies on the $xy$-plane at distance 1 from the origin. Among parallelepipeds satisfying the above conditions, the maximum volume is (17-1)(17-2).

On the coordinate plane, there is an annular region formed by the intersection of the exterior of the circle $x ^ { 2 } + y ^ { 2 } = 3$ and the interior of the circle $x ^ { 2 } + y ^ { 2 } = 4$ . A person wants to use a straight scanning rod of length 1 to scan a certain region $R$ above the $x$-axis of this annular region. He designs the scanning rod with black and white ends moving respectively on the semicircles $C _ { 1 } : x ^ { 2 } + y ^ { 2 } = 3 ( y \geq 0 )$ and $C _ { 2 } : x ^ { 2 } + y ^ { 2 } = 4 ( y \geq 0 )$ . Initially, the black end of the scanning rod is at point $A ( \sqrt { 3 } , 0 )$ and the white end is at point $B$ on $C _ { 2 }$ . Then the black and white ends move counterclockwise along $C _ { 1 }$ and $C _ { 2 }$ respectively until the white end reaches point $B ^ { \prime } ( - 2,0 )$ on $C _ { 2 }$ , at which point scanning stops.
What are the coordinates of point $B$? (Single-choice question, 3 points)
(1) $( 0,2 )$
(2) $( 1 , \sqrt { 3 } )$
(3) $( \sqrt { 2 } , \sqrt { 2 } )$
(4) $( \sqrt { 3 } , 1 )$
(5) $( 2,0 )$

On the coordinate plane, there is an annular region formed by the intersection of the exterior of the circle $x ^ { 2 } + y ^ { 2 } = 3$ and the interior of the circle $x ^ { 2 } + y ^ { 2 } = 4$ . A person wants to use a straight scanning rod of length 1 to scan a certain region $R$ above the $x$-axis of this annular region. He designs the scanning rod with black and white ends moving respectively on the semicircles $C _ { 1 } : x ^ { 2 } + y ^ { 2 } = 3 ( y \geq 0 )$ and $C _ { 2 } : x ^ { 2 } + y ^ { 2 } = 4 ( y \geq 0 )$ . Initially, the black end of the scanning rod is at point $A ( \sqrt { 3 } , 0 )$ and the white end is at point $B$ on $C _ { 2 }$ . Then the black and white ends move counterclockwise along $C _ { 1 }$ and $C _ { 2 }$ respectively until the white end reaches point $B ^ { \prime } ( - 2,0 )$ on $C _ { 2 }$ , at which point scanning stops.
Let $O$ be the origin. When the scanning rod stops, the positions of the black and white ends are $A ^ { \prime }$ and $B ^ { \prime }$ respectively. In the diagram area of the answer sheet, use hatching to indicate the region $R$ swept by the scanning rod; and in the solution area, find $\cos \angle O A ^ { \prime } B ^ { \prime }$ and the polar coordinates of point $A ^ { \prime }$ . (Non-multiple choice question, 6 points)

On the coordinate plane, there is an annular region formed by the intersection of the exterior of the circle $x ^ { 2 } + y ^ { 2 } = 3$ and the interior of the circle $x ^ { 2 } + y ^ { 2 } = 4$ . A person wants to use a straight scanning rod of length 1 to scan a certain region $R$ above the $x$-axis of this annular region. He designs the scanning rod with black and white ends moving respectively on the semicircles $C _ { 1 } : x ^ { 2 } + y ^ { 2 } = 3 ( y \geq 0 )$ and $C _ { 2 } : x ^ { 2 } + y ^ { 2 } = 4 ( y \geq 0 )$ . Initially, the black end of the scanning rod is at point $A ( \sqrt { 3 } , 0 )$ and the white end is at point $B$ on $C _ { 2 }$ . Then the black and white ends move counterclockwise along $C _ { 1 }$ and $C _ { 2 }$ respectively until the white end reaches point $B ^ { \prime } ( - 2,0 )$ on $C _ { 2 }$ , at which point scanning stops.
(Continuing from Question 19) Let $\Omega$ denote the region swept by the scanning rod in the first quadrant. Find the areas of $\Omega$ and $R$ respectively. (Non-multiple choice question, 6 points)