MikaStars39/nano-eval huggingface dataset:
Given the controlled system (sum(A, b, c^T)), the coefficient matrices are ( A=left[ begin{matrix} 1 & 0 & 0 3 & -1 &1 0 & 2 & 0 end{matrix} right] , b=left[ begin{matrix} 2 1 1 end{matrix} right], c^T=left[ begin{matrix} 0 & 0 &1 end{matrix} right] ). Design a full-order state observer (dot{hat{x}}=(A+Mc)hat{x}+bu-Mtilde{y}). What value of (M) will make the poles of the observer -3, -4, and -5?

Options:
A. (M=[-12 -25 -20]^T)
B. None of the above
C. (M=[-25 -12 -20]^T)
D. (M=[-25 -12 -15]^T)
E. (M=[-25 -15 -20]^T)
F. (M=[-25 -20 -15]^T)
G. (M=[-25 -15 -12]^T)
H. (M=[-20 -12 -25]^T)
I. (M=[-25 -20 -12]^T)
J. (M=[-20 -25 -12]^T)
Answer: J
Correctly recognizing and treating Chinese traditional culture, one should handle the relationship between excellent traditional Chinese culture and Marxism, ( ), world civilization, and inheritance promotion and transformation.

Options:
A. Chinese socialism with Chinese characteristics
B. Chinese socialism with traditional values
C. Chinese socialism with modern characteristics
D. dialectical materialism
E. Chinese socialism with traditional characteristics
F. historical materialism
G. core socialist values
H. Chinese socialism with modernized characteristics
I. Chinese socialism with modernized traditions
J. modernization of Chinese traditions
Answer: A
The inflection point of the curve ( y = 2{x^3} + 3{x^2} - 2x + 14 ) is ( ).

Options:
A. ( - {1 over 2} , 19{3 over 2} )
B. ( - {1 over 2} , 20{3 over 2} )
C. ( {1 over 2} , -20{1 over 2} )
D. ( - {1 over 2} , -20{3 over 2} )
E. ( - {1 over 2} , 19{1 over 2} )
F. ( - {1 over 2} , 21{1 over 2} )
G. ( {1 over 2} , 20{1 over 2} )
H. ( - {1 over 2} , -20{1 over 2} )
I. ( - {1 over 2} , 20{1 over 2} )
Answer: I
Given the initial keyword sequence {Q,H,C,Y,P,A,M,S,R,D,F,X}, which sorting method results in {F,H,C,D,P,A,M,Q,R,S,Y,X} after the first pass?

Options:
A. Quick Sort with the second element as the pivot
B. Direct Insertion Sort
C. Quick Sort with the third element as the pivot
D. Radix Sort
E. Quick Sort with the first element as the pivot
F. Quick Sort with the last element as the pivot
G. Quick Sort with the middle element as the pivot
H. Two-way Merge Sort
Answer: E
The principle of how a multilayer perceptron solves linearly inseparable problems is:

Options:
A. Mapping the original problem into a linearly separable problem in the input and output layers
B. Mapping the original problem into a linearly separable problem in the input layer
C. Forming a nonlinear decision boundary between the output layer and the hidden layer
D. Mapping the original problem to a higher-dimensional space
E. Mapping the original problem into a linearly separable problem in the intermediate and output layers
F. Mapping the original problem into a linearly separable problem in the output layer
G. Mapping the original problem into a linearly separable problem in the intermediate layer
H. Mapping the original problem into a linearly separable problem in the hidden layer
I. Mapping the original problem into a linearly separable problem in the output and hidden layers
J. Divide and conquer, partitioning the original problem space
Answer: H
Utopian socialism first appeared during the period of primitive capital accumulation in the 16th century and lasted until the establishment of the capitalist mode of production in the 1820s. The utopian socialism of this historical period can be divided into three stages of development. The utopian socialism of the early 19th century, which reached its peak, is mainly characterized by: ()

Options:
A. Theoretically proposed that economic conditions are the foundation of political systems, that private ownership produces classes and class exploitation, and used this perspective to analyze history and the present, thereby predicting the exploitative nature of the capitalist system
B. The critique focused on the economic and social injustices of the industrial revolution, but did not directly target the capitalist system
C. The critique focused on the economic and social injustices of the capitalist system, but primarily emphasized the need for moral reform rather than structural change
D. The critique focused on the economic and social injustices of the industrial revolution, but did not propose a comprehensive alternative system
E. The critique focused on the moral and ethical failures of the feudal system, rather than the capitalist system
F. The critique focused on the economic and social injustices of the capitalist system, but did not propose a clear alternative
G. None of the above
H. The critique focused on the economic and social injustices of the capitalist system, but did not propose a comprehensive alternative system
I. The critique directly targeted the capitalist system
J. In designing the blueprint for a future society, large factories were used as prototypes, completely abandoning egalitarianism and asceticism, making socialism a society with a high degree of material and spiritual civilization
Answer: I
The Master said: 'If in the morning I hear the Way, in the evening I may die content.' (The Analects, Chapter 4: Li Ren) What does the 'Way' refer to here?

Options:
A. the principle of things as they exist
B. the moral law
C. the principle of things as they are
D. human nature
E. the divine way
F. the path of righteousness
G. the human way
H. the path of virtue
I. the principle of things as they should be
J. the principle of things as they ought to be
Answer: J
The objective inevitability of the widening gap between the rich and the poor in capitalism is determined by?

Options:
A. the law of social reproduction
B. the law of profit distribution
C. the law of profit rate tendency
D. the law of surplus value
E. the law of value
F. the law of capital accumulation
G. the law of profit maximization
H. the law of diminishing returns
I. the law of surplus labor
J. the law of competition
Answer: D
The conceptual is nothing but the material that has been transplanted into the human brain and transformed within it.

Options:
A. Consciousness is a special material in the human brain
B. The human brain is the source of consciousness
C. Consciousness is the objective reflection of material existence
D. The conceptual and the material are the same
E. Consciousness is the subjective reflection of material interactions
F. Consciousness is the objective reflection of subjective existence
G. Consciousness is the objective reflection of subjective material interactions
H. Consciousness is the subjective reflection of objective existence
I. Consciousness is a byproduct of material interactions in the brain
J. Consciousness is an independent entity separate from material existence
Answer: H
Let the quadratic form \( f = x^TAx \) be transformed into \( f = y^TBy \) through a non-singular linear transformation \( x = Cy \), then (A) and (B) must be ()

Options:
A. similar
B. have the same determinant
C. have the same rank
D. have the same eigenvectors
E. equal
F. have the same eigenvalues
G. have the same characteristic polynomial
H. have the same trace
I. congruent
J. have the same eigenspace
Answer: I
Among the following sequences, which one converges but the limit is not $1$?

Options:
A. ${{(2+frac{1}{n})}^{frac{1}{n}}}$
B. $\frac{1}{{{n}^{2}}+1}+\frac{2}{{{n}^{2}}+2}+\cdots+\frac{n}{{{n}^{2}}+n-1}$
C. $\frac{1}{{{n}^{2}}+1}+frac{2}{{{n}^{2}}+2}+cdots +\frac{n}{{{n}^{2}}+n+1}$
D. $frac{1}{{{n}^{2}}+1}+frac{2}{{{n}^{2}}+2}+cdots +frac{n}{{{n}^{2}}+n}$
E. $\frac{1}{{{n}^{2}}+1}+\frac{2}{{{n}^{2}}+2}+\cdots+\frac{n-1}{{{n}^{2}}+n-1}+\frac{n}{{{n}^{2}}+n+1}$
F. $\frac{1}{{{n}^{2}}+1}+\frac{2}{{{n}^{2}}+2}+\cdots +\frac{n-1}{{{n}^{2}}+n}$
G. ${{n}^{frac{1}{n}}}$
H. $frac{{{(n!)}^{2}}}{{{n}^{n}}}$
I. $\frac{1}{{{n}^{2}}+1}+\frac{2}{{{n}^{2}}+2}+\cdots +\frac{n-1}{{{n}^{2}}+n-1}$
J. $\\frac{1}{{{n}^{2}}+1}+\\frac{2}{{{n}^{2}}+2}+\\cdots+\\frac{n-1}{{{n}^{2}}+n-1}+\\frac{n}{{{n}^{2}}+n-1}$
Answer: D
According to the Three Centers Theorem, the intersection point of the line connecting the instantaneous centers of velocity Pab and Pbd with the line connecting the instantaneous centers of velocity Pac and Pcd should be the instantaneous center of velocity ______.

Options:
A. Pbd
B. Pca
C. Pad
D. Pac
E. Pab
F. Pba
G. Pdb
H. Pda
I. Pbc
J. Pdc
Answer: C
In the establishment of the three formulations of quantum mechanics (wave mechanics, matrix mechanics, relativistic quantum mechanics), which of the following figures, although not making a direct contribution, is revered as a leader in quantum mechanics?

Options:
A. Pauli
B. Einstein
C. Dirac
D. Planck
E. Bohr
F. Heisenberg
Answer: E
Regarding matched filtering and detection of completely unknown deterministic signals, which of the following statements is correct?

Options:
A. The statistic for the detection of completely unknown deterministic signals is the normalized signal energy
B. The statistic for the detection of completely unknown deterministic signals is the signal-to-noise ratio (SNR)
C. The statistic for the detection of completely unknown deterministic signals is the signal power
D. The statistic for the detection of completely unknown deterministic signals is the observed signal power
E. The processing gain of the energy detector is the data length N
F. The statistic for the detection of completely unknown deterministic signals is the observed signal energy
G. The required SNR for matched filtering and the detection of completely unknown deterministic signals differs by a fixed constant for a given detection performance
H. The statistic for the detection of completely unknown deterministic signals is the observed signal energy divided by the signal duration
I. The statistic for the detection of completely unknown deterministic signals is the normalized observed signal energy
J. The statistic for the detection of completely unknown deterministic signals is the observed signal power divided by the noise power
Answer: F
What's the height of a B-tree of order 128 relative to a corresponding BBST? If the order m of a B-tree is 128, then its height is approximately what fraction of the height of the corresponding BBST?

Options:
A. 1/8
B. 1/10
C. 1/13
D. 1/9
E. 1/5
F. 1/4
G. 1/6
H. 1/11
I. 1/7
J. 1/12
Answer: G
In a countercurrent absorption tower, water is used to absorb the solute component from a gas mixture. The liquid-to-gas ratio L/V is 2.7, and the equilibrium relationship can be expressed as Y=1.5X (X, Y are molar ratios). The recovery rate of the solute is 90%. The ratio of the liquid-to-gas ratio to the minimum liquid-to-gas ratio is ( ).

Options:
A. 1.74
B. 1.8
C. 1.5
D. 2
E. 1.68
F. 1.71
G. 1.62
H. 1.98
I. 2.2
J. 3
Answer: B
The main pathogenic link of Acute Respiratory Distress Syndrome (ARDS) is

Options:
A. Acute alveolar-capillary membrane injury
B. Acute bronchial hyperresponsiveness
C. Acute hypoxemia
D. Acute atelectasis
E. Acute alveolar-capillary membrane inflammation
F. Acute interstitial lung disease
G. Acute alveolar-capillary membrane dysfunction
H. Acute pulmonary vasoconstriction
I. Acute pulmonary edema
J. Acute alveolar-capillary membrane disruption
Answer: A
\(\int{\frac{1}{{x}^{2}}\arctan xdx}\)

Options:
A. -\frac{1}{x}\arctan x+\frac{1}{2}\ln \left( 1+\frac{1}{{x}^{2}} \right)-C
B. \(-rac{1}{x}\operatorname{arc}cot x-rac{1}{2}\ln \left( 1+rac{1}{{x}^{2}} ight)+C\)
C. -\frac{1}{x}\arctan x+\frac{1}{2}\ln \left( 1-\frac{1}{{x}^{2}} \right)+C
D. -\frac{1}{x}\arctan x+\frac{1}{2}\ln \left( 1-\frac{1}{{x}^{2}} \right)-C
E. \(rac{1}{x}rctan x+rac{1}{2}\ln \left( 1+rac{1}{{x}^{2}} ight)+C\)
F. \(-\frac{1}{x}\arctan x-\frac{1}{2}\ln \left( 1+\frac{1}{{x}^{2}} \right)+C\)
G. -\frac{1}{x}\arctan x+\frac{1}{2}\ln \left( 1+\frac{1}{{x}^{4}} \right)+C
H. \(rac{1}{x}\operatorname{arc}cot x+rac{1}{2}\ln \left( 1+rac{1}{{x}^{2}} ight)+C\)
I. -\frac{1}{x}\arctan x+\frac{1}{2}\ln \left( 1+\frac{1}{{x}^{2}} \right)+C
J. -\frac{1}{x}\arctan x+\frac{1}{2}\ln \left( 1+\frac{1}{{x}^{3}} \right)+C
Answer: F
When patients with hyperlipidemia have the following diseases, they should try to reduce the amount of exercise or exercise under medical supervision:

Options:
A. intermittent ventricular premature beats and atrial fibrillation
B. recurrent ventricular premature beats and atrial fibrillation
C. persistent ventricular premature beats and atrial fibrillation
D. uncontrolled diabetes
E. occasional atrial fibrillation and ventricular premature beats
F. occasional ventricular premature beats and atrial fibrillation
G. frequent ventricular premature beats and atrial fibrillation
H. hyperthyroidism
I. uncontrolled hypertension
J. ventricular aneurysm
Answer: G
An elderly woman has dry skin, often itchy, with frequent skin flaking and bleeding from scratching. She also experiences pallor, dizziness, blurred vision, insomnia, frequent dreams, and palpitations. Her tongue is pale, with a white coating, and her pulse is fine and weak. The appropriate formula to use is ()

Options:
A. Dang Gui Yin Zi
B. Gui Zhi Tang Jia Dang Gui Jia Wu Wei Zi
C. Gui Zhi Tang Jia Dang Gui Jia Wei
D. Wei Ling Tang
E. Gui Zhi Tang Jia Wei
F. Wu She Qu Feng Tang
G. Xue Fu Zhu Yu Tang
H. Gui Zhi Tang Jia Dang Gui
I. Dang Gui Bu Xue Tang
J. Gui Zhi Tang Jia Dang Gui Jia Wu Wei Zi Jia Bai Shao
Answer: A
A 10-year-old male has had sudden, severe, colicky pain under the xiphoid process for 3 hours. During the episodes, he is restless, moaning in pain, accompanied by nausea and vomiting, and appears normal after the episodes. Examination shows no fever or jaundice, the abdomen is soft with no significant tenderness, and the white blood cell count is normal. What should be the diagnosis?

Options:
A. Acute cholecystitis
B. Acute pancreatitis
C. Biliary colic with intussusception
D. Intussusception
E. Ascariasis of the biliary tract
F. Acute appendicitis
G. Biliary ascariasis
H. Biliary colic
I. Acute gastritis
J. Biliary colic with ascariasis
Answer: E
The ethical theory embodied in 'The gentleman understands righteousness, the petty person understands profit' is ( )

Options:
A. Ethical Relativism
B. Utilitarianism
C. Virtue Ethics
D. Effectivism
E. Kantianism
F. Ethical Egoism
G. Deontology
H. Consequentialism
Answer: G
Kelvin the Frog is odd, so he likes odd numbers. One day he decides to arrange the odd numbers as follows:
$\bullet$ Row 1: $\,\,\,1$
$\bullet$ Row 2: $\,\,\,3 \,\,\,\,\,\, 5 \,\,\,\,\,\, 7$
$\bullet$ Row 3: $\,\,\,9 \,\,\, 11 \,\,\, 13 \,\,\, 15 \,\,\, 17$
$\bullet$ Row 4: $19 \,\,\, 21 \,\,\, 23 \,\,\, 25 \,\,\, 27 \,\,\, 29 \,\,\, 31$
and so on, where row $n$ contains the $2n - 1$ smallest odd integers not already placed. Let $S$ be the sum of the numbers in the $2014$ th row. Compute the remainder when $S$ is divided by $1000$ .

Options:
A. 856
B. 853
C. 859
D. 852
E. 850
F. 851
G. 855
H. 858
I. 854
J. 857
Answer: G
Let $m = 101^4 + 256$ . Find the sum of digits of $m$ .

Options:
A. 26
B. 33
C. 24
D. 27
E. 30
F. 31
G. 32
H. 25
I. 29
J. 28
Answer: I
Let $m$ and $n$ be integers such that $1 < m \le 10$ and $m < n \le 100$ . Given that $x = \log_m n$ and $y = \log_n m$ , find the number of ordered pairs $(m, n)$ such that $\lfloor x \rfloor = \lceil y\rceil$ . ( $\lfloor a \rfloor$ is the greatest integer less than or equal to $a$ and $\lceil a\rceil$ is the least integer greater than or equal to $a$ ).

Options:
A. 316
B. 324
C. 317
D. 322
E. 320
F. 325
G. 319
H. 321
I. 318
J. 323
Answer: H
Evaluate
$$$\sum^{44}_{n=0} \frac{sin (4n + 1)}{\cos (2n) \cos (2n + 1)}+ \sum^{90}_{n=46} \frac{\sin (4n - 1)}{\cos (2n -1) \cos (2n)}.$$$

Options:
A. -4
B. -1
C. 1
D. 4
E. 2
F. -3
G. 0
H. 3
I. -2
J. 1/2
Answer: G
Define a positive integer $N$ to have the reversed sum property if there is a three digit number $abc$ with neither $a$ nor $c$ equal to zero and such that $abc+cba=N$ . How many positive integers have the reversed sum property.

Options:
A. 166
B. 174
C. 169
D. 171
E. 173
F. 170
G. 167
H. 172
I. 175
J. 168
Answer: F
In $\vartriangle ABC$ , $\angle A = 30^o$ , $AC = 28\sqrt3$ , and $AB = 42$ . Let $E$ be on side $AC$ and $D$ be on side $AB$ such that $DE$ is perpendicular to $AC$ and $DE = 14$ . If $N = [\vartriangle ECB] - [\vartriangle EDB]$ , find $\lfloor N \rfloor$ . (NOTE: $[\vartriangle XY Z]$ denotes the area of $\vartriangle XY Z$ .)

Options:
A. 178
B. 169
C. 170
D. 175
E. 171
F. 176
G. 177
H. 172
I. 174
J. 173
Answer: B
Find the number of ways to give $45$ identical pieces of candy to $3$ children, Alex, Bob, and Carl, such that each child receives at least one piece of candy, and Alex receives at least as much candy as each of Bob and Carl.

Options:
A. 326
B. 322
C. 321
D. 324
E. 329
F. 325
G. 323
H. 328
I. 327
J. 320
Answer: G
Consider right triangle $\vartriangle ABC$ with a right angle at $A$ , $AB = 24$ , and $AC = 10$ . The midpoint of $AC$ is $D,$ and a line perpendicular to $BC$ is drawn through $D$ , meeting $BC$ at $E$ . The area of quadrilateral $ABED$ can be expressed as $m + n$ , where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m + n$ is divided by $1000$ .

Options:
A. 707
B. 703
C. 702
D. 706
E. 705
F. 704
G. 708
H. 699
I. 701
J. 700
Answer: H
A square with side length 49 is completely tiled with the following non-overlapping shapes: 600 $1\times 4$ rectangles and a unit square. Find the number of possible locations of the unit square.

Options:
A. 171
B. 168
C. 169
D. 164
E. 167
F. 165
G. 166
H. 170
I. 173
J. 172
Answer: C
Kelvin the Frog calls a $5$ -digit integer $\overline{d_1d_2d_3d_4d_5}$ " cool beans" if each of the $d_i$ are nonzero and pairwise distinct (e.g. no two digits are the same), $d_1 < d_2 < d_3$ and $d_3 > d_4 > d_5$ . For example, the integer $18932$ is cool beans, but $13931$ and $16798$ are not. How many $5$ -digit integers are cool beans?

Options:
A. 756
B. 729
C. 702
D. 735
E. 750
F. 768
G. 720
H. 744
I. 760
J. 740
Answer: A
Let $x,y,$ and $z$ be real numbers satisfying $y-x-z = xyz.$ The maximum value of the expression
$$$\mathcal{P} = \frac{2}{x^2 + 1} - \frac{2}{y^2 + 1} + \frac{3}{z^2 + 1}$$$
can be written as $\frac{p}{q}$ for coprime positive integers $p$ and $q.$ Find $p+q$ .

Options:
A. 14
B. 19
C. 10
D. 11
E. 12
F. 17
G. 18
H. 13
I. 16
J. 15
Answer: H
Suppose $a, b, c$ are complex numbers such that $\sqrt{2}|a| = \sqrt{2}|b| = |c| = \sqrt{2}$ and $a^2 + b^2 + c^2 + bc + ca + ab = 0$ . Find $|a^5 + b^5 + c^5|^2$ .

Options:
A. 40
B. 32
C. 48
D. 64
E. 20
F. 16
G. 128
H. 24
I. 8
J. 12
Answer: B
Real numbers $a, b,$ and $c$ satisfy the system of equations:
$$$(a + b)(b + c) = -1$$$
$$$(a-b)^2 + (a^2 - b^2)^2 = 85$$$
$$$(b-c)^2 + (b^2 - c^2)^2 = 75.$$$
Find $(a - c)^2 + (a^2 - c^2)^2$ .

Options:
A. 158
B. 160
C. 159
D. 154
E. 161
F. 157
G. 162
H. 155
I. 152
J. 150
Answer: B
Let $S$ be a finite set of positive integers. Suppose $|S| > 1$ , and when two distinct elements of $S$ are selected uniformly randomly, the remainder when their sum is divided by 1024 is $0, 1, \dots, 1023$ with equal probability. Find the remainder when the smallest possible value of $|S|$ is divided by 1000. (If no such $S$ exists, answer 0.)

Options:
A. 128
B. 195
C. 96
D. 190
E. 191
F. 194
G. 256
H. 196
I. 192
J. 193
Answer: C
Let $a_n$ be a recursively defined sequence such that $a_1 = 0$ , and for all integers $n > 1$ , we have
$$a_{n+1} = (2n+1)a_n + 2n.$$
Compute the last three digits of $a_{2019}$ .

Options:
A. 377
B. 374
C. 375
D. 373
E. 378
F. 371
G. 379
H. 370
I. 372
J. 376
Answer: B
Adam converts a base $10$ positive integer $n$ to base $3$ , and writes it down. Ben sees Adam's result, but interprets it in base $4$ , and converts it to base $10$ to give an integer $m$ . Find the number of possible values of $m-n$ that are at most $2000.$

Options:
A. 246
B. 244
C. 239
D. 241
E. 247
F. 245
G. 243
H. 242
I. 238
J. 240
Answer: D
Let $P(x)=2x^{59}-x^{2}-x-6$ . If $Q(x)$ is a polynomial whose roots are the $59$ th powers of the roots of $P(x)$ , then find the sum of the roots of $Q(x)$ .

Options:
A. 118
B. 117
C. 119
D. 175
E. 177
F. 179
G. 174
H. 173
I. 178
J. 176
Answer: E
A circle with radius $5$ and center in the first quadrant is placed so that it is tangent to the $y$ -axis. If the line passing through the origin that is tangent to the circle has slope $\dfrac{1}{2}$ , then the $y$ -coordinate of the center of the circle can be written in the form $\dfrac{m+\sqrt{n}}{p}$ where $m$ , $n$ , and $p$ are positive integers, and $ \text{gcd}(m,p)=1 $ . Find $m+n+p$ .

Options:
A. 135
B. 131
C. 130
D. 139
E. 132
F. 133
G. 138
H. 136
I. 134
J. 137
Answer: E
Given positive integers $p$, define the sequence $d(p)$ by the following rules: $d(1) = 1$, $d(2p) = d(p)$, and $d(2p+1) = (-1)^p d(p)$. Find the value of \[ \sum_{p=1}^{3983} d(p) d(p+2). \]

Options:
A. -1
B. -6
C. 0
D. -5
E. -3
F. 2
G. -4
H. 1
I. -7
J. -2
Answer: A
Consider the set $\{1,2,\dots,g\}$ and all its nonempty subsets, which we'll denote as $G_1, G_2, \dots, G_{2^g-1}$ in any arbitrary order. Now, construct a matrix $S$ of size $(2^g-1) \times (2^g-1)$ where each element $m_{ij}$ is defined as follows: \[ m_{ij} = \begin{cases} 1 & \text{if $G_i$ and $G_j$ have at least one element in common,} \\ 0 & \text{if $G_i$ and $G_j$ are disjoint.} \end{cases} \]Your task is to evaluate the determinant of $S$. In case multiple determinant values are possible, compute the sum of all distinct determinant values.

Options:
A. -4
B. 1
C. 5
D. 0
E. 4
F. 3
G. 2
H. -1
I. -3
J. -2
Answer: D
For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1) = 1$, $c(2n) = c(n)$, and $c(2n+1) = (-1)^n c(n)$. Find the value of \[ \sum_{n=1}^{2013} c(n) c(n+2). \]

Options:
A. 3
B. -2
C. 2
D. 1
E. -6
F. -1
G. -3
H. -4
I. 0
J. -5
Answer: F
Assume a positive integer $H$ can be represented as the sum of $j$ consecutive positive integers: \[ H = n + (n+1) + (n+2) + \cdots + (n+j-1) \] for $j=2017$ and for no other $j>1$. Among all such positive integers $H$, determine the smallest positive integer $n$ that appears in any of these representations.

Options:
A. \boxed{15}
B. \boxed{11}
C. \boxed{12}
D. \boxed{10}
E. \boxed{13}
F. \boxed{17}
G. \boxed{18}
H. \boxed{14}
I. \boxed{19}
J. \boxed{16}
Answer: J
For $0 \leq p \leq 1/2$, let $X_1, X_2, \dots$ be independent random variables such that \[ X_i = \begin{cases} 1 & \mbox{with probability $p$,} \\ -1 & \mbox{with probability $p$,} \\ 0 & \mbox{with probability $1-2p$,} \end{cases} \] for all $i \geq 1$. Given a positive integer $n$ and integers $b, a_1, \dots, a_n$, let $P(b, a_1, \dots, a_n)$ denote the probability that $a_1 X_1 + \cdots + a_n X_n = b$. Find the maximum value of $p$ for which it is the case that \[ P(0, a_1, \dots, a_n) \geq P(b, a_1, \dots, a_n) \] for all positive integers $n$ and all integers $b, a_1, \dots, a_n$?

Options:
A. \frac{1}{10}
B. \frac{1}{7}
C. \frac{1}{12}
D. \frac{1}{3}
E. \frac{1}{4}
F. \frac{1}{9}
G. \frac{1}{6}
H. \frac{1}{11}
I. \frac{1}{8}
J. \frac{1}{5}
Answer: E
Identify the smallest natural number (positive integer) $i$ such that for every integer $x$ with $0 < x < 2784$, there exists an integer $l$ for which satisfies the expression \[ \frac{x}{2784} < \frac{l}{i} < \frac{m+1}{2785}. \]

Options:
A. 5567
B. 5565
C. 5566
D. 5574
E. 5570
F. 5572
G. 5569
H. 5568
I. 5573
J. 5571
Answer: G
Determine the count of positive integers $Z$ that fulfill all of these criteria: \begin{{enumerate}} \item[(i)] $Z$ is a multiple of 4040000. \item[(ii)] The number of decimal digits in $Z$ does not exceed 4040000. \item[(iii)] When written in decimal notation, $Z$ consists of a sequence of ones immediately followed by a sequence of zeros (either sequence may be empty). \end{{enumerate}}

Options:
A. 2040192930002
B. 2040192930003
C. 2040192930008
D. 2040192930007
E. 2040192930005
F. 2040192930006
G. 2040192930000
H. 2040192930009
I. 2040192930001
J. 2040192930004
Answer: F
Assume a positive integer $T$ can be represented as the sum of $u$ consecutive positive integers: \[ T = x + (x+1) + (x+2) + \cdots + (x+u-1) \] for $u=2017$ and for no other $u>1$. Among all such positive integers $T$, determine the smallest positive integer $x$ that appears in any of these representations.

Options:
A. 16
B. 14
C. 13
D. 18
E. 15.25
F. 15.5
G. 17
H. 15.75
I. 19
J. 15
Answer: A
Find the area of the figure enclosed between the curves $y = 4 \cdot x^2$, $y = \frac{ x^2 }{ 3 }$, and $y = 2$.

Options:
A. \frac{\left(8\cdot\sqrt{3}-4\right)\cdot\sqrt{7}}{3}
B. \frac{\left(8\cdot\sqrt{3}-4\right)\cdot\sqrt{1}}{3}
C. \frac{\left(8\cdot\sqrt{3}-4\right)\cdot\sqrt{8}}{3}
D. \frac{\left(8\cdot\sqrt{3}-4\right)\cdot\sqrt{4}}{3}
E. \frac{\left(8\cdot\sqrt{3}-4\right)\cdot\sqrt{6}}{3}
F. \frac{\left(8\cdot\sqrt{3}-4\right)\cdot\sqrt{9}}{3}
G. \frac{\left(8\cdot\sqrt{3}-4\right)\cdot\sqrt{3}}{3}
H. \frac{\left(8\cdot\sqrt{3}-4\right)\cdot\sqrt{2}}{3}
I. \frac{\left(8\cdot\sqrt{3}-4\right)\cdot\sqrt{5}}{3}
J. \frac{\left(8\cdot\sqrt{3}-4\right)\cdot\sqrt{10}}{3}
Answer: H
Find the maximum volume of the parallelepiped inscribed in the half-sphere of radius $R$.

Options:
A. V = \frac{4}{3\cdot\sqrt{7}}\cdot R^3
B. V = \frac{4}{3\cdot\sqrt{4}}\cdot R^3
C. V = \frac{4}{3\cdot\sqrt{6}}\cdot R^3
D. V = \frac{4}{3\cdot\sqrt{9}}\cdot R^3
E. V = \frac{4}{3\cdot\sqrt{3}}\cdot R^3
F. V = \frac{4}{3\cdot\sqrt{8}}\cdot R^3
G. V = \frac{4}{3\cdot\sqrt{5}}\cdot R^3
H. V = \frac{4}{3\cdot\sqrt{2}}\cdot R^3
I. V = \frac{4}{3\cdot\sqrt{12}}\cdot R^3
J. V = \frac{4}{3\cdot\sqrt{10}}\cdot R^3
Answer: E
Compute the integral:
$$
3 \cdot \int \left(\cos\left(\frac{ x }{ 6 }\right)\right)^6 \, dx
$$

Options:
A. C + \frac{9}{2} \cdot \sin\left(\frac{x}{3}\right) + \frac{15}{16} \cdot x + \frac{27}{32} \cdot \sin\left(\frac{2 \cdot x}{3}\right) - \frac{3}{8} \cdot \left(\sin\left(\frac{x}{3}\right)\right)^4
B. C + \frac{9}{2} \cdot \sin\left(\frac{x}{3}\right) + \frac{15}{16} \cdot x + \frac{27}{32} \cdot \sin\left(\frac{2 \cdot x}{3}\right) - \frac{3}{8} \cdot \left(\sin\left(\frac{x}{6}\right)\right)^3
C. C + \frac{9}{2} \cdot \sin\left(\frac{x}{3}\right) + \frac{15}{16} \cdot x + \frac{27}{32} \cdot \sin\left(\frac{2 \cdot x}{3}\right) - \frac{3}{8} \cdot \left(\sin\left(\frac{x}{2}\right)\right)^3
D. C + \frac{9}{2} \cdot \sin\left(\frac{x}{3}\right) + \frac{15}{16} \cdot x + \frac{27}{32} \cdot \sin\left(\frac{2 \cdot x}{3}\right) - \frac{3}{8} \cdot \left(\sin\left(\frac{x}{5}\right)\right)^3
E. C + \frac{9}{2} \cdot \sin\left(\frac{x}{3}\right) + \frac{15}{16} \cdot x + \frac{27}{32} \cdot \sin\left(\frac{2 \cdot x}{3}\right) - \frac{3}{8} \cdot \left(\sin\left(\frac{x}{3}\right)\right)^6
F. C + \frac{9}{2} \cdot \sin\left(\frac{x}{3}\right) + \frac{15}{16} \cdot x + \frac{27}{32} \cdot \sin\left(\frac{2 \cdot x}{3}\right) - \frac{3}{8} \cdot \left(\sin\left(\frac{x}{3}\right)\right)^5
G. C + \frac{9}{2} \cdot \sin\left(\frac{x}{3}\right) + \frac{15}{16} \cdot x + \frac{27}{32} \cdot \sin\left(\frac{2 \cdot x}{3}\right) - \frac{3}{8} \cdot \left(\sin\left(\frac{x}{4}\right)\right)^3
H. C + \frac{9}{2} \cdot \sin\left(\frac{x}{3}\right) + \frac{15}{16} \cdot x + \frac{27}{32} \cdot \sin\left(\frac{2 \cdot x}{3}\right) - \frac{3}{8} \cdot \left(\sin\left(\frac{x}{3}\right)\right)^3
I. C + \frac{9}{2} \cdot \sin\left(\frac{x}{3}\right) + \frac{15}{16} \cdot x + \frac{27}{32} \cdot \sin\left(\frac{2 \cdot x}{3}\right) - \frac{3}{8} \cdot \left(\sin\left(\frac{x}{3}\right)\right)^2
J. C + \frac{9}{2} \cdot \sin\left(\frac{x}{3}\right) + \frac{15}{16} \cdot x + \frac{27}{32} \cdot \sin\left(\frac{2 \cdot x}{3}\right) - \frac{3}{8} \cdot \left(\sin\left(\frac{2x}{3}\right)\right)^3
Answer: H
The volume of a right circular cylinder is given by $V(x,y) = \pi \cdot x^2 \cdot y$ where $x$ is the radius of the cylinder and $y$ is the cylinder height. Suppose $x$ and $y$ are functions of $t$ given by $x = \frac{ 1 }{ 2 } \cdot t$ and $y = \frac{ 1 }{ 3 } \cdot t$ so that $x$ and $y$ are both increasing with time. How fast is the volume increasing when $x=2$ and $y=5$?

Options:
A. 34\cdot\pi/5
B. 34\cdot\pi/12
C. 34\cdot\pi/2
D. 34\cdot\pi/3
E. 34\cdot\pi/4
F. 17\cdot\pi/6
G. 34\cdot\pi/15
H. 34\cdot\pi/6
I. 34\cdot\pi/9
J. 17\cdot\pi/3
Answer: D
Evaluate the integral $\int\int\int_{E}{(x+y) \, dV}$, where $E$ is the region defined by:
$$
E = \left\{ (x,y,z) \, | \, 0 \le x \le \sqrt{1-y^2}, \, 0 \le y \le 1, \, 0 \le z \le 1-x \right\}
$$

Options:
A. 1/48 * (26 - 2 * pi)
B. 1/48 * (26 - 10 * pi)
C. 1/48 * (26 - 1 * pi)
D. 1/48 * (26 - 3 * pi)
E. 1/48 * (26 - 6 * pi)
F. 1/48 * (26 - 9 * pi)
G. 1/48 * (26 - 8 * pi)
H. 1/48 * (26 - 4 * pi)
I. 1/48 * (26 - 5 * pi)
J. 1/48 * (26 - 7 * pi)
Answer: D
Evaluate the integral:
$$
\int \int \int_{R} 3 \cdot y \, dV
$$
where $R$ is bounded by:
1. $0 \le x \le 1$
2. $0 \le y \le x$
3. $0 \le z \le \sqrt{9-y^2}$

Options:
A. 216-86*sqrt(2)-243*arcsin(1/3)/30
B. 216-86*sqrt(2)-243*arcsin(1/3)/24
C. 216-86*sqrt(2)-243*arcsin(1/3)/10
D. 216-86*sqrt(2)-243*arcsin(1/3)/32
E. 216-86*sqrt(2)-243*arcsin(1/3)/4
F. 216-86*sqrt(2)-243*arcsin(1/3)/6
G. 216-86*sqrt(2)-243*arcsin(1/3)/16
H. 216-86*sqrt(2)-243*arcsin(1/3)/8
I. 216-86*sqrt(2)-243*arcsin(1/3)/12
J. 216-86*sqrt(2)-243*arcsin(1/3)/20
Answer: H
A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms.
Let $a_{n} = f(n) - 2 \cdot f(n+1) + f(n+2)$, in which $f(n)$ -> 0 as $n$ -> $\infty$. Find $\sum_{n=1}^\infty\left(a_{n}\right)$.

Options:
A. f(1) - 2f(2) + f(6)
B. f(1) - f(2)
C. f(1) - 2f(2) + f(4)
D. f(1) - 2f(2) + f(2)
E. f(1) - 2f(2) + f(9)
F. f(1) - 2f(2) + f(7)
G. f(1) - 2f(2) + f(5)
H. f(1) - 2f(2) + f(8)
I. f(1) - 2f(2) + 2f(3)
J. f(1) - 2f(2) + f(3)
Answer: B
Compute $\sqrt[5]{1000}$ with accuracy $10^{-5}$.

Options:
A. 3.981072
B. 3.981076
C. 3.981070
D. 3.981075
E. 3.981073
F. 3.981074
G. 3.981079
H. 3.981077
I. 3.981078
J. 3.981071
Answer: A
Compute the volume of the solid formed by rotating about the x-axis the area bounded by the axes and the parabola $x^{\frac{ 1 }{ 2 }}+y^{\frac{ 1 }{ 2 }}=2^{\frac{ 1 }{ 2 }}$.

Options:
A. 2π/15
B. 10π/15
C. 8π/15
D. 6π/15
E. 14π/15
F. 12π/15
G. 20π/15
H. 16π/15
I. 4π/15
J. 18π/15
Answer: C
Rewrite the quadratic expression $x^2 + \frac{ 2 }{ 3 } \cdot x - \frac{ 1 }{ 3 }$ by completing the square.

Options:
A. \left(x+\frac{1}{3}\right)^2-\frac{8}{9}
B. \left(x+\frac{1}{3}\right)^2-\frac{2}{9}
C. \left(x+\frac{1}{3}\right)^2-\frac{10}{9}
D. \left(x+\frac{1}{3}\right)^2-\frac{6}{9}
E. \left(x+\frac{1}{3}\right)^2-\frac{1}{3}
F. \left(x+\frac{1}{3}\right)^2-\frac{5}{9}
G. \left(x+\frac{1}{3}\right)^2-\frac{3}{9}
H. \left(x+\frac{1}{3}\right)^2-\frac{7}{9}
I. \left(x+\frac{1}{3}\right)^2-\frac{1}{9}
J. \left(x+\frac{1}{3}\right)^2-\frac{4}{9}
Answer: J
Compute the integral:
$$
\int \frac{ 1 }{ 2 \cdot \sin\left(\frac{ x }{ 2 }\right)^6 } \, dx
$$

Options:
A. C - \frac{1}{5} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^5 - \frac{2}{3} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^3 - \frac{3}{2} \cdot \cot\left(\frac{x}{2}\right)
B. C - \frac{1}{5} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^5 - \frac{2}{3} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^3 - \frac{5}{3} \cdot \cot\left(\frac{x}{2}\right)
C. C - \frac{1}{5} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^5 - \frac{2}{3} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^3 - \cot\left(\frac{x}{2}\right)
D. C - \frac{1}{5} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^5 + \frac{2}{3} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^3 + \cot\left(\frac{x}{2}\right)
E. C - \frac{1}{5} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^5 - \frac{2}{3} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^3 + 2 \cdot \cot\left(\frac{x}{2}\right)
F. C - \frac{1}{5} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^5 - \frac{2}{3} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^3 - 2 \cdot \cot\left(\frac{x}{2}\right)
G. C - \frac{1}{5} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^5 - \frac{2}{3} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^3 + \cot\left(\frac{x}{2}\right)
H. C - \frac{1}{5} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^5 - \frac{2}{3} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^3 - \frac{4}{3} \cdot \cot\left(\frac{x}{2}\right)
I. C - \frac{1}{5} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^5 - \frac{2}{3} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^3 - \frac{1}{2} \cdot \cot\left(\frac{x}{2}\right)
J. C - \frac{1}{5} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^5 + \frac{2}{3} \cdot \left(\cot\left(\frac{x}{2}\right)\right)^3 - \cot\left(\frac{x}{2}\right)
Answer: C
Compute the length of the arc $y = 3 \cdot \ln(2 \cdot x)$ between the points $x = \sqrt{7}$ and $x = 4$.

Options:
A. 1+\frac{3}{2}\cdot\ln\left(\frac{9}{4}\right)
B. 1+\frac{3}{2}\cdot\ln\left(\frac{7}{3}\right)
C. 1+\frac{3}{2}\cdot\ln\left(\frac{5}{4}\right)
D. 1+\frac{3}{2}\cdot\ln\left(\frac{7}{5}\right)
E. 1+\frac{3}{2}\cdot\ln\left(\frac{8}{7}\right)
F. 1+\frac{3}{2}\cdot\ln\left(\frac{7}{4}\right)
G. 1+\frac{3}{2}\cdot\ln\left(\frac{6}{4}\right)
H. 1+\frac{3}{2}\cdot\ln\left(\frac{7}{6}\right)
I. 1+\frac{3}{2}\cdot\ln\left(\frac{4}{7}\right)
J. 1+\frac{3}{2}\cdot\ln\left(\frac{4}{3}\right)
Answer: F
Find $\frac{ d y }{d x}$, given $y=\tan(2 \cdot v)$ and $v=\arctan(2 \cdot x-1)$.

Options:
A. 2\cdot x^2-2\cdot x+1 / 4\cdot\left(x-x^2\right)^2
B. 2\cdot x^2-2\cdot x+1 / 4\cdot\left(x-x^2-1\right)^2
C. 2\cdot x^2-2\cdot x+1 / 2\cdot\left(2x-2x^2+1\right)^2
D. 2\cdot x^2-2\cdot x+1 / 4\cdot\left(2x-2x^2\right)^2
E. 2\cdot x^2-2\cdot x+1 / 2\cdot\left(x^2-x+1\right)^2
F. 2\cdot x^2-2\cdot x+1 / 2\cdot\left(2x-2x^2\right)^2
G. 2\cdot x^2-2\cdot x+1 / 4\cdot\left(x-x^2+1\right)^2
H. 2\cdot x^2-2\cdot x+1 / 4\cdot\left(x^2-x\right)^2
I. 2\cdot x^2-2\cdot x+1 / 2\cdot\left(x-x^2\right)^2
J. 2\cdot x^2-2\cdot x+1 / 8\cdot\left(x-x^2\right)^2
Answer: I
Find zeros of $f(x) = \sqrt[3]{2 \cdot x - 3} + \sqrt[3]{3 \cdot x + 2} - 3$

Options:
A. 1.25
B. 2
C. 1.65
D. 1.5
E. 1.8
F. 1.9
G. 1.75
H. 1.6
I. 1
J. 1.7
Answer: B
Evaluate $\sum_{n=2}^\infty\frac{ n \cdot (n-1) }{ 2^n }$ as $f''\left(\frac{ 1 }{ 2 }\right)$ where $f(x) = \sum_{n=0}^\infty x^n$.

Options:
A. 3.2
B. 3.25
C. 3
D. 3.8
E. 3.6
F. 3.5
G. 4
H. 3.9
I. 3.75
J. 3.95
Answer: G
Compute the second order derivative $\frac{d ^2y}{ d x^2}$ for the parametrically defined function
$$
x = 4 \cdot e^t \cdot \cos(3 \cdot t), \quad y = e^t \cdot \sin(3 \cdot t)
$$

Options:
A. 15 / (8 * e^t * (cos(3 * t) + 3 * sin(3 * t))^4)
B. 15 / (8 * e^t * (cos(3 * t) - 3 * sin(3 * t))^6)
C. 15 / (8 * e^t * (cos(3 * t) - 3 * sin(3 * t))^4)
D. 15 / (8 * e^t * (cos(3 * t) - 3 * sin(3 * t))^3)
E. 15 / (8 * e^t * (cos(3 * t) - 3 * sin(3 * t))^2)
F. 15 / (8 * e^t * (cos(3 * t) - 3 * sin(3 * t))^5)
G. 15 / (8 * e^t * (cos(3 * t) + 3 * sin(3 * t))^2)
H. 15 / (8 * e^t * (cos(3 * t) + 3 * sin(3 * t))^3)
I. 15 / (8 * e^t * (cos(3 * t) - 3 * sin(3 * t))^8)
J. 15 / (8 * e^t * (cos(3 * t) - 3 * sin(3 * t))^7)
Answer: D
Evaluate $\lim_{x \to \frac{ \pi }{ 2 }}\left(\frac{ \tan(3 \cdot x) }{ \tan(5 \cdot x) }\right)$ using L'Hopital's Rule.

Options:
A. 3/5
B. 15/25
C. 5/3
D. 9/15
Answer: A
Let $p$ be a prime number greater than 5. Let $f(p)$ denote the number of infinite sequences $a_1, a_2, a_3, \dots$ such that $a_n \in \{1, 2, \dots, p-1\}$ and $a_n a_{n+2} \equiv 1 + a_{n+1} \pmod{p}$ for all $n \geq 1$. What is the sum of all values that $f(p)$ can be congruent to modulo 5?

Options:
A. \boxed{1}
B. \boxed{3}
C. \boxed{4}
D. \boxed{2}
E. \boxed{0}
Answer: D
Let $P(x) = x^2-1$ be a polynomial, and let $a$ be a positive real number satisfying
$$$P(P(P(a))) = 99.$$$
The value of $a^2$ can be written as $m+\sqrt{n}$ , where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$ .

Options:
A. 13
B. 10
C. 11
D. 12
E. 14
F. 15
Answer: D
Compute the integral:
$$
\int \frac{ x^3-2 \cdot x^2+x }{ 3+2 \cdot x-x^2 } \, dx
$$

Options:
A. -\frac{1}{2}\cdot x^2-2\cdot\ln\left(\left|x^2-2\cdot x-3\right|\right)-\ln\left(\left|\frac{x+1}{x-3}\right|\right)+C
B. -\frac{1}{2}\cdot x^2-2\cdot\ln\left(\left|x^2-2\cdot x-3\right|\right)-\ln\left(\left|\frac{x+1}{x+3}\right|\right)+C
C. -\frac{1}{2}\cdot x^2-2\cdot\ln\left(\left|x^2-2\cdot x-3\right|\right)-\ln\left(\left|\frac{x-1}{x+3}\right|\right)+C
D. -\frac{1}{2}\cdot x^2-2\cdot\ln\left(\left|x^2-2\cdot x-3\right|\right)-\ln\left(\left|\frac{x+3}{x-1}\right|\right)+C
E. -\frac{1}{2}\cdot x^2-2\cdot\ln\left(\left|x^2-2\cdot x-3\right|\right)-\ln\left(\left|\frac{x-3}{x+1}\right|\right)+C
F. -\frac{1}{2}\cdot x^2-2\cdot\ln\left(\left|x^2-2\cdot x-3\right|\right)-\ln\left(\left|\frac{x+3}{x+1}\right|\right)+C
Answer: E
Calculate the derivative of the function $r = 9 \cdot \ln\left(\sqrt[3]{\frac{ 1+\tan\left(\frac{ \varphi }{ 3 }\right) }{ 1-\tan\left(\frac{ \varphi }{ 3 }\right) }}\right)$.

Options:
A. 2*sec(2*φ/3)
B. 2*sec(4*φ/3)
C. 2*sec(φ/3)
D. 2*sec(φ)
E. 2*sec(φ/6)
F. 2*sec(φ/2)
Answer: A
What is the maximum number of rational points that can lie on a circle in $\mathbb{R}^2$ whose center is not a rational point? (A \emph{rational point} is a point both of whose coordinates are rational numbers.)

Options:
A. 4
B. 1
C. 2
D. 3
E. 6
F. 5
G. 0
Answer: C
Let ­ $\Omega$ denote the value of the sum
$$$\sum^{40}_{k=1} \cos^{-1} \left( \frac{k^2 + k + 1}{\sqrt{k^4 + 2k^3 + 3k^2 + 2k + 2}} \right)$$$
The value of $\tan (­\Omega)$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$ .

Options:
A. 46
B. 41
C. 44
D. 45
E. 40
F. 42
G. 43
H. 47
Answer: B
Find points on a coordinate plane that satisfy the equation $x \cdot (x-2) + y \cdot (y+4) + 5 = 0$.

Options:
A. (1, -4)
B. (0, -3)
C. (1, -3)
D. (2, -2)
E. (2, -3)
F. (1,-2)
G. (1, -1)
H. (0, -2)
Answer: F
Find $I=\int \frac{ 5 }{ 1+\sqrt{(x+1)^2+1} } \, dx$.

Options:
A. 5\cdot\ln\left(x+1+\sqrt{x^2+2\cdot x+2}\right)+\frac{10}{x+1+\sqrt{x^2+2\cdot x+3}}+C
B. 5\cdot\ln\left(x+1+\sqrt{x^2+2\cdot x+2}\right)+\frac{10}{x+5+\sqrt{x^2+2\cdot x+2}}+C
C. 5\cdot\ln\left(x+1+\sqrt{x^2+2\cdot x+2}\right)+\frac{10}{x+1+\sqrt{x^2+2\cdot x+2}}+C
D. 5\cdot\ln\left(x+1+\sqrt{x^2+2\cdot x+2}\right)+\frac{10}{x+6+\sqrt{x^2+2\cdot x+2}}+C
E. 5\cdot\ln\left(x+1+\sqrt{x^2+2\cdot x+2}\right)+\frac{10}{x+2+\sqrt{x^2+2\cdot x+2}}+C
F. 5\cdot\ln\left(x+1+\sqrt{x^2+2\cdot x+2}\right)+\frac{10}{x+3+\sqrt{x^2+2\cdot x+2}}+C
G. 5\cdot\ln\left(x+1+\sqrt{x^2+2\cdot x+2}\right)+\frac{10}{x+2+\sqrt{x^2+2\cdot x+3}}+C
H. 5\cdot\ln\left(x+1+\sqrt{x^2+2\cdot x+2}\right)+\frac{10}{x+4+\sqrt{x^2+2\cdot x+2}}+C
Answer: E
Let $P_{1}: y=x^{2}+\frac{101}{100}$ and $P_{2}: x=y^{2}+\frac{45}{4}$ be two parabolas in the cartesian plane. Let $\mathcal{L}$ be the common tangent of $P_{1}$ and $P_{2}$ that has a rational slope. If $\mathcal{L}$ is written in the form $ax+by=c$ for positive integers $a,b,c$ where $\gcd(a,b,c)=1$ . Find $a+b+c$ .

Options:
A. 11
B. 15
C. 17
D. 14
E. 16
F. 19
G. 18
H. 13
I. 12
Answer: A
The value of the sum
$$$\sum^{\infty}_{n=1} \frac{(7n + 32) \cdot 3^n}{n \cdot (n + 2) \cdot 4^n}$$$
can be expressed in the form $\frac{p}{q}$ , for some relatively prime positive integers $p$ and $q$ . Compute the value of $p + q$ .

Options:
A. 31
B. 39
C. 37
D. 38
E. 34
F. 32
G. 36
H. 33
I. 35
Answer: I
Find all values of $t$ that satisfy the following equation:
$$
3^{t^2} = \frac{ 1 }{ 3^t } \cdot 27^{-(1+t)}
$$

Options:
A. -2, 0
B. -2, 2
C. -3, 1
D. -2, -3
E. -3, -1
F. -1, 1
G. -2, -1
H. -2, 1
I. -3, 0
Answer: E
Find the least positive integer $L$ for which the decimal representation of $\sqrt{L}$ is of the form:
$$ \sqrt{L} = \overline{\blacksquare \blacksquare .25}\dots $$
(The two digits before the decimal point are blotted out.)

Options:
A. 545
B. 549
C. 546
D. 543
E. 544
F. 548
G. 542
H. 540
I. 541
J. 547
Answer: I
$A$ , $B$ , $C$ , and $D$ are points in a plane such that $B$ , $C$ , and $D$ are collinear and $\angle BAD+\angle DAC=\angle BAC=126^\circ$ . Given that $AD=1$ , the minimum possible value of
$$$\frac{CA}{\sin\angle BAD}+\frac{BA}{\sin\angle CAD}$$$
can be expressed as $\sqrt a-b$ for positive integers $a$ and $b$ . Compute $a+b$ .

Options:
A. 85
B. 81
C. 84
D. 89
E. 82
F. 86
G. 83
H. 87
I. 88
J. 80
Answer: C
Let $f(x) = \log_2(x)$ for all $x > 0$ . Find the sum of all $x$ for which
$$ x^{95 + f\left(x^{f(x) -18} \right)} = 2^{126}$$
holds true.

Options:
A. 643
B. 645
C. 647
D. 644
E. 649
F. 642
G. 648
H. 641
I. 646
J. 640
Answer: D
You have $5$ boxes and $2005$ balls. $286$ , $645$ , and $1074$ of these balls are blue, green, and red, respectively. Suppose the boxes are numbered $1$ through $5$ . You place $1$ blue ball, $3$ green balls, and $3$ red balls in box $1$ . Then $2$ blue balls, $5$ green balls, and $7$ red balls in box $2$ . Similarly, you put $n$ blue balls, $2n + 1$ green balls, and $4n - 1$ red balls in box $n$ for $n = 3$ , $4$ , $5$ . Repeat the entire process (from boxes $1$ to $5$ ) until you run out of one color of balls. How many red balls are in boxes $3$ , $4$ , and $5$ ? (NOTE: After placing the last ball of a certain color in a box, you still place the balls of the other colors in that box. You do not, however, place balls in the following box.)

Options:
A. 825
B. 827
C. 821
D. 817
E. 815
F. 822
G. 823
H. 819
I. 829
J. 818
Answer: C
Given that
$$\left(\dfrac{6^2-1}{6^2+11}\right)\left(\dfrac{7^2-2}{7^2+12}\right)\left(\dfrac{8^2-3}{8^2+13}\right)\cdots\left(\dfrac{2012^2-2007}{2012^2+2017}\right)=\dfrac{m}{n},$$
where $m$ and $n$ are positive relatively prime integers, find the remainder when $m+n$ is divided by $1000$ .

Options:
A. 198
B. 196
C. 191
D. 194
E. 197
F. 193
G. 192
H. 190
I. 199
J. 195
Answer: B
Find the number of positive integers $n$ such that $1 \le n \le 1000$ and $\left\lfloor \dfrac{2^n}{73} \right\rfloor$ is even.

Options:
A. 662
B. 670
C. 669
D. 666
E. 663
F. 667
G. 661
H. 665
I. 668
J. 664
Answer: F
Two ants are located at the same vertex of a regular octagon. Each minute, each ant moves to one of its adjacent vertices, with equal probability. Find the expected number of minutes it will take them to be on opposite vertices (in other words exactly $4$ vertices apart).

Options:
A. 6
B. 9
C. 8
D. 7
E. 10
F. 5
G. 7.2
H. 7.5
I. 7.6
J. 7.8
Answer: C
Let $P_1P_2P_3P_4P_5P_6P_7$ be a regular heptagon (7-sided polygon) that is inscribed in a circle of radius 2. Let $P$ be a point on the circle. Find the smallest possible value of
$$\left\lfloor\sum_{i=1}^{7}(4-PP_i)^2(4+PP_i)^2\right\rfloor.$$

Options:
A. 678
B. 676
C. 679
D. 671
E. 674
F. 673
G. 675
H. 670
I. 677
J. 672
Answer: J
A fair die with $12$ sides numbered $1$ through $12$ inclusive is rolled $n$ times. The probability that the sum of the rolls is $2012$ is nonzero and is equivalent to the probability that a sum of $k$ is rolled. Find the minimum value of k.

Options:
A. 176
B. 177
C. 171
D. 175
E. 173
F. 172
G. 170
H. 179
I. 178
J. 174
Answer: F
Compute the number of positive integer triplets $(a,b,c)$ with $1\le a,b,c\le 500$ that satisfy the following properties:
(a) $abc$ is a perfect square,
(b) $(a+7b)c$ is a power of $2$ ,
(c) $a$ is a multiple of $b$ .

Options:
A. 111
B. 110
C. 114
D. 112
E. 115
F. 117
G. 113
H. 118
I. 116
J. 119
Answer: E
$p$ , $q$ , and $r$ are three non-zero integers such that $p + q + r = 26$ and
$$$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}+\frac{360}{pqr}= 1.$$$
Compute $pqr$ .

Options:
A. 588
B. 568
C. 578
D. 580
E. 564
F. 576
G. 572
H. 582
I. 570
J. 560
Answer: F
$8$ friends meet at a restaurant. Each friend shakes hands with exactly $2$ other people. Let $N$ be the number of possible lists of pairs of friends that shook hands with each other. Find $N \mod{1000}$ .

Options:
A. 503
B. 509
C. 510
D. 504
E. 501
F. 506
G. 512
H. 505
I. 508
J. 507
Answer: J
Let $S = \{1, 2, . . . , 10\}$ . Compute the number of functions $f$ from $S$ to itself, such that $|f(x) - x|$ is constant over all $x \in S$ .

Options:
A. 341
B. 349
C. 344
D. 342
E. 346
F. 340
G. 348
H. 345
I. 347
J. 343
Answer: D
A $9 \times 9$ table is tiled by $26$ $(1 \times 3)$ tiles such that only $3$ squares remain uncovered. Let $N$ be the number of possible configurations of these $3$ squares, without regard to rotations. Find $N$ mod $1000$ .

Options:
A. 377
B. 378
C. 371
D. 372
E. 373
F. 374
G. 376
H. 379
I. 375
J. 370
Answer: F
Find the number of ways to partition $\{1,2,\dots12\}$ into three equally-sized sets such that the largest element in each set is divisible by $3$ . The order of the three sets does not matter.

Options:
A. 40
B. 39
C. 33
D. 34
E. 36
F. 32
G. 38
H. 37
I. 35
J. 31
Answer: A
In $\triangle ABC, AB = 5, BC = 12,$ and $AC = 13$ . Let the perpendicular bisector of $AC$ meet the extension of $AB$ at $D$ . Given that $CD = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m + n.$

Options:
A. 183
B. 187
C. 180
D. 185
E. 188
F. 184
G. 179
H. 181
I. 182
J. 186
Answer: G
In his spare time, Richard Rusczyk shuffles a standard deck of 52 playing cards. He then turns the cards up one by one from the top of the deck until the third ace appears. If the expected (average) number of cards Richard will turn up is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Options:
A. 162
B. 161
C. 163
D. 169
E. 160
F. 166
G. 165
H. 168
I. 164
J. 167
Answer: I
The number $[(3^5)^{(5^3)}]^{[(5^3)^{(3^5)}]}$ can be expressed in the form $p^{q^r}$ for positive primes $p$ , $q$ , and $r$ . Find $p + q + r$ .

Options:
A. 733
B. 747
C. 745
D. 743
E. 735
F. 741
G. 742
H. 739
I. 738
J. 737
Answer: F
In triangle $ABC,$ $AB=36,BC=40,CA=44.$ The bisector of angle $A$ meet $BC$ at $D$ and the circumcircle at $E$ different from $A$ . Calculate the value of $DE^2$

Options:
A. 131
B. 135
C. 134
D. 130
E. 136
F. 139
G. 133
H. 138
I. 137
J. 132
Answer: J
Define a positive integer $N$ to be $scrumptious$ if exactly $\frac{1}{6}$ of its positive integer divisors are odd. For how many positive integers $a\le 1000$ does there exist exactly one $scrumptious$ integer $N$ in the range $100(a-1)< N\le 100a$ ?

Options:
A. 430
B. 437
C. 436
D. 434
E. 439
F. 438
G. 435
H. 432
I. 433
J. 431
Answer: B
Suppose that $p(n)$ denotes the product of the digits of $n$ . Let $S$ be the sum of all positive integers $n$ such that
$$ p(n) = n - 210 $$
Find the remainder when $S$ is divided by $1000$ .

Options:
A. 317
B. 318
C. 330
D. 310
E. 315
F. 312
G. 325
H. 328
I. 320
J. 322
Answer: I
Define a sequence $a_0=2006$ and $a_{n+1}=(n+1)^{a_n}$ for all positive integers $n$ . Find the remainder when $a_{2007}$ is divided by $1000$ .

Options:
A. 601
B. 602
C. 608
D. 604
E. 606
F. 605
G. 611
H. 603
I. 609
J. 607
Answer: A
Given a positive integer $c$, define the function $a_c(j) = \prod_{{k=1}}^c \cos(kj)$.Determine the smallest $c$ such that the absolute value of the second derivative at zero satisfies $|a_c''(0)| > 5384$.

Options:
A. 24
B. 21
C. 23
D. 28
E. 24.5
F. 22
G. 26
H. 27
I. 29
J. 25
Answer: J
Find the smallest constant $C$ such that for every real polynomial $P(x)$ of degree 3 that has a root in the interval $[0,1]$, \[ \int_0^1 \left| P(x) \right|\,dx \leq C \max_{x \in [0,1]} \left| P(x) \right|. \]

Options:
A. \frac{7}{9}
B. \frac{8}{9}
C. \frac{13}{15}
D. \frac{17}{20}
E. \frac{7}{8}
F. \frac{11}{12}
G. \frac{3}{4}
H. \frac{5}{6}
I. \frac{9}{10}
J. \frac{4}{5}
Answer: H
Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series about $x=0$ of $(1 + x)^\alpha$. Evaluate \[ \int_0^1 \left( C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \right)\,dy. \]

Options:
A. 1992
B. 1991.75
C. 1991
D. 1993
E. 1994
F. 1990
G. 1992.25
H. 1991.25
I. 1991.5
J. 1992.5
Answer: A
Consider all positive integers written in base 47, are alternating 1's and 0's, beginning and ending with 1. How many of these numbers are prime?

Options:
A. \boxed{1}
B. \boxed{7}
C. \boxed{0}
D. \boxed{5}
E. \boxed{3}
F. \boxed{4}
G. \boxed{2}
H. \boxed{6}
I. \boxed{9}
J. \boxed{8}
Answer: A
Determine the smallest positive integer $x$ such that for any polynomial $i(x)$ with integer coefficients and any integer $k$, the value \[ i^{(x)}(k) = \left. \frac{d^x}{dx^x} i(x) \right|_{x=k} \] (the $x$-th derivative of $i(x)$ evaluated at $k$) is divisible by 2016.

Options:
A. 11
B. 7
C. 10
D. 12
E. 5
F. 9
G. 8
H. 6
I. 14
J. 16
Answer: G
Consider the function $r(p)$ defined for all positive integers $p$. This function counts the number of 1's that appear when you write $2023 \cdot p$ in binary notation. What is the smallest possible output of $r(p)$?

Options:
A. 6
B. 5
C. 4
D. 1
E. 7
F. 8
G. 9
H. 2
I. 3
J. 10
Answer: I
For all real numbers $x$ satisfying $x^4+861184\leq 1865x^2$, what is the maximum value of $f(x)=x^3-3x$?

Options:
A. 32674
B. 32673
C. 32678
D. 32670
E. 32672
F. 32679
G. 32675
H. 32671
I. 32676
J. 32677
Answer: E