MikaStars39/nano-eval huggingface dataset:
$A=\left( \begin{array}{ll}3 \quad 5 \\ -5 \quad 3 \end{array} \right)$, then a real fundamental solution matrix of the system $\frac{dx}{dt}=Ax$ is ( )

Options:
A. $left( \begin{array}{ll} {{e}^{3t}}\cos 5t \quad -{{e}^{3t}}\sin 5t \\ {{e}^{3t}}\sin 5t \quad {{e}^{3t}}\cos 5t \end{array} \right)$
B. $left( egin{array}{ll}1 \quad 0 \ 0 \quad 1 \end{array} ight)$
C. $left( egin{array}{ll}{{e}^{3t}}\cos 5t \quad -{{e}^{3t}}\sin 5t \ {{e}^{3t}}\sin 5t \quad {{e}^{3t}}\cos 5t \end{array} ight)$
D. $left( \begin{array}{ll} {{e}^{3t}}\cos 5t \quad {{e}^{3t}}\sin 5t \\ -{{e}^{3t}}\sin 5t \quad {{e}^{3t}}\cos 5t \end{array} \right)$
E. $left( \begin{array}{ll} {{e}^{3t}}\cos 5t \quad {{e}^{3t}}\sin 5t \\ -{{e}^{3t}}\cos 5t \quad {{e}^{3t}}\sin 5t \end{array} \right)$
F. $left( \begin{array}{ll} {{e}^{3t}}\cos 5t \quad -{{e}^{3t}}\sin 5t \\ -{{e}^{3t}}\sin 5t \quad -{{e}^{3t}}\cos 5t \end{array} \right)$
G. $left( egin{array}{ll}{{e}^{3t}}\cos 5t \quad {{e}^{3t}}\sin 5t \ {{e}^{3t}}\sin 5t \quad {{e}^{3t}}\cos 5t \end{array} ight)$
H. $left( \begin{array}{ll} {{e}^{3t}}\cos 5t \quad -{{e}^{3t}}\sin 5t \\ {{e}^{3t}}\sin 5t \quad -{{e}^{3t}}\cos 5t \end{array} \right)$
I. $left( \begin{array}{ll} {{e}^{3t}}\cos 5t \quad {{e}^{3t}}\sin 5t \\ {{e}^{3t}}\sin 5t \quad -{{e}^{3t}}\cos 5t \end{array} \right)$
J. $left( \begin{array}{ll} {{e}^{3t}}\cos 5t \quad -{{e}^{3t}}\sin 5t \\ -{{e}^{3t}}\sin 5t \quad {{e}^{3t}}\cos 5t \end{array} \right)$
Answer: D
Hereditary angioedema is caused by which type of complement deficiency?

Options:
A. C1r deficiency
B. C1s deficiency
C. C1 inhibitor deficiency
D. C1q deficiency
E. C9 deficiency
F. C1 inhibitor deficiency (partial)
G. C4 deficiency
H. C2 deficiency
I. C3 deficiency
J. C1INH deficiency
Answer: J
In the experiment, if the voltage values measured for the Hall element under the conditions of (+IS, +IM), (+IS, -IM), (-IS, -IM), (-IS, +IM) are 4.02mV, 2.15mV, -2.14mV, -4.06mV respectively, when only considering the influence of the unequal potential effect, the unequal potential difference between the two Hall electrodes is ( ).

Options:
A. 3.11mV
B. 2.04mV
C. 3.08mV
D. 2.02mV
E. 3.09mV
F. 0mV
G. 3.10mV
H. 0.95mV
I. 3.12mV
J. 3.07mV
Answer: E
In the Bin Packing problem, if the First Fit method is used, placing items sequentially into the first bin that can accommodate them, the known upper bound for the number of bins used is closest to (OPT is the theoretical optimal solution):

Options:
A. 1.5 OPT
B. 1.8 OPT
C. 1.2 OPT
D. 2.0 OPT
E. 3.0 OPT
F. 1.9 OPT
G. 2.2 OPT
H. 2.5 OPT
I. 2.3 OPT
J. 2.1 OPT
Answer: D
If $f(x)$ is infinitely differentiable in some neighborhood of $x_0$, then the sum function of the power series $\sum\limits_{n=0}^{\infty} \left[ \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n \right]$

Options:
A. may exist but not equal $f(x)$
B. is not necessarily $f(x)$
C. may not exist anywhere
D. may exist but not equal $f(x)$ for all $x$ in the neighborhood of $x_0$
E. is definitely $f(x)$ for all $x$ in the neighborhood of $x_0$
F. is definitely $f(x)$
G. is definitely $f(x)$ for all $x$ outside the neighborhood of $x_0$
H. is definitely $f(x)$ for some $x$ outside the neighborhood of $x_0$
I. is definitely $f(x)$ for all $x$ in the interval of convergence of the series
J. is definitely not $f(x)$
Answer: B
The primordium of the reproductive system is

Options:
A. somites
B. urogenital ridge
C. nephric ridge
D. genital ridge
E. gonadal ridge
F. nephrogenic cord
G. mesonephric duct
H. mesonephric ridge
I. mesonephros
J. urogenital fold
Answer: D
Which of the following reflects the 'revolutionary' aspect of Kant's 'Copernican Revolution'?

Options:
A. Conforming the object to thought while maintaining universal necessity
B. Conforming the object to thought
C. Conforming thought to the universal necessity of knowledge
D. Conforming thought to the object
E. Conforming the object to thought while maintaining the universal necessity of knowledge
F. Setting boundaries for knowledge
G. Conforming the object to thought while setting boundaries for universal necessity
H. Conforming thought to the object while maintaining universal necessity
I. Defending the universal necessity of knowledge
J. Conforming the object to thought while setting boundaries for knowledge
Answer: B
Let $f(x)$ be a polynomial over the field $F$, and $K$ be a field containing $F$. Then which of the following statements is correct?

Options:
A. If $f(x)$ is reducible over $K$, then $f(x)$ is irreducible over a subfield of $F$.
B. If $f(x)$ is irreducible over $K$, then $f(x)$ is also irreducible over $F$;
C. If $f(x)$ is reducible over $K$, then $f(x)$ is also reducible over $F$;
D. If $f(x)$ is irreducible over $F$, then $f(x)$ is reducible over $K$;
E. The reducibility of $f(x)$ is independent of the field.
F. If $f(x)$ is reducible over $F$, then $f(x)$ is also reducible over $K$;
G. If $f(x)$ is reducible over $K$, then $f(x)$ is irreducible over a subfield of $K$.
H. If $f(x)$ is irreducible over $F$, then $f(x)$ is also irreducible over $K$;
I. If $f(x)$ is reducible over $K$, then $f(x)$ is irreducible over $F$;
J. If $f(x)$ is irreducible over $F$, then $f(x)$ is reducible over a subfield of $K$.
Answer: B
\( \int{\frac{2x-1}{\sqrt{1- { { x}^{2}}}}dx} \)

Options:
A. \(2\sqrt{1- { { x}^{2}}}+rcsin x+C\)
B. \(\sqrt{1- { { x}^{2}}}-rcsin x+C\)
C. \(\sqrt{1- { { x}^{2}}}+rcsin x+C\)
D. \(-2\sqrt{1- { { x}^{2}}}-\arcsin x+C\)
E. -2\sqrt{1- { { x}^{2}}}+\arcsin x+C
F. -2\sqrt{1- { { x}^{2}}}+2\arcsin x+C
G. -\sqrt{1- { { x}^{2}}}-2\arcsin x+C
H. -\sqrt{1- { { x}^{2}}}+2\arcsin x+C
I. -\sqrt{1- { { x}^{2}}}-\arcsin x+C
J. -\sqrt{1- { { x}^{2}}}-\arcsin x+2C
Answer: D
Let `A,B` be invertible matrices of the same order, then ( )

Options:
A. there exist invertible matrices `P,Q` such that `P^{-1}AQ=B`
B. there exists an invertible matrix `C` such that `C^TAC=B`
C. there exist invertible matrices `P,Q` such that `P^{-1}A^{-1}Q=B`
D. there exist invertible matrices `P,Q` such that `PAQ=B`
E. there exist invertible matrices `P,Q` such that `P^{-1}A^{-1}Q^{-1}=B`
F. there exist invertible matrices `P,Q` such that `P^{-1}A^{-1}Q^{-1}=B^{-1}`
G. there exist invertible matrices `P,Q` such that `PA^{-1}Q=B`
H. there exists an invertible matrix `P` such that `P^{-1}AP=B`
I. there exist invertible matrices `P,Q` such that `P^{-1}AQ^{-1}=B`
J. `AB=BA`
Answer: D
u and v are two vertices in a graph. After performing DFS, dTime(u)&lt;dTime(v)&lt;fTime(v)&lt;fTime(u), the relationship between u and v in the DFS forest is:

Options:
A. v is an ancestor of u
B. v is a descendant of u
C. u is an ancestor of v
D. are siblings
E. belong to different connected components
Answer: C
Xi Jinping's Thought on Socialism with Chinese Characteristics for a New Era deepens the understanding of ( ) with a brand new perspective.

Options:
A. the laws of socialist economic construction
B. the laws of economic development in socialism
C. the laws of socialist construction
D. the laws of socialist industrial development
E. the laws of governance by the Communist Party
F. the laws of development of objective things
G. the laws of socialist economic development
H. the laws of socialist infrastructure development
I. the laws of socialist infrastructure construction
J. the laws of human social development
Answer: C
In which work did Mao Zedong propose that "the establishment and development of the Red Army, guerrillas, and red regions are the highest form of peasant struggle under the leadership of the proletariat in semi-colonial China?"

Options:
A. Why Can the Red Regime Exist in China?
B. A Spark Can Start a Prairie Fire
C. The Strategic Problems of China's Revolutionary Warfare (1935 Edition)
D. Why Can the Red Regime Exist in China? (1930 Edition)
E. The Struggle on the Jinggang Mountains
F. The Strategic Problems of China's Revolutionary Warfare (1936 Edition)
G. Why Can the Red Regime Exist in China? (1928 Edition)
H. The Strategic Problems of China's Revolutionary War
I. Against Dogmatism
J. The Strategic Problems of China's Revolutionary Warfare
Answer: B
Let $A$ be an $m \times n$ matrix, $B$ be an $n \times m$ matrix, and $AB$ equals the identity matrix. Then

Options:
A. $A$'s column vector set is linearly independent, $B$'s row vector set is linearly dependent;
B. $A$'s row vector set is linearly dependent, $B$'s column vector set is linearly independent;
C. $A$'s row vector set is linearly independent, $B$'s column vector set is linearly independent;
D. $A$'s row vector set is linearly dependent, $B$'s column vector set is linearly dependent;
E. $A$'s row vector set is linearly dependent, $B$'s row vector set is linearly dependent;
F. $A$'s column vector set is linearly independent, $B$'s row vector set is linearly independent;
G. $A$'s column vector set is linearly independent, $B$'s column vector set is linearly independent.
H. $A$'s column vector set is linearly dependent, $B$'s row vector set is linearly independent;
I. $A$'s row vector set is linearly independent, $B$'s row vector set is linearly dependent;
J. $A$'s row vector set is linearly independent, $B$'s row vector set is linearly independent;
Answer: C
Among the following improper integrals, which one is divergent?

Options:
A. \int_0^{ + \infty } { \frac{1}{x^{0.5}} dx}
B. \( \int_0^1 { rac{1}{\sqrt{x}} dx} \)
C. \( \int_1^{ + \infty } { \frac{1}{\sqrt{x}} dx} \)
D. \int_0^{ + \infty } { \frac{1}{x^{1.5}} dx}
E. \int_0^{ + \infty } { \frac{1}{x} dx}
F. \( \int_{ - \infty }^0 { e^x dx} \)
G. \int_0^{ + \infty } { \frac{1}{x^2} dx}
H. \int_1^{ + \infty } { \frac{1}{x^{1.1}} dx}
I. \int_1^{ + \infty } { \frac{1}{x^{1.05}} dx}
J. \( \int_0^{ + \infty } { e^{ - 100x} dx} \)
Answer: C
A 50-year-old male patient presents with cold pain in the waist, restricted flexion and extension, a feeling of heaviness, which worsens on rainy days. His tongue coating is white and greasy, and his pulse is deep and slow. What is the preferred treatment formula?

Options:
A. Gan Jiang Ling Shu Tang (with added White Atractylodes)
B. Gan Jiang Ling Shu Tang (with added Drynaria)
C. Gan Jiang Ling Shu Tang (with added Cinnamon)
D. Ling Gui Zhu Gan Tang
E. Qiang Huo Sheng Shi Tang
F. Gan Jiang Ling Shu Tang
G. Gan Jiang Ling Shu Tang (with added Ginger)
H. Gan Jiang Ling Shu Tang (with added Poria)
I. Si Miao Wan
J. Du Huo Ji Sheng Tang
Answer: F
Regarding the monotonicity of the function \( f(x) = 2x^3 - 6x^2 - 18x - 7 \), which of the following statements is correct?

Options:
A. \( x \in (1,4) \) is monotonically decreasing
B. \( x \in (-\infty, -1) \) is monotonically decreasing
C. \( x \in (-1,3) \) is monotonically decreasing
D. \( x \in (2,5) \) is monotonically decreasing
E. \( x \in (3, +\infty) \) is monotonically increasing
F. \( x \in (-2,2) \) is monotonically decreasing
G. \( x \in (-1,3) \) is monotonically increasing
H. \( x \in (-\infty, -1) \) is monotonically increasing
I. \( x \in (3, +\infty) \) is monotonically decreasing
J. \( x \in (0,2) \) is monotonically decreasing
Answer: C
If \( \int_0^k \frac{x}{x^2+4} dx = \frac{1}{2} \ln 4 \), then \( k = \)

Options:
A. \( 2\sqrt{2} \)
B. \( \sqrt{8} \)
C. \( 2\sqrt{3} \)
D. \sqrt{3}
E. \\( \\sqrt{10} \\)
F. \( \sqrt{14} \)
G. \( 2 \)
H. \( \sqrt{2} \)
I. \( \sqrt{12} \)
J. \( rac{1}{2} \tan(\ln \sqrt{2}) \)
Answer: I
The spin Hall effect of photons produces a self-selection split on the order of ( ) under general conditions, which is a subtle physical effect.

Options:
A. millimeter
B. angstrom
C. micrometer
D. femtometer
E. nanometer
F. picometer
Answer: E
Marx pointed out: 'Capitalist accumulation constantly, and in proportion to its capacity and scale, produces a relative, i.e., surplus or additional, population of workers that exceeds the average needs of capital for its own growth.' 'This surplus population is a necessary product of accumulation or the development of wealth on a capitalist basis, but this surplus population in turn becomes a lever for capitalist accumulation, and even a condition for the existence of the capitalist mode of production.' The above argument indicates that

Options:
A. Unemployment generated by capital accumulation is a result of the increasing demand for higher-skilled labor due to technological advancements.
B. The cyclical nature of capitalist production requires a law of relative surplus population to adapt to it.
C. Unemployment generated by capital accumulation is a result of the decreasing demand for labor due to technological advancements.
D. Unemployment generated by capital accumulation is caused by the increase in the organic composition of capital due to the capitalist pursuit of cost reduction.
E. Capitalist accumulation leads to an increasing trend in the scale of unemployment in capitalist society.
F. Unemployment generated by capital accumulation is caused by the increase in the organic composition of capital due to the capitalist pursuit of efficiency.
G. Unemployment generated by capital accumulation is caused by the decrease in the organic composition of capital due to the capitalist pursuit of cost reduction.
H. Unemployment generated by capital accumulation is caused by the decrease in the organic composition of capital due to the capitalist pursuit of surplus value.
I. Unemployment generated by capital accumulation is caused by the increase in the organic composition of capital due to the capitalist pursuit of surplus value.
J. Relative surplus population is the supply of labor exceeding the demand for it by capital.
Answer: I
The Fibonacci sequence, also known as the golden ratio sequence, was introduced by the mathematician Leonardo da Fibonacci using the example of rabbit reproduction, hence it is also called the "rabbit sequence". It refers to the following sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, ... . Regarding the properties of the Fibonacci sequence, which of the following statements is correct?

Options:
A. Starting from the second term, the square of each even term is 2 less than the product of the terms before and after it
B. Starting from the third term, the square of each even term is 1 less than the product of the terms before and after it
C. Starting from the third term, the square of each even term is 1 more than the product of the terms before and after it
D. Starting from the third term, the square of each odd term is 1 more than the product of the terms before and after it
E. Starting from the second term, the square of each even term is 1 less than the product of the terms before and after it
F. Starting from the third term, the square of each even term is 2 more than the product of the terms before and after it
G. Starting from the second term, the square of each even term is 1 more than the product of the terms before and after it
H. Starting from the third term, the square of each odd term is 1 less than the product of the terms before and after it
I. Starting from the third term, the square of each even term is 2 less than the product of the terms before and after it
J. Starting from the second term, the square of each even term is 2 more than the product of the terms before and after it
Answer: E
When P32-labeled phosphate molecules are added to the cell culture medium, and ATP is isolated from the cells in a short time, it is found that the amount of ATP does not change much, but some of the terminal P of ATP has been labeled with P32. This phenomenon indicates:

Options:
A. The P second farthest from A in ATP is prone to detach
B. The P farthest from A in ATP is prone to attach
C. ATP is the direct energy source in cells
D. The P closest to A in ATP is prone to detach
E. The P farthest from A in ATP is prone to detach
F. Some P32-labeled ATP is resynthesized
G. The P second from A in ATP is prone to detach
H. The P farthest from A in ATP is prone to reattach
I. The P farthest from A in ATP is prone to reattach and detach
J. In this process, there is both synthesis and hydrolysis of ATP
Answer: E
For patients with stroke characterized by pale complexion, incontinence of urine and stool, shortness of breath, and scattered or weak pulse, which of the following acupuncture prescriptions should be selected:

Options:
A. Neiguan, Shuigou, Fenglong, Quchi
B. Neiguan, Shuigou, Guanyuan, Qihai, Zhongwan
C. Neiguan, Shuigou, Guanyuan, Qihai, Shenque
D. Neiguan, Shuigou, Guanyuan, Qihai, Taichong
E. Neiguan, Shuigou, Taixi, Fengchi
F. Neiguan, Shuigou, Guanyuan, Qihai, Zhaohai
G. Neiguan, Shuigou, Guanyuan, Qihai, Yongquan
H. Neiguan, Shuigou, Guanyuan, Qihai, Zusanli
I. Neiguan, Shuigou, Twelve Well Points, Taichong, Weizhong
J. Neiguan, Shuigou, Sanyinjiao, Chize, Weizhong
Answer: C
An immature member of the B-cell lineage, developing in the bone marrow, displays a surface receptor with two heavy chains attached to surrogate light chains. This receptor can receive information that triggers ______ in/of the cell.

Options:
A. mutation of the light chain hypervariable loops
B. apoptosis if the receptor binds antigen
C. clonal expansion if the receptor binds antigen
D. exit from the bone marrow
E. rearrangement of a light chain gene
F. mutation of the light chain variable region
G. mutation of the light chain constant region
H. rearrangement of a heavy chain gene
I. mutation of the heavy chain hypervariable loops
Answer: E
The mechanism of penicillin's antibacterial action is:

Options:
A. Interfering with bacterial protein synthesis
B. Inhibiting the linkage between the pentapeptide side chain and the tetrapeptide side chain
C. Inhibiting the linkage between the pentapeptide side chain and the tripeptide cross-bridge
D. Inhibiting the glycan backbone of peptidoglycan
E. Inhibiting the linkage between the tetrapeptide side chain and the pentapeptide cross-bridge
F. Inhibiting the linkage between the pentapeptide side chain and the tetrapeptide cross-bridge
G. Inhibiting the linkage between the tetrapeptide side chain and the tripeptide cross-bridge
H. Interfering with bacterial DNA replication
I. Damaging the cell membrane
J. Inhibiting the linkage between the tripeptide side chain and the pentapeptide cross-bridge
Answer: E
A computer memory is byte-addressed and uses little-endian format to store data. Assume the compiler specifies the lengths of int and short types as 32 bits and 16 bits, respectively, and data is stored aligned to boundaries. Consider the following C language code segment: struct { int a; char b; short c; } record; record.a = 273; If the starting address of the record variable is 0xC008, then the content at 0xC008 and the address of record.c are ( )

Options:
A. 0x00, 0xC00E
B. 0x11, 0xC00D
C. 0x11, 0xC00B
D. 0x00, 0xC00D
E. 0x00, 0xC00F
F. 0x11, 0xC010
G. 0x11, 0xC00A
H. 0x11, 0xC00E
I. 0x11, 0xC00C
J. 0x11, 0xC00F
Answer: H
The demand curve for factors of production slopes downward to the right because

Options:
A. the marginal product of the factor increases
B. the marginal product of the factor increases at a decreasing rate
C. the marginal product of the factor decreases
D. the marginal cost of the factor decreases
E. the marginal revenue product of the factor decreases
F. the marginal revenue of the factor decreases
G. the factor's contribution to production exhibits decreasing returns to scale
H. the marginal utility of the product produced by the factor decreases
I. none of the above
J. the marginal cost of the factor increases
Answer: C
Hume, through his skepticism about causality, ultimately believed that the causal relationship is ( )

Options:
A. necessary and relative
B. innate
C. hypothetical
D. possible
E. necessary and probable
F. necessary
G. probable
H. absolute
I. contingent
J. relative
Answer: I
A 48-year-old female. She has had excessive thirst and increased drinking for more than half a month, with dry mouth and tongue, frequent and large amounts of urination, red tongue at the edges and tip, yellow coating, and a large, rapid pulse. The appropriate treatment method is ()

Options:
A. Clear and purge the lungs and stomach
B. Clear heat and nourish the kidneys, generate body fluids and quench thirst
C. Clear heat and moisten the lungs, generate body fluids and quench thirst
D. Clear heat and nourish the lungs, generate body fluids and reduce urination
E. Clear stomach fire and nourish yin to preserve body fluids
F. Clear heat and nourish the lungs, generate body fluids and reduce thirst
G. Clear heat and nourish the lungs, generate body fluids and quench thirst
H. Clear heat and nourish the lungs, consolidate body fluids and quench thirst
I. Nourish yin and moisten the lungs, generate body fluids and quench thirst
J. Nourish yin and consolidate the kidneys
Answer: C
The function $f(x)=\frac{x \sin^2 x}{1+\sin x}$, then the integral $\int_0^{\pi} f(x)dx=$().

Options:
A. $\frac{4\pi-\pi^2}{5}$
B. $\frac{4\pi+\pi^2}{4}$
C. $\frac{4\pi-\pi^2}{8}$
D. $0$
E. $\frac{4\pi-\pi^2}{6}$
F. $2\pi$
G. $\frac{4\pi-\pi^2}{2}$
H. $\frac{4\pi-\pi^2}{4}$
I. $\frac{4\pi-\pi^2}{3}$
J. $rac{4\pi+\pi^2}{2}$
Answer: G
Let \(A\) be an \(m \times n\) matrix, and \(Ax=0\) be the homogeneous linear equation system corresponding to the non-homogeneous linear equation system \(Ax=b\). Which of the following conclusions is correct?

Options:
A. If \(Ax=b\) has infinitely many solutions, then \(Ax=0\) has non-trivial solutions
B. If \(Ax=0\) has non-trivial solutions, then \(Ax=b\) has infinitely many solutions
C. If \(Ax=b\) has a unique solution, then \(Ax=0\) has non-trivial solutions
D. If \(Ax=b\) has no solution, then \(Ax=0\) has only the trivial solution
E. If \(Ax=0\) has only the trivial solution, then \(Ax=b\) has a unique solution
F. If \(Ax=b\) has no solution, then \(Ax=0\) has non-trivial solutions
G. If \(Ax=0\) has non-trivial solutions, then \(Ax=b\) has a unique solution
H. If \(Ax=b\) has infinitely many solutions, then \(Ax=0\) has only the trivial solution
I. If \(Ax=b\) has a unique solution, then \(Ax=0\) has only the trivial solution
J. If \(Ax=0\) has non-trivial solutions, then \(Ax=b\) has no solution
Answer: A
The cause of inbreeding depression is

Options:
A. an increase in homozygous sites in individuals, leading to the exposure of recessive harmful genes
B. an increase in homozygous sites in individuals, leading to the expression of recessive beneficial genes
C. the appearance of phenomena affecting reproductive performance
D. an increase in homozygous sites in individuals, leading to the expression of dominant harmful genes
E. the generation of new harmful gene mutations
F. reduced adaptability to the environment during individual development
G. an increase in heterozygous sites in individuals, leading to the masking of recessive harmful genes
H. an increase in heterozygous sites in individuals, leading to the exposure of dominant harmful genes
I. an increase in homozygous sites in individuals, leading to the expression of dominant beneficial genes
J. an increase in homozygous sites in individuals, leading to the expression of recessive harmful genes
Answer: A
Let $L$ be the circle $\left\{ \begin{array}{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=1 \\ y=xtan \alpha (\alpha \in (0,\frac{\pi }{2})) \end{array} \right.$, viewed from the positive direction of the $x$-axis, its direction is counterclockwise, then $\int_{L}{(y-z)dx+(z-x)dy+(x-y)dz}=$

Options:
A. 2\pi (cos\alpha -sin\alpha )
B. \pi (2cos\alpha - sin\alpha)
C. \pi (2sin\alpha - cos\alpha)
D. \pi (cos\alpha + 2sin\alpha)
E. \pi (2cos\alpha - 2sin\alpha)
F. \pi (2sin\alpha + cos\alpha)
G. \pi (coslpha -sinlpha )
H. \pi (coslpha +sinlpha )
I. 2\pi (coslpha +sinlpha )
J. \pi (2cos\alpha + sin\alpha)
Answer: A
A chemical plant produces a pesticide that can render small mammals, such as otters, infertile. After the plant began operations, the incidence of infertility in otters living in a nearby stream increased rapidly. Therefore, the plant must have polluted the river when producing the pesticide.

Options:
A. A low-calcium diet can lead to a decrease in egg production in poultry. After chickens on a farm were allowed to forage in the spring, their egg production significantly decreased. Therefore, the decrease in egg production must have been due to a decrease in physical activity.
B. A low-calcium diet can lead to a decrease in egg production in poultry. After chickens on a farm were allowed to forage in the spring, their egg production significantly decreased. Therefore, the decrease in egg production must have been due to a lack of protein.
C. A low-calcium diet can lead to a decrease in egg production in poultry. After chickens on a farm were allowed to forage in the spring, their egg production significantly decreased. Therefore, the decrease in egg production must have been due to a decrease in the availability of calcium-rich food sources.
D. Malnourished animals are prone to disease, and the animals in large city zoos are not malnourished. Therefore, they must be unlikely to get sick.
E. A characteristic of apes is having opposable thumbs and no tail. Recently, the remains of an unknown animal were found, and since this animal had opposable thumbs, it must be an ape.
F. A low-calcium diet can lead to a decrease in egg production in poultry. After chickens on a farm were allowed to forage in the spring, their egg production significantly decreased. Therefore, the decrease in egg production must have been due to a lack of sunlight.
G. A low-calcium diet can lead to a decrease in egg production in poultry. After chickens on a farm were allowed to forage in the spring, their egg production significantly decreased. Therefore, the decrease in egg production must have been due to a decrease in water intake.
H. The bacteria that cause tetanus live in the digestive tract of horses, and tetanus is a highly contagious disease. Therefore, horses must be more susceptible to tetanus than most other animals.
I. A low-calcium diet can lead to a decrease in egg production in poultry. After chickens on a farm were allowed to forage in the spring, their egg production significantly decreased. Therefore, the food they found and consumed must have been low in calcium.
J. A low-calcium diet can lead to a decrease in egg production in poultry. After chickens on a farm were allowed to forage in the spring, their egg production significantly decreased. Therefore, the decrease in egg production must have been due to a decrease in environmental temperature.
Answer: I
Let ${{J}_{k}}=\int_{{}}^{{}}{\frac{dx}{{{[(x+a)^{2}+b^{2}]}^{k}}}}\quad (b\ne 0)$, then ${{J}_{k}}$ satisfies ( ).

Options:
A. ${{J}_{k+1}}=rac{1}{2{{b}^{2}}}\left[ (x+a){{\left( {{(x+a)}^{2}}+{{b}^{2}} ight)}^{-k}}+(2k-1){{J}_{k}} ight],\quad (k\ge 2)$
B. {{J}_{k+1}}=\frac{1}{2k{{b}^{2}}}\left[ (x+a){{\left( {{(x+a)}^{2}}+{{b}^{2}} \right)}^{-k}}-(2k-1){{J}_{k}} \right],\quad (k\ge 2)
C. {{J}_{k+1}}=\frac{1}{2k{{b}^{2}}}\left[ (x+a){{\left( {{(x+a)}^{2}}+{{b}^{2}} \right)}^{-k}}-(2k+1){{J}_{k}} \right],\quad (k\ge 2)
D. {{J}_{k+1}}=\frac{1}{2k{{b}^{2}}}\left[ (x+a){{\left( {{(x+a)}^{2}}+{{b}^{2}} \right)}^{-k}}+(2k-3){{J}_{k}} \right],\quad (k\ge 2)
E. {{J}_{k+1}}=\frac{1}{2k{{b}^{2}}}\left[ (x+a){{\left( {{(x+a)}^{2}}+{{b}^{2}} \right)}^{-k}}+(2k+1){{J}_{k}} \right],\quad (k\ge 2)
F. ${{J}_{k+1}}=rac{1}{2k{{b}^{2}}}\left[ (x+a){{\left( {{(x+a)}^{2}}+{{b}^{2}} ight)}^{-k}}-(2k-1){{J}_{k}} ight],\quad (k\ge 2)$
G. {{J}_{k+1}}=\frac{1}{2k{{b}^{2}}}\left[ (x+a){{\left( {{(x+a)}^{2}}+{{b}^{2}} \right)}^{-k}}-(2k-2){{J}_{k}} \right],\quad (k\ge 2)
H. {{J}_{k+1}}=\frac{1}{2k{{b}^{2}}}\left[ (x+a){{\left( {{(x+a)}^{2}}+{{b}^{2}} \right)}^{-k}}+(2k-2){{J}_{k}} \right],\quad (k\ge 2)
I. ${{J}_{k+1}}=\frac{1}{2k{{b}^{2}}}\left[ (x+a){{\left( {{(x+a)}^{2}}+{{b}^{2}} \right)}^{-k}}+(2k-1){{J}_{k}} \right],\quad (k\ge 2)$
J. ${{J}_{k+1}}=rac{1}{2{{b}^{2}}}\left[ (x+a){{\left( {{(x+a)}^{2}}+{{b}^{2}} ight)}^{-k}}+(2k+1){{J}_{k}} ight],\quad (k\ge 2)$
Answer: I
The ( ) of the function $f(x)=x+sin x$.

Options:
A. The convex interval is $(2n\pi,(2n+1)\pi)$, the concave interval is $((2n-1)\pi,2n\pi)$
B. The convex interval is $(n\pi,(n+1)\pi)$, the concave interval is $((n-1)\pi,n\pi)$
C. The convex interval is $((2n-1)\pi,2n\pi)$, the concave interval is $(2n\pi,(2n+1)\pi)$
D. The convex interval is $((n-1)\pi,n\pi)$, the concave interval is $(n\pi,(n+1)\pi)$
E. The convex interval is $((2n-2)\pi,(2n-1)\pi)$, the concave interval is $((2n-1)\pi,2n\pi)$
Answer: A
In 1947, the Chinese Communist Party held a national land conference in Pingshan County, Hebei Province, and the programmatic document that openly abolished the feudal and semi-feudal land system was ( )

Options:
A. The Instructions on Liquidation, Rent Reduction, and Land Issues
B. The Chinese Land Law Reform Outline
C. The Chinese Land Reform Act
D. The Xingguo Land Law
E. The Chinese Land Law Reform Act
F. The Chinese Land Reform Law Outline
G. The Chinese Land Reform Outline
H. The Chinese Land Reform Law
I. The Chinese Land Law Outline
J. The Chinese Land Law Reform
Answer: I
Let the population $Xsim N(mu, sigma^{2})$, and $X_{1}, X_{2},cdots, X_{n}$ be a sample from the population. When $mu$ is unknown, to test the hypothesis $H_{0}: sigma^{2}geqsigma_{0}^{2}, H_{1}: sigma^{2}<sigma_{0}^{2}$, the rejection region is ( )

Options:
A. $chi^{2}<chi^{2}_{1-alpha}(n-4)$
B. $chi^{2}<chi^{2}_{alpha}(n)$
C. $chi^{2}<chi^{2}_{alpha}(n-1)$
D. $chi^{2}>chi^{2}_{1-alpha}(n-1)$
E. $chi^{2}<chi^{2}_{1-alpha}(n-2)$
F. $chi^{2}>chi^{2}_{1-alpha}(n)$
G. $chi^{2}<chi^{2}_{1-alpha}(n)$
H. $chi^{2}<chi^{2}_{1-alpha}(n-3)$
I. $chi^{2}<chi^{2}_{1-alpha}(n-1)$
J. $chi^{2}<chi^{2}_{1-alpha}(n+1)$
Answer: I
In the field of navigation, the concept of Geometric Dilution of Precision (GDOP) has been introduced, which physically represents the ( ) of the shape formed by the unit vectors from the target receiver to the navigation stations involved in the positioning solution.

Options:
A. area
B. inverse square
C. inverse of the square root
D. inverse cube
E. product
F. inverse
G. volume
H. inverse square root
I. square root
J. reciprocal
Answer: J
The regulation that divided the schools across the country into primary education and vocational education during the late Qing dynasty's New Policies is ( )

Options:
A. 'Imperially Decreed School Establishment'
B. 'Strange Indeed, What Is Called the Question of the National System'
C. 'Imperially Decreed School Regulations'
D. 'Imperially Decreed School Organization'
E. 'Imperially Decreed School Ordinances'
F. 'Provisional New Criminal Code'
G. 'Imperially Decreed School Reforms'
H. Regulations for the Establishment of Schools
I. 'Imperially Decreed School Ordinances for the Establishment of Schools'
J. 'Imperially Decreed School System'
Answer: H
For high-speed, low-torque, frequent forward and reverse rotation, and two shafts with impact vibration, it is advisable to use ( ) coupling.

Options:
A. Flange Coupling
B. Flexible Disc Coupling
C. Flexible Bushing Pin Coupling
D. Diaphragm Coupling
E. Flexible Bushing Coupling
F. Elastic Bushing Pin Coupling
G. Flexible Bushing Pin Coupling with increased stiffness
H. Flexible Bushing Pin Coupling with reduced dampening
I. Flexible Bushing Pin Coupling with additional dampening
J. Universal Joint Coupling
Answer: F
Let A be a symmetric transformation of a finite-dimensional Euclidean space $V$. Which of the following statements is correct?

Options:
A. Eigenvectors of A corresponding to the same eigenvalue are not necessarily orthogonal
B. Eigenvectors of A corresponding to different eigenvalues are not orthogonal
C. The matrix of A under an orthonormal basis is a symmetric matrix
D. The matrix of A under any basis is a symmetric matrix
E. The orthogonal complement of an A-invariant subspace is also an A-invariant subspace
F. Eigenvectors of A corresponding to different eigenvalues are not always orthogonal
G. Eigenvectors of A corresponding to the same eigenvalue are always orthogonal
H. Eigenvectors of A corresponding to the same eigenvalue are orthogonal
I. Eigenvectors of A corresponding to different eigenvalues are orthogonal
J. Eigenvectors of A corresponding to different eigenvalues are not necessarily orthogonal
Answer: I
$int_{{}}^{{}}{frac{2{{x}^{2}}+2x+13}{(x-2){{({{x}^{2}}+1)}^{2}}}dx}=$

Options:
A. $frac{1}{2}cdot frac{3-4x}{{{x}^{2}}+1}+frac{1}{2}ln frac{{{(x-2)}^{2}}}{{{x}^{2}}+1}-3arctan x+C$
B. $frac{1}{2}cdot frac{3-4x}{{{x}^{2}}+1}-frac{1}{2}ln frac{{{(x-2)}^{2}}}{{{x}^{2}}+1}-arctan x+C$
C. $frac{1}{2}cdot frac{3-4x}{{{x}^{2}}+1}-frac{1}{2}ln frac{{{(x-2)}^{2}}}{{{x}^{2}}+1}-4arctan x+C$
D. $frac{1}{2}cdot frac{3-4x}{{{x}^{2}}+1}-frac{1}{2}ln frac{{{(x-2)}^{2}}}{{{x}^{2}}+1}-2arctan x+C$
E. $frac{1}{2}cdot frac{3-4x}{{{x}^{2}}+1}+frac{1}{2}ln frac{{{(x-2)}^{2}}}{{{x}^{2}}+1}-4arctan x+C$
F. $frac{1}{2}cdot frac{3-4x}{{{x}^{2}}+1}+frac{1}{2}ln frac{{{(x-2)}^{2}}}{{{x}^{2}}+1}+arctan x+C$
G. $frac{1}{2}cdot frac{3-4x}{{{x}^{2}}+1}-frac{1}{2}ln frac{{{(x-2)}^{2}}}{{{x}^{2}}+1}+arctan x+C$
H. $frac{1}{2}cdot frac{3-4x}{{{x}^{2}}+1}-frac{1}{2}ln frac{{{(x-2)}^{2}}}{{{x}^{2}}+1}+4arctan x+C$
I. $frac{1}{2}cdot frac{3-4x}{{{x}^{2}}+1}+frac{1}{2}ln frac{{{(x-2)}^{2}}}{{{x}^{2}}+1}+4arctan x+C$
J. $frac{1}{2}cdot frac{3-4x}{{{x}^{2}}+1}+frac{1}{2}ln frac{{{(x-2)}^{2}}}{{{x}^{2}}+1}-2arctan x+C$
Answer: E
An important condition for Clostridium tetani infection is

Options:
A. the microenvironment of the wound is partially anaerobic
B. the anaerobic microenvironment of the wound
C. the body has no immunity
D. the presence of oxygen in the wound
E. the microenvironment of the wound is completely anaerobic
F. dysbiosis
G. the spores of the bacterium contaminate the wound
H. the vegetative cells of the bacterium contaminate the wound
I. the microenvironment of the wound is hypoxic
J. the aerobic microenvironment of the wound
Answer: B
The 'mark-to-market' fluctuations in foreign exchange reserves mainly come from the exchange rate gains and losses and the price fluctuations of () assets.

Options:
A. non-USD assets
B. RMB and USD assets
C. non-RMB assets
D. USD assets
E. non-RMB and non-USD assets
F. RMB assets
G. RMB and non-USD assets
H. non-USD and RMB assets
I. non-USD and USD assets
J. USD and non-USD assets
Answer: A
The Fourier transform of the signal $f(t)=Asin(omega_0t)varepsilon(t)$ is

Options:
A. $frac{jApi}{2}[delta(omega+omega_0)-delta(omega-omega_0)]+frac{omega_0A}{omega^2+2omega_0^2}$
B. $frac{jApi}{2}[delta(omega+omega_0)-delta(omega-omega_0)]-frac{omega_0A}{omega^2-omega_0^2}$
C. $frac{jApi}{2}[delta(omega+omega_0)-delta(omega-omega_0)]+frac{omega_0A}{omega^2-omega_0^2}$
D. $frac{jApi}{2}[delta(omega+omega_0)-delta(omega-omega_0)]-frac{omega_0A}{2omega^2-omega_0^2}$
E. $frac{jApi}{2}[delta(omega+omega_0)-delta(omega-omega_0)]-frac{omega_0A}{omega^2-2omega_0^2}$
F. $frac{jApi}{2}[delta(omega+omega_0)-delta(omega-omega_0)]-frac{omega_0A}{omega^2+omega_0^2}$
G. $frac{Api}{2}[delta(omega+omega_0)-delta(omega-omega_0)]-frac{omega_0A}{omega^2-omega_0^2}$
H. $frac{jApi}{2}[delta(omega+omega_0)-delta(omega-omega_0)]+frac{omega_0A}{2omega^2-omega_0^2}$
I. $frac{jApi}{2}[delta(omega+omega_0)-delta(omega-omega_0)]-frac{omega_0A}{omega^2+2omega_0^2}$
J. $frac{jApi}{2}[delta(omega+omega_0)-delta(omega-omega_0)]+frac{omega_0A}{omega^2+omega_0^2}$
Answer: B
Wang wrote a novel and authorized Company A to publish it in installments on its website. Without Wang's permission, Publisher B published the novel, Li adapted the novel into a TV script, and Company C made the novel into a movie. According to the Copyright Law and related regulations, which of the following statements is correct?

Options:
A. Li infringes Wang's right of adaptation
B. Publisher B infringes Wang's right of adaptation
C. Publisher B infringes Wang's right of communication to the public
D. Publisher B infringes Wang's right of public performance
E. Publisher B infringes Wang's right of modification
F. Publisher B infringes Wang's right of distribution
G. Publisher B infringes Wang's right of translation
H. Publisher B infringes Wang's right of publication
I. Company C infringes Wang's right of reproduction
J. Publisher B infringes Wang's right of reproduction
Answer: J
A primigravida with regular menstrual cycles is now at 32 weeks of gestation. In the past 3 weeks, there has been no significant increase in weight, and the uterine height is 23 cm. Type B ultrasound suggests oligohydramnios (amniotic fluid index is 43 mm). The management plan for this case is:

Options:
A. Actively promote fetal lung maturity and then terminate the pregnancy
B. Wait until 34 weeks and then immediately terminate the pregnancy
C. Immediate termination of pregnancy
D. Wait for spontaneous delivery
E. Actively promote fetal lung maturity and wait until 33 weeks to terminate the pregnancy
F. Actively promote fetal lung maturity and wait until 36 weeks to terminate the pregnancy
G. Actively promote fetal lung maturity and wait until 35 weeks to terminate the pregnancy
H. Actively promote fetal lung maturity and wait until 34 weeks to terminate the pregnancy
I. Actively promote fetal lung maturity and wait until 32 weeks to terminate the pregnancy
J. Wait until full term and then immediately terminate the pregnancy
Answer: A
When the soil is in an elastic equilibrium state without shear failure, the relationship between the Mohr's circle of stress at each point in the soil and the envelope is

Options:
A. intersecting at one point
B. intersecting at a single point
C. intersecting at the origin
D. coincident
E. intersecting at two points
F. intersecting
G. tangent
H. separated
I. perpendicular
J. parallel
Answer: G
The CD molecule that contains ITIM in the cytoplasmic region and terminates T cell activation upon binding to B7 is ( )

Options:
A. CD28
B. CD152
C. CD154
D. CTLA-4
E. CD3
F. CD4
G. CD8
Answer: D
Let $\mathcal{S}_{n}$ be the set of strings with only $0$ 's or $1$ 's with length $n$ such that any $3$ adjacent place numbers sum to at least 1. For example, $00100$ works, but $10001$ does not. Find the number of elements in $\mathcal{S}_{11}$ .

Options:
A. 928
B. 926
C. 907
D. 891
E. 890
F. 919
G. 917
H. 927
I. 918
J. 925
Answer: H
A $\frac 1p$ -array is a structured, infinite, collection of numbers. For example, a $\frac 13$ -array is constructed as follows:
$$\begin{align*} 1 \qquad \frac 13\,\ \qquad \frac 19\,\ \qquad \frac 1{27} \qquad &\cdots\\ \frac 16 \qquad \frac 1{18}\,\ \qquad \frac{1}{54} \qquad &\cdots\\ \frac 1{36} \qquad \frac 1{108} \qquad &\cdots\\ \frac 1{216} \qquad &\cdots\\ &\ddots \end{align*}$$
In general, the first entry of each row is $\frac{1}{2p}$ times the first entry of the previous row. Then, each succeeding term in a row is $\frac 1p$ times the previous term in the same row. If the sum of all the terms in a $\frac{1}{2008}$ -array can be written in the form $\frac mn$ , where $m$ and $n$ are relatively prime positive integers, find the remainder when $m+n$ is divided by $2008$ .

Options:
A. 2003
B. 2007
C. 2002
D. 2010
E. 2009
F. 2004
G. 1
H. 2008
I. 2005
J. 2006
Answer: G
Define the sequence of integers $V_n$ such that $V_1 = 5000, V_2 = 4999,$ and
$$$V_n = \lfloor (V_{n-1})^{\frac{2}{3}} \cdot \lfloor(V_{n-2})^{\frac{1}{3}}\rfloor \rfloor$$$
for all integers $n > 2.$ Find the last three digits of $V_{1000}.$

Options:
A. 914
B. 910
C. 912
D. 915
E. 919
F. 918
G. 916
H. 911
I. 917
J. 913
Answer: J
Let $T$ be the sum of all positive integers of the form $2^r\cdot3^s$ , where $r$ and $s$ are nonnegative integers that do not exceed $4$ . Find the remainder when $T$ is divided by $1000$ .

Options:
A. 754
B. 758
C. 756
D. 750
E. 751
F. 759
G. 752
H. 757
I. 753
J. 755
Answer: E
Let $A=(11,4), B=(15,19),$ and $C=(3,18)$ be points in the coordinate plane. A point $D$ is located inside $\triangle ABC$ such that the areas of $\triangle ABD, \triangle BCD, \triangle CAD$ are in a $3:3:2$ ratio. Find the product of the coordinates of $D$ .

Options:
A. 117.25
B. 116.5
C. 115
D. 117.5
E. 116
F. 114
G. 117
H. 119
I. 116.25
J. 118
Answer: G
$\triangle ABC$ has $AB=3$ , $BC=2007$ , and $CA=2008$ . Point $E$ is on $AC$ such that $\angle ABE=\angle EBC$ . The parallel to $BC$ through $E$ hits $AB$ at $F$ . If $BE$ hits $CF$ at $D$ , then find $\frac{CD}{DF}$ .

Options:
A. 671
B. 669.5
C. 670
D. 669
E. 670.75
F. 670.25
G. 672
H. 670.125
I. 670.5
J. 668
Answer: C
Let $m_1$ and $m_2$ be the slopes of the lines through the origin that are tangent to the circle $(x- 2)^2 + (y - 2006)^2 = 1$ . Find $\left\lfloor \frac{m_1 + m_2}{10} \right\rfloor$ .
.

Options:
A. 262
B. 263
C. 269
D. 266
E. 265
F. 260
G. 267
H. 261
I. 264
J. 268
Answer: G
Let $\omega_1$ be a circle of radius $1$ that is internally tangent to a circle $\omega_2$ of radius $2$ at point $A$ . Suppose $\overline{AB}$ is a chord of $\omega_2$ with length $2\sqrt3$ that intersects $\omega_1$ at point $C\ne A$ . If the tangent line of $\omega_1$ at $C$ intersects $\omega_2$ at points $D$ and $E$ , find $CD^4 + CE^4$ .

Options:
A. 68
B. 62
C. 65
D. 66
E. 60
F. 63
G. 67
H. 61
I. 69
J. 64
Answer: F
Function $f(x)$ is a polynomial defined by the equation
$$\begin{align*} f(x) & = 1011+\sum_{n=0}^{1010} \left((2n+1)x^{2n+2}-(2n+2)x^{2n+1}\right)\\ & = 1011+(x^2-2x)+(3x^4-4x^3)+...+(2021x^{2022}-2022x^{2021}). \end{align*}$$
Then the graph of $y = \tfrac{f(x)}{(x-1)^2}$ is a polynomial function with a hole at the point $(1,y)$ . Compute the remainder when $y$ is divided by $1000$ .

Options:
A. 443
B. 442
C. 449
D. 448
E. 447
F. 440
G. 446
H. 444
I. 445
J. 441
Answer: G
Let $P_n(x)=1+x+x^2+\cdots+x^n$ and $Q_n(x)=P_1\cdot P_2\cdots P_n$ for all integers $n\ge 1$ . How many more distinct complex roots does $Q_{1004}$ have than $Q_{1003}$ ?

Options:
A. 526
B. 530
C. 528
D. 531
E. 524
F. 527
G. 529
H. 522
I. 523
J. 525
Answer: C
Find the number of positive integers $x$ less than $2500$ such that both $\lfloor \log_4 x\rfloor$ and $\lfloor \log_3 x\rfloor$ are prime.

Options:
A. 395
B. 393
C. 397
D. 390
E. 398
F. 399
G. 392
H. 394
I. 391
J. 396
Answer: I
An ant starts at $(1,1)$ in the Cartesian plane. Every second, the ant moves from its position $(x,y)$ to either $(x+1,y)$ or $(x,y+1)$ . However, it cannot ever move to a point whose $x-$ and $y-$ coordinates are both divisible by $3$ . Let $N$ be the number of paths that the ant can take from $(1,1)$ to $(11,11)$ . Find the remainder when $N$ is divided by $1000$ .

Options:
A. 491
B. 499
C. 494
D. 493
E. 498
F. 497
G. 496
H. 490
I. 492
J. 495
Answer: C
Let the center of regular polygon $A_1A_2\cdots A_{200}$ be $O$ . Suppose $N$ is the number of ways to assign a distinct number $1\leq a_i\leq 200$ to each of its vertices $A_i$ , such that $a_i,a_j$ are consecutive integers if and only if $i,j$ are consecutive integers or $A_i,A_j,O$ are collinear, and $A_1=1$ . Find the remainder when $N$ is divided by $1000$ .

Options:
A. 300
B. 150
C. 500
D. 100
E. 110
F. 120
G. 350
H. 200
I. 250
J. 400
Answer: D
A monic polynomial $Q(x)$ with degree $4$ is defined with positive real number roots $a,b,c,$ and $d$ such that $2^a = 3,$ $6^b = 5,$ $30^c = 7,$ and $210^d = 11.$ Find $\left \lfloor Q(-1) \right \rfloor.$

Options:
A. 9
B. 12
C. 15
D. 10
E. 10.75
F. 14
G. 10.25
H. 11
I. 13
J. 10.5
Answer: H
Compute the sum of the distinct prime factors of $a^4\left((a-b^3)b+1\right)+1$ if $a=9$ and $b=2$ .

Options:
A. 65
B. 63
C. 66
D. 61
E. 62
F. 68
G. 64
H. 60
I. 67
J. 69
Answer: A
Given that $a$ , $b$ , $c$ , and $d$ are real numbers satisfying
(i) $a(b+c)=d(b-c)$ ,
(ii) $a+b=c+d$ ,
(iii) $b(d+a)=c(d-a)$ , and
(iv) $abc=-3456$ ,
then find the value of $a+b+c+d$ .

Options:
A. 20
B. 21
C. 26
D. 28
E. 24
F. 27
G. 25
H. 22
I. 23
J. 18
Answer: E
Forestria is a world made when two regular tetrahedrons of side length 100 km are combined among a common face. Fox lives on the center of one face, and Squirrel lives on the center of another face that does not share an edge with Fox's face. Fox wishes to ambush Squirrel, and can only travel on the surface of Forestria. What is the shortest distance, in km, that Fox must travel in order to reach Squirrel's lair?

Options:
A. 110
B. 100
C. 140
D. 115
E. 175
F. 130
G. 120
H. 150
I. 105
J. 125
Answer: B
Let $S$ denote the value of the sum
$$$\left( \frac23 \right)^{2005} \sum^{2005}_{k=1} \frac{k^2}{2^k} \cdot {2005 \choose k}.$$$
Determine the remainder obtained when $S$ is divided by $1000$ .

Options:
A. 112
B. 110
C. 116
D. 113
E. 114
F. 115
G. 111
H. 117
I. 119
J. 118
Answer: F
Let $n$ be the number of non-congruent equiangular hexagons with all integer side lengths that have all of their side lengths less than $2017$ . Find the remainder when $n$ is divided by $1000$ .
Author: Vismay Sharan

Options:
A. 379
B. 375
C. 371
D. 378
E. 374
F. 373
G. 370
H. 376
I. 372
J. 377
Answer: H
Determine the smallest positive integer $l$ such that for any polynomial $x(x)$ with integer coefficients and any integer $k$, the value \[ x^{(l)}(k) = \left. \frac{d^l}{dx^l} x(x) \right|_{x=k} \] (the $l$-th derivative of $x(x)$ evaluated at $k$) is divisible by 2016.

Options:
A. \boxed{9}
B. \boxed{16}
C. \boxed{5}
D. \boxed{7}
E. \boxed{6}
F. \boxed{14}
G. \boxed{8}
H. \boxed{11}
I. \boxed{12}
J. \boxed{10}
Answer: G
Given positive integers $d$, define the sequence $s(d)$ by the following rules: $s(1) = 1$, $s(2d) = s(d)$, and $s(2d+1) = (-1)^d s(d)$. Find the value of \[ \sum_{d=1}^{3881} s(d) s(d+2). \]

Options:
A. 0
B. -4
C. -6
D. 2
E. -1
F. 1
G. -2
H. -7
I. -5
J. -3
Answer: E
Let $a_0 = 5/2$ and $a_k = a_{k-1}^2 - 2$ for $k \geq 1$. Compute \[ \prod_{k=0}^\infty \left(1 - \frac{1}{a_k} \right) \] in closed form.

Options:
A. \frac{3}{10}
B. \frac{3}{8}
C. \frac{4}{9}
D. \frac{2}{5}
E. \frac{3}{7}
F. \frac{5}{14}
G. \frac{3}{11}
H. \frac{4}{7}
I. \frac{5}{11}
J. \frac{5}{12}
Answer: E
The infinite sequence of 2's and 3's \begin{align*} &2,3,3,2,3,3,3,2,3,3,3,2,3,3,2,3,3, \\ &3,2,3,3,3,2,3,3,3,2,3,3,2,3,3,3,2,\dots \end{align*} has the property that, if one forms a second sequence that records the number of 3's between successive 2's, the result is identical to the given sequence. Find the real number $r$ such that, for any $n$, the $n$th term of the sequence is 2 if and only if $n = 1 + \lfloor rm \rfloor$ for some nonnegative integer $m$. (Note: $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$.)

Options:
A. 2 + \sqrt{7}
B. 2 + \sqrt{8}
C. 2 + \sqrt{3}
D. 2 + \sqrt{5}
E. 2 + \sqrt{11}
F. 2 + \sqrt{1}
G. 2 + \sqrt{4}
H. 2 + \sqrt{10}
I. 2 + \sqrt{2}
J. 2 + \sqrt{6}
Answer: C
Consider the sequence defined by $j_0=1$, $j_1=2$, and $j_c=4j_{c-1}-j_{c-2}$ for $c\geq 2$. Determine the smallest odd prime factor of $j_{155}$.

Options:
A. 182
B. 177
C. 171
D. 173
E. 181
F. 183
G. 179
H. 175
I. 178
J. 187
Answer: E
Given a list of the positive integers $1,2,3,4,\dots$, take the first three numbers $1,2,3$ and their sum $6$ and cross all four numbers off the list. Repeat with the three smallest remaining numbers $4,5,7$ and their sum $16$. Continue in this way, crossing off the three smallest remaining numbers and their sum, and consider the sequence of sums produced: $6, 16, 27, 36, \dots$. Find a number in the sequence whose base 10 representation ends with $2015$. \,

Options:
A. 42011
B. 42012
C. 42010
D. 42014
E. 42016
F. 42015
G. 42019
H. 42018
I. 42013
J. 42017
Answer: F
Given any convex pentagon whose vertices (no three of which are collinear) have integer coordinates find the minimum possible areas it can cover?

Options:
A. \frac{13}{2}
B. \frac{9}{2}
C. \frac{10}{2}
D. \frac{7}{2}
E. \frac{4}{2}
F. \frac{3}{2}
G. \frac{5}{2}
H. \frac{6}{2}
I. \frac{8}{2}
J. \frac{11}{2}
Answer: G
Evaluate $\int\int\int_{E}{z \, dV}$, where $E$ is the region defined by:
$$
E = \left\{ (x,y,z) \mid -y \le x \le y, \, 0 \le y \le 1, \, 0 \le z \le 1-x^4-y^4 \right\}
$$

Options:
A. 113/490
B. 113/480
C. 113/410
D. 113/455
E. 113/440
F. 113/430
G. 113/420
H. 113/470
I. 113/460
J. 113/450
Answer: J
Compute the derivative of the function $f(x) = \sqrt[x]{3} + \frac{ 1 }{ 6^{5 \cdot x} } + 4^{\sqrt{x}}$ at $x = 1$.

Options:
A. -0.5244
B. -0.5242
C. -0.5246
D. -0.5249
E. -0.5240
F. -0.5245
G. -0.5248
H. -0.5247
I. -0.5241
J. -0.5243
Answer: A
Calculate $E = \left(\sin\left(\frac{ \pi }{ 8 }\right)\right)^4 + \left(\sin\left(\frac{ 3 \cdot \pi }{ 8 }\right)\right)^4 + \left(\sin\left(\frac{ 5 \cdot \pi }{ 8 }\right)\right)^4 + \left(\sin\left(\frac{ 7 \cdot \pi }{ 8 }\right)\right)^4$

Options:
A. 7/8
B. 3/2
C. 5/4
D. 7/4
E. 15/8
F. 5/2
G. 13/8
H. 9/4
I. 17/8
J. 11/4
Answer: B
Find the derivative of the function: $f\left(\alpha\right) = \ln\left(\sqrt{\frac{ 3-\sin\left(\alpha\right) }{ 4+2 \cdot \sin\left(\alpha\right) }}\right)$ at $\alpha = \frac{ \pi }{ 4 }$.

Options:
A. -0.2846
B. -0.2851
C. -0.2844
D. -0.2845
E. -0.2843
F. -0.2850
G. -0.2848
H. -0.2847
I. -0.2849
J. -0.2842
Answer: G