MikaStars39/nano-eval huggingface dataset:
Find the minimum value of \[\frac{(x+1/x)^6-(x^6+1/x^6)-2}{(x+1/x)^3+(x^3+1/x^3)}\] for $x>0$.

Options:
A. 8
B. 5.9
C. 7
D. 4
E. 6
F. 5
G. 5.5
H. 5.2
I. 5.8
J. 5.7
Answer: E
Find the average value of the function $f(x,y) = \arctan(x \cdot y)$ over the region $R = [0,1] \times [0,1]$.

Options:
A. \frac{\pi-\ln(4)}{4}-\frac{\pi^2}{54}
B. \frac{\pi-\ln(4)}{4}-\frac{\pi^2}{40}
C. \frac{\pi-\ln(4)}{4}-\frac{\pi^2}{42}
D. \frac{\pi-\ln(4)}{4}-\frac{\pi^2}{32}
E. \frac{\pi-\ln(4)}{4}-\frac{\pi^2}{56}
F. \frac{\pi-\ln(4)}{4}-\frac{\pi^2}{50}
G. \frac{\pi-\ln(4)}{4}-\frac{\pi^2}{64}
H. \frac{\pi-\ln(4)}{4}-\frac{\pi^2}{44}
I. \frac{\pi-\ln(4)}{4}-\frac{\pi^2}{52}
J. \frac{\pi-\ln(4)}{4}-\frac{\pi^2}{48}
Answer: J
Find the Taylor polynomial $P_{5}$ for the function $f(x) = x \cdot \cos\left(x^2\right)$.

Options:
A. x - \frac{1}{3} \cdot x^4
B. x - \frac{1}{4} \cdot x^6
C. x - \frac{1}{2} \cdot x^6
D. x - \frac{1}{5} \cdot x^5
E. x - \frac{1}{2} \cdot x^4
F. x - \frac{1}{3} \cdot x^6
G. x - \frac{1}{2} \cdot x^5
H. x - \frac{1}{4} \cdot x^5
I. x - \frac{1}{6} \cdot x^5
J. x - \frac{1}{3} \cdot x^5
Answer: G
Simplify the expression $\frac{ \cos(t) }{ \sin(t) }+\frac{ \sin(t) }{ 1+\cos(t) }$ by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

Options:
A. sec(t) + tan(t)
B. sec(t) + cot(t) - 1
C. sec(t) - csc(t)
D. sec(t) + cot(t) + 1
E. sec(t) + csc(t)
F. sec(t) + cot(t) - csc(t)
G. sec(t)
H. sec(t) + cot(t)
I. csc(t)
J. sec(t) - tan(t)
Answer: I
Calculate $I=\int_{\pi}^{\frac{ 5 }{ 4 } \cdot \pi}{\frac{ \sin(2 \cdot x) }{ \left(\cos(x)\right)^4+\left(\sin(x)\right)^4 } \, dx}$

Options:
A. I=\frac{\pi}{12}
B. I=\frac{\pi}{4}
C. I=\frac{\pi}{3}
D. I=\frac{\pi}{9}
E. I=\frac{\pi}{7}
F. I=\frac{\pi}{5}
G. I=\frac{\pi}{2}
H. I=\frac{\pi}{6}
I. I=\frac{\pi}{10}
J. I=\frac{\pi}{8}
Answer: B
Compute the integral:
$$
\int \frac{ 8 }{ 7 \cdot x^2 \cdot \sqrt{5 \cdot x^2-2 \cdot x+1} } \, dx
$$

Options:
A. C - \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} - \ln\left(\left|\frac{1}{x} - \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} - 2\right|\right)
B. C - \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} - \ln\left(\left|\frac{1}{x} + \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} + 1\right|\right)
C. C - \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} - \ln\left(\left|\frac{1}{x} + \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} - 1\right|\right)
D. C - \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} - \ln\left(\left|\frac{1}{x} + \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} + 2\right|\right)
E. C - \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} - \ln\left(\left|\frac{1}{x} - \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} + 2\right|\right)
F. C - \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} - \ln\left(\left|\frac{1}{x} - \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} - 1\right|\right)
G. C - \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} - \ln\left(\left|\frac{1}{x} + \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} + 3\right|\right)
H. C - \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} - \ln\left(\left|\frac{1}{x} + \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} - 2\right|\right)
I. C - \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} - \ln\left(\left|\frac{1}{x} - \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} + 1\right|\right)
J. C - \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} - \ln\left(\left|\frac{1}{x} + \sqrt{5 + \frac{1}{x^2} - \frac{2}{x}} - 3\right|\right)
Answer: C
Find the product $C=\left(-\frac{ 2 }{ 3 }-\frac{ 1 }{ 2 } \cdot i\right) \cdot \left(\frac{ 1 }{ 5 }+\frac{ 4 }{ 5 } \cdot i\right)$.

Options:
A. C=\frac{4}{15}+\frac{19}{30}\cdot i
B. C=\frac{4}{15}-\frac{21}{30}\cdot i
C. C=\frac{4}{15}-\frac{23}{30}\cdot i
D. C=\frac{4}{15}-\frac{19}{30}\cdot i
E. C=\frac{4}{15}-\frac{22}{30}\cdot i
F. C=\frac{4}{15}-\frac{13}{30}\cdot i
G. C=\frac{4}{15}+\frac{17}{30}\cdot i
H. C=\frac{4}{15}-\frac{17}{30}\cdot i
I. C=\frac{4}{15}+\frac{13}{30}\cdot i
J. C=\frac{4}{15}+\frac{23}{30}\cdot i
Answer: D
Compute the integral:
$$
\int \frac{ x }{ \left(x^2-4 \cdot x+8\right)^2 } \, dx
$$

Options:
A. C + \frac{x-2}{2\cdot\left(8+2\cdot(x-2)^2\right)} + \frac{1}{8}\cdot\arctan\left(\frac{1}{2}\cdot(x-2)\right) - \frac{1}{7\cdot\left(x^2-4\cdot x+8\right)}
B. C + \frac{x-2}{2\cdot\left(8+2\cdot(x-2)^2\right)} + \frac{1}{8}\cdot\arctan\left(\frac{1}{2}\cdot(x-2)\right) - \frac{1}{10\cdot\left(x^2-4\cdot x+8\right)}
C. C + \frac{x-2}{2\cdot\left(8+2\cdot(x-2)^2\right)} + \frac{1}{8}\cdot\arctan\left(\frac{1}{2}\cdot(x-2)\right) - \frac{1}{11\cdot\left(x^2-4\cdot x+8\right)}
D. C + \frac{x-2}{2\cdot\left(8+2\cdot(x-2)^2\right)} + \frac{1}{8}\cdot\arctan\left(\frac{1}{2}\cdot(x-2)\right) - \frac{1}{6\cdot\left(x^2-4\cdot x+8\right)}
E. C + \frac{x-2}{2\cdot\left(8+2\cdot(x-2)^2\right)} + \frac{1}{8}\cdot\arctan\left(\frac{1}{2}\cdot(x-2)\right) - \frac{1}{4\cdot\left(x^2-4\cdot x+8\right)}
F. C + \frac{x-2}{2\cdot\left(8+2\cdot(x-2)^2\right)} + \frac{1}{8}\cdot\arctan\left(\frac{1}{2}\cdot(x-2)\right) - \frac{1}{9\cdot\left(x^2-4\cdot x+8\right)}
G. C + \frac{x-2}{2\cdot\left(8+2\cdot(x-2)^2\right)} + \frac{1}{8}\cdot\arctan\left(\frac{1}{2}\cdot(x-2)\right) - \frac{1}{12\cdot\left(x^2-4\cdot x+8\right)}
H. C + \frac{x-2}{2\cdot\left(8+2\cdot(x-2)^2\right)} + \frac{1}{8}\cdot\arctan\left(\frac{1}{2}\cdot(x-2)\right) - \frac{1}{5\cdot\left(x^2-4\cdot x+8\right)}
I. C + \frac{x-2}{2\cdot\left(8+2\cdot(x-2)^2\right)} + \frac{1}{8}\cdot\arctan\left(\frac{1}{2}\cdot(x-2)\right) - \frac{1}{3\cdot\left(x^2-4\cdot x+8\right)}
J. C + \frac{x-2}{2\cdot\left(8+2\cdot(x-2)^2\right)} + \frac{1}{8}\cdot\arctan\left(\frac{1}{2}\cdot(x-2)\right) - \frac{1}{2\cdot\left(x^2-4\cdot x+8\right)}
Answer: J
Evaluate $\int\int\int_{E}{\left(y \cdot \ln(x)+z\right) d V}$, where $E$ is the region defined by:
$$
E = \left\{(x,y,z) | 1 \le x \le e, 0 \le y \le \ln(x), 0 \le z \le 1\right\}
$$

Options:
A. 7/2 - 5e/2
B. 7/2 - 3e
C. 7/2 - e/2
D. 7/2 - 2e^2
E. 7/2 - 4e
F. 7/2 - 2e
G. 7/2 - e^2
H. 7/2 - 3e/2
I. 7/2 - e/4
J. 7/2 - e
Answer: J
Find the area bounded by the curves $y = 3 \cdot x + 3$, $y = 3 \cdot \cos(x)$, and $y = 0$.

Options:
A. 4.2
B. 4.6
C. 4.1
D. 4.7
E. 4.4
F. 4.5
G. 4.9
H. 4.8
I. 4.3
J. 4.0
Answer: F
Calculate the double integral:
$$
\int\int_{R}{\left(x \cdot y \cdot \sqrt{x^2+y^2}\right) d A}
$$
where $R=\left\{(x,y)|0 \le x \le 1,0 \le y \le 2\right\}$.

Options:
A. frac{25\cdot\sqrt{5}-32}{15}
B. frac{25\cdot\sqrt{5}-38}{15}
C. frac{25\cdot\sqrt{5}-31}{15}
D. frac{25\cdot\sqrt{5}-33}{15}
E. frac{25\cdot\sqrt{5}-34}{15}
F. frac{25\cdot\sqrt{5}-39}{15}
G. frac{25\cdot\sqrt{5}-36}{15}
H. frac{25\cdot\sqrt{5}-35}{15}
I. frac{25\cdot\sqrt{5}-37}{15}
J. frac{25\cdot\sqrt{5}-30}{15}
Answer: D
Find the area of the region bounded by the parametric curves and the indicated values of the parameter:
$$
x = a \cdot \sin(2 \cdot t), \quad y = b \cdot \sin(t), \quad 0 \le t < 2 \cdot \pi
$$
(the "hourglass").

Options:
A. 8/3 * a * b
B. 18/3 * a * b
C. 16/3 * a * b
D. 6/3 * a * b
E. 4/3 * a * b
F. 2/3 * a * b
G. 10/3 * a * b
H. 12/3 * a * b
I. 14/3 * a * b
J. 11/3 * a * b
Answer: A
Solve the integral:
$$
\int \tan(x)^4 \, dx
$$

Options:
A. C + \frac{1}{3} \cdot (\tan(x))^3 + \arctan(\tan(x)) - \frac{1}{4}\tan(x)
B. C + \frac{1}{3} \cdot (\tan(x))^3 + \arctan(\tan(x)) - \frac{1}{3}\tan(x)
C. C + \frac{1}{3} \cdot (\tan(x))^3 + \arctan(\tan(x)) - \frac{5}{4}\tan(x)
D. C + \frac{1}{3} \cdot (\tan(x))^3 + \arctan(\tan(x)) - \frac{1}{5}\tan(x)
E. C + \frac{1}{3} \cdot (\tan(x))^3 + \arctan(\tan(x)) - \frac{3}{2}\tan(x)
F. C + \frac{1}{3} \cdot (\tan(x))^3 + \arctan(\tan(x)) - \frac{1}{2}\tan(x)
G. C + \frac{1}{3} \cdot (\tan(x))^3 + \arctan(\tan(x)) - \tan(x)
H. C + \frac{1}{3} \cdot (\tan(x))^3 + \arctan(\tan(x)) - \frac{2}{3}\tan(x)
I. C + \frac{1}{3} \cdot (\tan(x))^3 + \arctan(\tan(x)) - \frac{4}{3}\tan(x)
J. C + \frac{1}{3} \cdot (\tan(x))^3 + \arctan(\tan(x)) - 2\tan(x)
Answer: G
If it takes $6$ years for an exponentially growing quantity to triple in size, then by what percentage does it increase every year? Round your answer to two decimal places, and omit the % sign.

Options:
A. 21.15
B. 20.25
C. 19.95
D. 20.09
E. 19.85
F. 19.05
G. 19.15
H. 20.90
I. 18.92
J. 20.15
Answer: D
Using the fact that:
$$
e^x = \sum_{n=0}^\infty \left(\frac{ x^n }{ n! }\right)
$$
Evaluate:
$$
\sum_{n=0}^\infty \left(\frac{ 3 \cdot n }{ n! } \cdot 7^{1-3 \cdot n}\right)
$$

Options:
A. 3/49 * e^(1/1372)
B. 3/49 * e^(1/117649)
C. 3/49 * e^(1/125)
D. 3/49 * e^(1/7)
E. 3/49 * e^(1/16807)
F. 3/49 * e^(1/1225)
G. 3/49 * e^(1/49)
H. 3/49 * e^(1/216)
I. 3/49 * e^(1/343)
J. 3/49 * e^(1/2401)
Answer: I
Let $a_0=1$, $a_1=2$, and $a_n=4a_{n-1}-a_{n-2}$ for $n\geq 2$. Find the least odd prime factor of $a_{2015}$.

Options:
A. 182
B. 189
C. 181
D. 187
E. 191
F. 193
G. 199
H. 197
I. 183
J. 185
Answer: C
Compute the integral:
$$
\int \frac{ \sqrt{4+x^2} }{ x } \, dx
$$

Options:
A. C + \sqrt{4 + x^2} + \ln\left(\left|\frac{\sqrt{4 + x^2} + 2}{2 + \sqrt{4 + x^2}}\right|\right)
B. C + \sqrt{4 + x^2} + \ln\left(\left|\frac{\sqrt{4 + x^2} + 2}{2 - \sqrt{4 + x^2}}\right|\right)
C. C + \sqrt{4 + x^2} + \ln\left(\left|\frac{\sqrt{4 + x^2} - 2}{2 + \sqrt{4 + x^2}}\right|\right)
D. C + \sqrt{4 + x^2} + \ln\left(\left|\frac{\sqrt{4 + x^2} - 2}{2 - \sqrt{4 + x^2}}\right|\right)
Answer: C
In acute triangle $ABC,$ $\ell$ is the bisector of $\angle BAC$ . $M$ is the midpoint of $BC$ . a line through $M$ parallel to $\ell$ meets $AC,AB$ at $E,F,$ respectively. Given that $AE=1,EF=\sqrt{3}, AB=21,$ the sum of all possible values of $BC$ can be expressed as $\sqrt{a}+\sqrt{b},$ where $a,b$ are positive integers. What is $a+b$ ?

Options:
A. 890
B. 892
C. 894
D. 891
E. 893
Answer: A
Find the solutions of $(x+2)^4 + (x+5)^4 = 17$.

Options:
A. x_1=-4, x_2=-3
B. x_1=-3.5, x_2=-3.5
C. x_1=-3.5, x_2=-3
D. x_1=-3.5, x_2=-4.5
E. x_1=-3.5, x_2=-4
Answer: A
Find $L=\lim_{n \to \infty}\left(x_{n}\right)$, where $x_{n} = \frac{ \sqrt{n^2+1} + \sqrt{n} }{ \sqrt[4]{n^3+n} - \sqrt{n} }$ is the general term of a sequence.

Options:
A. 0
B. 2
C. 1
D. -∞
E. -1
F. ∞
Answer: F
Compute the integral:
$$
\int \frac{ \sqrt{25+x^2} }{ 5 \cdot x } \, dx
$$

Options:
A. C + \frac{1}{2} \cdot \ln\left(\left|\frac{\sqrt{25+x^2}-5}{5+\sqrt{25+x^2}}\right|\right) + \frac{1}{5} \cdot \sqrt{25+x^2}
B. C + \frac{1}{2} \cdot \ln\left(\left|\frac{\sqrt{25+x^2}+5}{5-\sqrt{25+x^2}}\right|\right) - \frac{1}{5} \cdot \sqrt{25+x^2}
C. C + \frac{1}{2} \cdot \ln\left(\left|\frac{\sqrt{25+x^2}+5}{5+\sqrt{25+x^2}}\right|\right) + \frac{1}{5} \cdot \sqrt{25+x^2}
D. C + \frac{1}{2} \cdot \ln\left(\left|\frac{\sqrt{25+x^2}-5}{5-\sqrt{25+x^2}}\right|\right) - \frac{1}{5} \cdot \sqrt{25+x^2}
E. C + \frac{1}{2} \cdot \ln\left(\left|\frac{\sqrt{25+x^2}-5}{5-\sqrt{25+x^2}}\right|\right) + \frac{1}{5} \cdot \sqrt{25+x^2}
F. C + \frac{1}{2} \cdot \ln\left(\left|\frac{\sqrt{25+x^2}+5}{5-\sqrt{25+x^2}}\right|\right) + \frac{1}{5} \cdot \sqrt{25+x^2}
Answer: A
In the coordinate plane, an ant begins at the point $(0,0).$ When the ant is at $(x,y),$ it can move to either $(x+1,y)$ or $(x,y+1),$ but cannot move to a point whose coordinates leave the same nonzero remainder when divided by $4$ . The ant can take a total of $N$ paths to $(68,68)$ . Find the remainder when $N$ is divided by $1000$ .

Options:
A. 140
B. 100
C. 20
D. 40
E. 120
F. 60
G. 80
Answer: F
Find a real number $c$ and a positive number $L$ for which \[ \lim_{r\to\infty} \frac{r^c \int_0^{\pi/2} x^r \sin x \,dx}{\int_0^{\pi/2} x^r \cos x \,dx} = L \] and return the product of $c$ and $L$.

Options:
A. \frac{-1}{2\pi}
B. \frac{-1}{\pi^2}
C. \frac{-4}{\pi}
D. \frac{-3}{\pi}
E. \frac{-2}{\pi}
F. \frac{-1}{\pi}
G. \frac{-3}{2\pi}
Answer: E
Solve the integral:
$$
-\int \frac{ 1 }{ \cos(x)^3 \cdot \sin(x)^2 } \, dx
$$

Options:
A. C + \frac{\sin(x)}{2\cdot\left(\left(\sin(x)\right)^2-1\right)} + \frac{3}{4}\cdot\ln\left(\left|\sin(x)-1\right|\right) - \frac{1}{\sin(x)} - \frac{3}{4}\cdot\ln\left(\left|1+\sin(x)\right|\right)
B. C + \frac{\sin(x)}{2\cdot\left(\left(\sin(x)\right)^2-1\right)} + \frac{3}{4}\cdot\ln\left(\left|\sin(x)-1\right|\right) - \frac{1}{2\sin(x)} + \frac{3}{4}\cdot\ln\left(\left|1+\sin(x)\right|\right)
C. C + \frac{\sin(x)}{2\cdot\left(\left(\sin(x)\right)^2-1\right)} - \frac{3}{4}\cdot\ln\left(\left|\sin(x)-1\right|\right) + \frac{1}{\sin(x)} + \frac{3}{4}\cdot\ln\left(\left|1+\sin(x)\right|\right)
D. C + \frac{\sin(x)}{2\cdot\left(\left(\sin(x)\right)^2-1\right)} + \frac{3}{4}\cdot\ln\left(\left|\sin(x)-1\right|\right) + \frac{1}{\sin(x)} - \frac{3}{4}\cdot\ln\left(\left|1+\sin(x)\right|\right)
E. C + \frac{\sin(x)}{2\cdot\left(\left(\sin(x)\right)^2-1\right)} + \frac{3}{4}\cdot\ln\left(\left|\sin(x)-1\right|\right) - \frac{1}{2\sin(x)} - \frac{3}{4}\cdot\ln\left(\left|1+\sin(x)\right|\right)
F. C + \frac{\sin(x)}{2\cdot\left(\left(\sin(x)\right)^2-1\right)} + \frac{3}{4}\cdot\ln\left(\left|\sin(x)-1\right|\right) + \frac{1}{2\sin(x)} - \frac{3}{4}\cdot\ln\left(\left|1+\sin(x)\right|\right)
G. C + \frac{\sin(x)}{2\cdot\left(\left(\sin(x)\right)^2-1\right)} - \frac{3}{4}\cdot\ln\left(\left|\sin(x)-1\right|\right) - \frac{1}{\sin(x)} + \frac{3}{4}\cdot\ln\left(\left|1+\sin(x)\right|\right)
Answer: D
Evaluate $\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$, where $\mathrm{Arccot}\,t$ for $t \geq 0$ denotes the number $\theta$ in the interval $0 < \theta \leq \pi/2$ with $\cot \theta = t$.

Options:
A. \frac{\pi}{4}
B. \frac{3\pi}{4}
C. \frac{2\pi}{3}
D. \frac{5\pi}{6}
E. \frac{\pi}{6}
F. \frac{\pi}{3}
G. \frac{\pi}{2}
H. \frac{3\pi}{8}
Answer: G
Find all values of $x$ that satisfy the following equation:
$$
\left|\left(x^4-4\right)-\left(x^2+2\right)\right|=\left|x^4-4\right|-\left|x^2+2\right|
$$

Options:
A. |x| >= sqrt(3)
B. |x| >= sqrt(2)
C. |x| >= 1
D. |x| <= sqrt(3)
E. |x| <= 2
F. |x| <= sqrt(2)
G. |x| <= 1
H. |x| >= 2
Answer: A
Compute the number of ordered pairs of positive integers $(m,n)$ with $m + n \leq 64$ such that there exists at least one complex number $z$ such that $|z| = 1$ and $z^m + z^n + \sqrt{2} = 0$ .

Options:
A. 194
B. 198
C. 195
D. 199
E. 196
F. 197
G. 200
H. 193
I. 192
Answer: G
Find the curvature for the vector function: $\vec{r}(t) = \left\langle \sqrt{2} \cdot e^t, \sqrt{2} \cdot e^{-t}, 2 \cdot t \right\rangle$.

Options:
A. 2 * e^(2 * t) / (2 + 2 * e^(4 * t) + 4 * e^(-2 * t))
B. 2 * e^(2 * t) / (2 + 2 * e^(4 * t) + 4 * e^(2 * t))
C. 2 * e^(2 * t) / (2 + 2 * e^(4 * t) - 4 * e^(2 * t))
D. 2 * e^(2 * t) / (2 + 2 * e^(4 * t) + 4 * e^(-t))
E. 2 * e^(2 * t) / (2 + 2 * e^(4 * t) + 4 * e^(t))
F. 2 * e^(2 * t) / (2 + 2 * e^(4 * t) + 4 * e^(3 * t))
G. 2 * e^(2 * t) / (2 + 2 * e^(4 * t) + 4 * e^(t * 3))
H. 2 * e^(2 * t) / (2 + 2 * e^(4 * t) + 4 * e^(t * 2))
I. 2 * e^(2 * t) / (2 + 2 * e^(4 * t) - 4 * e^(-2 * t))
Answer: B
In a cube of side length $4$ , an equilateral triangle is drawn with its vertices a subset of the cube's. Let the area of the cross-section between the cube and a plane parallel to the triangle be $A$ . Find the square of the maximum value of $A$ .

Options:
A. 426
B. 432
C. 431
D. 429
E. 433
F. 435
G. 430
H. 428
I. 436
J. 434
Answer: B
Consider isosceles right triangle $\triangle ABC$ with a right angle at $B$ . There exists a line $\ell$ such that the distances from $A$ , $B$ , and $C$ to $\ell$ are $81$ , $64$ , and $49$ , respectively, and $A$ , $B$ , and $C$ all lie on the same side of $\ell$ . Find the area of $\triangle ABC$ .

Options:
A. 258
B. 256
C. 259
D. 251
E. 255
F. 257
G. 254
H. 252
I. 253
J. 250
Answer: F
Let $d(n)$ denote the sum of the digits of a positive integer $n$ in base ten. Evaluate the remainder when
$$$\sum^{2006}_{k=10} d(k)$$$
is divided by $1000$ .

Options:
A. 988
B. 987
C. 989
D. 992
E. 994
F. 993
G. 990
H. 985
I. 986
J. 991
Answer: G
Eleven nonparallel lines lie on a plane, and their pairwise intersections meet at angles of integer degree. How many possible values are there for the smallest of these angles?

Options:
A. 18
B. 17
C. 16
D. 15
E. 13
F. 12
G. 14
H. 11
I. 10
J. 19
Answer: C
Find the number of ways to divide the set $\{1,2,3,\dots, 20\}$ into non-empty subsets such that the elements in each subset are consecutive and sum to a multiple of $5$ .

Options:
A. 134
B. 126
C. 144
D. 136
E. 120
F. 124
G. 132
H. 112
I. 130
J. 128
Answer: J
Find the number of permutations $(a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9)$ of $(1,2,3,4,5,6,7,8,9)$ that satisfy $|a_i-i|\le 2$ for all integers $1\le i\le 9$ .

Options:
A. 929
B. 930
C. 924
D. 932
E. 928
F. 920
G. 922
H. 936
I. 926
J. 934
Answer: D
Let $N=\overline{a_1a_2\ldots a_n}$ be the largest positive integer that satisfies the following conditions:
(i) $a_1, a_2, \ldots, a_n$ are nonzero digits.
(ii) For all integers $i$ in the range $2$ to $n$ inclusive, $\overline{a_ia_{i+1}\ldots a_n}$ divides $\overline{a_{i-1}a_i\ldots a_n}.$
Find $\Bigl\lfloor \frac{N}{10^{n-3}}\Bigr\rfloor$ .
Author: Jason Tang

Options:
A. 951
B. 954
C. 952
D. 950
E. 955
F. 958
G. 956
H. 959
I. 957
J. 953
Answer: G
Let $n$ be the smallest positive integer for which there exist positive real numbers $a$ and $b$ such that $(a+bi)^n=(a-bi)^n$ , where $i=\sqrt{-1}.$ . Compute $\frac{b^2}{a^2}$ .

Options:
A. 2.25
B. 1
C. 4
D. 5
E. 2.75
F. 2
G. 3.5
H. 6
I. 2.5
J. 3
Answer: J
In triangle $ABC$ , the bisector of angle $A$ meets side $\overline{BC}$ at $D$ and the circumcircle of $ABC$ at $M$ . If $O_1$ and $O_2$ are the circumcenters of triangles $ABC$ and $ADC$ , respectively, and $O_1O_2 = 4$ , $MO_2 = 9$ , and $CO_2 = 7$ , compute $AC^2$ .

Options:
A. 107
B. 106
C. 98
D. 101
E. 100
F. 104
G. 108
H. 102
I. 103
J. 105
Answer: C
Let $N$ be the number of functions $f: \{1,2,3,...,10\}\rightarrow\{1,2,3,...,10\}$ such that $f_{10}(x)\neq x$ for $x\in (1,2,3,...,10)$ , where $f_1(x)=f(x)$ and $f_{n+1}=f(f_n(x))$ for $n\geq1$ . Find the remainder when $N$ is divided by $1000$ .

Options:
A. 397
B. 400
C. 394
D. 395
E. 398
F. 399
G. 392
H. 393
I. 396
J. 391
Answer: B
Let $N$ be the number of $15$ -letter strings that satisfy the following conditions:
The string consists of $5$ A's, $5$ B's, and $5$ C's
Any three consecutive letters contain at least one A.
Find the remainder when $N$ is divided by $1000.$

Options:
A. 286
B. 288
C. 294
D. 292
E. 298
F. 282
G. 290
H. 280
I. 284
J. 296
Answer: D
A sequence $a_i$ is defined by $2a_{n-1}a_n+2=4a_{n-1}+(a_n-1)a_{n-1}^2$ . In addition, $a_{100}=2$ , and $0<a_0<1$ . If $N$ is the amount of possible $a_0$ , compute the last $3$ digits of $N$ .

Options:
A. 343
B. 349
C. 345
D. 348
E. 347
F. 344
G. 342
H. 340
I. 346
J. 341
Answer: F
Determine the number of positive integer divisors of $6^{22}$ that are one more than a multiple of $7$ .

Options:
A. 91
B. 94
C. 89
D. 86
E. 92
F. 84
G. 82
H. 88
I. 96
J. 90
Answer: E
Let $f(n)$ denote the number of divisors of a positive integer $n$ . Evaluate $f(f(2006^{6002}))$ .

Options:
A. 115
B. 117
C. 111
D. 113
E. 110
F. 114
G. 119
H. 116
I. 112
J. 118
Answer: I
Eight boxes are numbered $A$ to $H$ . The number of ways you can put $16$ identical balls into the boxes such that none of them is empty is expressed as $\binom {a}{b}$ , where $b \le \frac{a}{2}$ . What is the remainder when $\binom {a}{b}$ is divided by $a + b$ ?

Options:
A. 14
B. 17
C. 12
D. 9
E. 11
F. 15
G. 13
H. 10
I. 16
J. 18
Answer: E
Find the number of $5$ digit numbers using only the digits $1,2,3,4,5,6,7,8$ such that every pair of adjacent digits is no more than $1$ apart. For instance, $12345$ and $33234$ are acceptable numbers, while $13333$ and $56789$ are not.

Options:
A. 496
B. 501
C. 504
D. 505
E. 500
F. 497
G. 502
H. 499
I. 498
J. 503
Answer: G
Let $\triangle ABC$ be a triangle, and let $I_A$ , $I_B$ , and $I_C$ be the points where the angle bisectors of $A$ , $B$ , and $C$ , respectfully, intersect the sides opposite them. Given that $AI_B=5$ , $CI_B=4$ , and $CI_A=3$ , then the ratio $AI_C:BI_C$ can be written in the form $m/n$ where $m$ and $n$ are positive relatively prime integers. Find $m+n$ .

Options:
A. 19
B. 17
C. 16
D. 10
E. 18
F. 15
G. 12
H. 14
I. 11
J. 13
Answer: I
How many trailing zeroes are in the decimal representation of $n = 1 + \sum_{k=1}^{2014}k! \cdot (k^3 + 2k^2 + 3k + 1)?$

Options:
A. 500
B. 508
C. 509
D. 502
E. 505
F. 503
G. 501
H. 507
I. 504
J. 506
Answer: I
There are 2010 boxes labeled $B_1, B_2, \dots, B_{2010}$, and $2010n$ balls have been distributed among them, for some positive integer $n$. You may redistribute the balls by a sequence of moves, each of which consists of choosing an $i$ and moving \emph{exactly} $i$ balls from box $B_i$ into any one other box. For which least value of $n$ is it possible to reach the distribution with exactly $n$ balls in each box, regardless of the initial distribution of balls?

Options:
A. 1000
B. 1007
C. 1005
D. 1008
E. 1001
F. 1006
G. 1003
H. 1002
I. 1004
J. 1009
Answer: C
Assume a positive integer $A$ can be represented as the sum of $p$ consecutive positive integers: \[ A = x + (x+1) + (x+2) + \cdots + (x+p-1) \] for $p=2017$ and for no other $p>1$. Among all such positive integers $A$, determine the smallest positive integer $x$ that appears in any of these representations.

Options:
A. 17
B. 13
C. 15.25
D. 15.5
E. 19
F. 15.75
G. 18
H. 16
I. 14
J. 15
Answer: H
Find the smallest positive integer $n$ such that for every integer $m$ with $0 < m < 1993$, there exists an integer $k$ for which \[ \frac{m}{1993} < \frac{k}{n} < \frac{m+1}{1994}. \]

Options:
A. 3986
B. 3984
C. 3981
D. 3989
E. 3988
F. 3983
G. 3980
H. 3987
I. 3985
J. 3982
Answer: H
Consider the set $\{1,2,\dots,a\}$ and all its nonempty subsets, which we'll denote as $F_1, F_2, \dots, F_{2^a-1}$ in any arbitrary order. Now, construct a matrix $R$ of size $(2^a-1) \times (2^a-1)$ where each element $m_{ij}$ is defined as follows: \[ m_{ij} = \begin{cases} 1 & \text{if $F_i$ and $F_j$ have at least one element in common,} \\ 0 & \text{if $F_i$ and $F_j$ are disjoint.} \end{cases} \]Your task is to evaluate the determinant of $R$. In case multiple determinant values are possible, compute the sum of all distinct determinant values.

Options:
A. -1
B. -2
C. 3
D. 4
E. 1
F. 5
G. -3
H. 0
I. -4
J. 2
Answer: H
Given positive integers $y$, define the sequence $l(y)$ by the following rules: $l(1) = 1$, $l(2y) = l(y)$, and $l(2y+1) = (-1)^y l(y)$. Find the value of \[ \sum_{y=1}^{1089} l(y) l(y+2). \]

Options:
A. 1
B. -2
C. -1
D. -7
E. -6
F. 2
G. -3
H. 0
I. -4
J. -5
Answer: C
For each real number $x$, let \[ f(x) = \sum_{n\in S_x} \frac{1}{2^n}, \] where $S_x$ is the set of positive integers $n$ for which $\lfloor nx \rfloor$ is even. What is the largest real number $L$ such that $f(x) \geq L$ for all $x \in [0,1)$? (As usual, $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.)

Options:
A. \frac{5}{8}
B. \frac{2}{7}
C. \frac{5}{9}
D. \frac{3}{5}
E. \frac{6}{7}
F. \frac{3}{8}
G. \frac{3}{7}
H. \frac{5}{7}
I. \frac{4}{7}
J. \frac{1}{7}
Answer: I
Compute \[ \log_2 \left( \prod_{a=1}^{2015} \prod_{b=1}^{2015} (1+e^{2\pi i a b/2015}) \right) \] Here $i$ is the imaginary unit (that is, $i^2=-1$).

Options:
A. 13721
B. 13720
C. 13725
D. 13727
E. 13723
F. 13726
G. 13724
H. 13722
I. 13728
J. 13729
Answer: C
Inscribe a rectangle of base $b$ and height $h$ in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what value of $h$ do the rectangle and triangle have the same area?

Options:
A. \frac{1}{\sqrt{2}}
B. \frac{1}{\sqrt{3}}
C. \frac{1}{\sqrt{10}}
D. \frac{3}{4}
E. \frac{3}{5}
F. \frac{1}{4}
G. \frac{1}{2}
H. \frac{2}{5}
I. \frac{1}{\sqrt{5}}
J. \frac{1}{3}
Answer: H
Divide the rational expressions:
$$
\frac{ q^2-36 }{ q^2+12 \cdot q+36 } \div \frac{ q^2-4 \cdot q-12 }{ q^2+4 \cdot q-12 }
$$

Options:
A. q-3/q+3
B. q-3/q-2
C. q-6/q+6
D. q-2/q+6
E. q-6/q-6
F. q-2/q-6
G. q-6/q+2
H. q-6/q-3
I. q-2/q+2
J. q-2/q-3
Answer: I
Determine a definite integral that represents the region common to $r = 3 \cdot \cos\left(\theta\right)$ and $r = 3 \cdot \sin\left(\theta\right)$.

Options:
A. 9\cdot\int_0^{\frac{\pi}{4}}\sin\left(2\theta\right)\cos\left(\theta\right)d\theta
B. 9\cdot\int_0^{\frac{\pi}{4}}\cos\left(2\theta\right)^2d\theta
C. 9\cdot\int_0^{\frac{\pi}{4}}\cos\left(\theta\right)^2d\theta
D. 9\cdot\int_0^{\frac{\pi}{4}}\sin\left(\theta\right)^2d\theta
E. 9\cdot\int_0^{\frac{\pi}{4}}\sin\left(2\theta\right)d\theta
F. 9\cdot\int_0^{\frac{\pi}{4}}\cos\left(\theta\right)\sin\left(\theta\right)d\theta
G. 9\cdot\int_0^{\frac{\pi}{4}}\cos\left(2\theta\right)d\theta
H. 9\cdot\int_0^{\frac{\pi}{4}}\sin\left(\theta\right)\cos\left(\theta\right)d\theta
I. 9\cdot\int_0^{\frac{\pi}{4}}\sin\left(\theta\right)\cos\left(2\theta\right)d\theta
J. 9\cdot\int_0^{\frac{\pi}{4}}\sin\left(2\theta\right)^2d\theta
Answer: D
Find vector $\vec{c} \times \left(\vec{a}+3 \cdot \vec{b}\right)$ where
$$
\vec{a}=\begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 5 & 0 & 9 \\ 0 & 1 & 0 \end{vmatrix}, \quad
\vec{b}=\begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 0 & -1 & 1 \\ 7 & 1 & -1 \end{vmatrix}, \quad \text{and} \quad
\vec{c}=\vec{i}-\vec{k}
$$

Options:
A. 21\cdot\vec{i}-17\cdot\vec{j}+19\cdot\vec{k}
B. 21\cdot\vec{i}-17\cdot\vec{j}-21\cdot\vec{k}
C. 21\cdot\vec{i}-17\cdot\vec{j}+13\cdot\vec{k}
D. 21\cdot\vec{i}-17\cdot\vec{j}+22\cdot\vec{k}
E. 21\cdot\vec{i}-17\cdot\vec{j}+10\cdot\vec{k}
F. 21\cdot\vec{i}-17\cdot\vec{j}+21\cdot\vec{k}
G. 21\cdot\vec{i}-17\cdot\vec{j}+14\cdot\vec{k}
H. 21\cdot\vec{i}-17\cdot\vec{j}+7\cdot\vec{k}
I. 21\cdot\vec{i}+17\cdot\vec{j}-21\cdot\vec{k}
J. 21\cdot\vec{i}+17\cdot\vec{j}+21\cdot\vec{k}
Answer: F
Simplify the expression $\tan(x)^2 + \sin(x) \cdot \csc(x)$ by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

Options:
A. 1 + sec(x)^2
B. 1 + sec(x)
C. 1 + tan(x)^2 - sin(x)
D. 1 + tan(x)^2 - cos(x)^2
E. 1 + tan(x)^2 - sin(x)^2
F. sec(x)^2
G. 1 + tan(x)^2 - sec(x)^2
H. 1 + tan(x)^2 - 1
I. 1 + tan(x)^2
J. 1 + tan(x)
Answer: F
Compute the limit:
$$
\lim_{x \to 5} \left( \frac{ 3 \cdot x }{ x-5 }-\frac{ 3 }{ \ln\left(\frac{ x }{ 5 }\right) } \right)
$$

Options:
A. 3/12
B. 3/8
C. 3/7
D. 3/3
E. 3/5
F. 3/2
G. 3/6
H. 3/4
I. 3/15
J. 3/10
Answer: F
Evaluate $\int\int\int_{E}{(x \cdot z+1) d V}$, where $E$ is the region defined by:
$$
E = \left\{(x,y,z) | 0 \le x \le \sqrt{y}, 0 \le y \le 2, 0 \le z \le 1-x^2-y^2\right\}
$$

Options:
A. 73/30 - 52*sqrt(2)/42
B. 73/30 - 52*sqrt(2)/41
C. 73/30 - 52*sqrt(2)/36
D. 73/30 - 52*sqrt(2)/40
E. 73/30 - 52*sqrt(2)/38
F. 73/30 - 52*sqrt(2)/45
G. 73/30 - 52*sqrt(2)/37
H. 73/30 - 52*sqrt(2)/50
I. 73/30 - 52*sqrt(2)/39
J. 73/30 - 52*sqrt(2)/35
Answer: J
Calculate the integral:
$$
I = \int 4 \cdot \cos\left(3 \cdot \ln(2 \cdot x)\right) \, dx
$$

Options:
A. 1/10 * (C + 4 * x * cos(3 * ln(2 * x)) - 6 * x * sin(3 * ln(2 * x)))
B. 1/10 * (C + 4 * x * cos(3 * ln(2 * x)) + 12 * x * sin(3 * ln(2 * x)))
C. 1/10 * (C + 4 * x * cos(3 * ln(2 * x)) + 14 * x * sin(3 * ln(2 * x)))
D. 1/10 * (C + 4 * x * cos(3 * ln(2 * x)) + 8 * x * sin(3 * ln(2 * x)))
E. 1/10 * (C + 4 * x * cos(3 * ln(2 * x)) + 6 * x * sin(3 * ln(2 * x)))
F. 1/10 * (C + 4 * x * cos(3 * ln(2 * x)) + 10 * x * sin(3 * ln(2 * x)))
G. 1/10 * (C + 4 * x * cos(3 * ln(2 * x)) - 12 * x * sin(3 * ln(2 * x)))
H. 1/10 * (C + 4 * x * cos(3 * ln(2 * x)) + 9 * x * sin(3 * ln(2 * x)))
I. 1/10 * (C + 4 * x * cos(3 * ln(2 * x)) + 13 * x * sin(3 * ln(2 * x)))
J. 1/10 * (C + 4 * x * cos(3 * ln(2 * x)) + 11 * x * sin(3 * ln(2 * x)))
Answer: B
Find a rectangular equation which is equivalent to the following parametric equations:
1. $x^2 = t^3 - 3 \cdot t^2 + 3 \cdot t - 1$
2. $y^2 = t^3 + 6 \cdot t^2 + 12 \cdot t + 8$

Options:
A. √[3]{x^2} - √[3]{y^2} = -4
B. √[3]{x^2} - √[3]{y^2} = 4
C. √[3]{x^2} - √[3]{y^2} = 1
D. √[3]{x^2} + √[3]{y^2} = 5
E. √[3]{x^2} - √[3]{y^2} = -1
F. √[3]{x^2} - √[3]{y^2} = 3
G. √[3]{x^2} - √[3]{y^2} = -3
H. √[3]{x^2} + √[3]{y^2} = -5
I. √[3]{x^2} - √[3]{y^2} = 2
J. √[3]{x^2} - √[3]{y^2} = -2
Answer: G
The electrical resistance $R$ produced by wiring resistors $R_{1}$ and $R_{2}$ in parallel can be calculated from the formula $\frac{ 1 }{ R }=\frac{ 1 }{ R_{1} }+\frac{ 1 }{ R_{2} }$. If $R_{1}$ and $R_{2}$ are measured to be $7$ ohm and $6$ ohm respectively, and if these measurements are accurate to within $0.05$ ohm, estimate the maximum possible error in computing $R$.

Options:
A. 0.02514794
B. 0.02514798
C. 0.02514791
D. 0.02514792
E. 0.02514797
F. 0.02514796
G. 0.02514799
H. 0.02514793
I. 0.02514795
J. 0.02514790
Answer: H
Consider points $P(3,7,-2)$ and $Q(1,1,-3)$. Determine the angle between vectors $\vec{OP}$ and $\vec{OQ}$. Express the answer in radians, rounded to two decimal places.

Options:
A. 1.05
B. 0.87
C. 0.89
D. 1.23
E. 1.12
F. 1.29
G. 1.01
H. 0.95
I. 1.19
J. 0.91
Answer: J
Compute the integral:
$$
\int \frac{ 1 }{ \left(\cos(2 \cdot x)\right)^3 } \, dx
$$

Options:
A. C + \frac{\sin(2\cdot x)}{4\cdot\left(\cos(2\cdot x)\right)^2} + \frac{1}{4}\cdot\ln\left(\left|\tan\left(\frac{1}{2}\cdot\left(2\cdot x + \pi\right)\right)\right|\right)
B. C + \frac{\sin(2\cdot x)}{4\cdot\left(\cos(2\cdot x)\right)^2} + \frac{1}{4}\cdot\ln\left(\left|\tan\left(\frac{1}{2}\cdot\left(2\cdot x + \frac{\pi}{6}\right)\right)\right|\right)
C. C + \frac{\sin(2\cdot x)}{4\cdot\left(\cos(2\cdot x)\right)^2} + \frac{1}{4}\cdot\ln\left(\left|\tan\left(\frac{1}{2}\cdot\left(2\cdot x + \frac{3\pi}{2}\right)\right)\right|\right)
D. C + \frac{\sin(2\cdot x)}{4\cdot\left(\cos(2\cdot x)\right)^2} + \frac{1}{4}\cdot\ln\left(\left|\tan\left(\frac{1}{2}\cdot\left(2\cdot x - \frac{\pi}{4}\right)\right)\right|\right)
E. C + \frac{\sin(2\cdot x)}{4\cdot\left(\cos(2\cdot x)\right)^2} + \frac{1}{4}\cdot\ln\left(\left|\tan\left(\frac{1}{2}\cdot\left(2\cdot x - \frac{\pi}{2}\right)\right)\right|\right)
F. C + \frac{\sin(2\cdot x)}{4\cdot\left(\cos(2\cdot x)\right)^2} + \frac{1}{4}\cdot\ln\left(\left|\tan\left(\frac{1}{2}\cdot\left(2\cdot x + \frac{\pi}{4}\right)\right)\right|\right)
G. C + \frac{\sin(2\cdot x)}{4\cdot\left(\cos(2\cdot x)\right)^2} - \frac{1}{4}\cdot\ln\left(\left|\tan\left(\frac{1}{2}\cdot\left(2\cdot x + \frac{\pi}{2}\right)\right)\right|\right)
H. C + \frac{\sin(2\cdot x)}{4\cdot\left(\cos(2\cdot x)\right)^2} - \frac{1}{4}\cdot\ln\left(\left|\tan\left(\frac{1}{2}\cdot\left(2\cdot x - \frac{\pi}{2}\right)\right)\right|\right)
I. C + \frac{\sin(2\cdot x)}{4\cdot\left(\cos(2\cdot x)\right)^2} + \frac{1}{4}\cdot\ln\left(\left|\tan\left(\frac{1}{2}\cdot\left(2\cdot x + \frac{\pi}{2}\right)\right)\right|\right)
J. C + \frac{\sin(2\cdot x)}{4\cdot\left(\cos(2\cdot x)\right)^2} + \frac{1}{4}\cdot\ln\left(\left|\tan\left(\frac{1}{2}\cdot\left(2\cdot x + \frac{\pi}{3}\right)\right)\right|\right)
Answer: I
Suppose that we have a series: $1-\frac{ 1 }{ 2 }+\frac{ 1 }{ 3 }-\cdots+(-1)^{n+1} \cdot \frac{ 1 }{ n }+\cdots$ and its sum is equal to $\ln(2)$. Find the sum of the series: $1-\frac{ 1 }{ 2 }-\frac{ 1 }{ 4 }+\frac{ 1 }{ 3 }-\frac{ 1 }{ 6 }-\frac{ 1 }{ 8 }+\frac{ 1 }{ 5 }-\frac{ 1 }{ 10 }-\frac{ 1 }{ 12 }+\cdots$.

Options:
A. S=\frac{1}{3}\cdot\ln(2)
B. S=\frac{1}{7}\cdot\ln(2)
C. S=\frac{1}{6}\cdot\ln(2)
D. S=\frac{3}{8}\cdot\ln(2)
E. S=\frac{1}{4}\cdot\ln(2)
F. S=\frac{2}{3}\cdot\ln(2)
G. S=\frac{1}{5}\cdot\ln(2)
H. S=\frac{1}{2}\cdot\ln(2)
I. S=\frac{3}{4}\cdot\ln(2)
J. S=\frac{1}{8}\cdot\ln(2)
Answer: H
For the function $y = \frac{ 2 \cdot x + 3 }{ 4 \cdot x + 5 }$ find the derivative $y^{(n)}$.

Options:
A. y^{(n)}=\frac{1}{2}\cdot(-1)^n\cdot\left(n!\right)\cdot4^{n+4}\cdot(4\cdot x+5)^{-(n+1)}
B. y^{(n)}=\frac{1}{2}\cdot(-1)^n\cdot\left(n!\right)\cdot4^{n+2}\cdot(4\cdot x+5)^{-(n+1)}
C. y^{(n)}=\frac{1}{2}\cdot(-1)^n\cdot\left(n!\right)\cdot4^{n+6}\cdot(4\cdot x+5)^{-(n+1)}
D. y^{(n)}=\frac{1}{2}\cdot(-1)^n\cdot\left(n!\right)\cdot4^{n-2}\cdot(4\cdot x+5)^{-(n+1)}
E. y^{(n)}=\frac{1}{2}\cdot(-1)^n\cdot\left(n!\right)\cdot4^{n+5}\cdot(4\cdot x+5)^{-(n+1)}
F. y^{(n)}=\frac{1}{2}\cdot(-1)^n\cdot\left(n!\right)\cdot4^{n-1}\cdot(4\cdot x+5)^{-(n+1)}
G. y^{(n)}=\frac{1}{2}\cdot(-1)^n\cdot\left(n!\right)\cdot4^{n+3}\cdot(4\cdot x+5)^{-(n+1)}
H. y^{(n)}=\frac{1}{2}\cdot(-1)^n\cdot\left(n!\right)\cdot4^{n-3}\cdot(4\cdot x+5)^{-(n+1)}
I. y^{(n)}=\frac{1}{2}\cdot(-1)^n\cdot\left(n!\right)\cdot4^{n+1}\cdot(4\cdot x+5)^{-(n+1)}
J. y^{(n)}=\frac{1}{2}\cdot(-1)^n\cdot\left(n!\right)\cdot4^n\cdot(4\cdot x+5)^{-(n+1)}
Answer: J
Compute
$$$\left \lfloor 100 \sin(0^{\circ}) \right \rfloor + \left \lceil 100 \sin(1^{\circ}) \right \rceil + \dots + \left \lceil 100 \sin(359^{\circ}) \right \rceil + \left \lfloor 100 \sin(360^{\circ}) \right \rfloor.$$$

Options:
A. 30
B. 33
C. 29
D. 37
E. 35
F. 34
G. 31
H. 32
I. 38
J. 36
Answer: G
Find the zeros of $\tan(x) + \tan\left(\frac{ \pi }{ 4 } + x\right) = -2$.

Options:
A. x_{1}=-\frac{ \pi }{ 4 }+\pi \cdot n, x_{2}=\frac{ \pi }{ 4 }+n \cdot \pi
B. x_{1}=-\frac{ \pi }{ 6 }+\pi \cdot n, x_{2}=\frac{ \pi }{ 6 }+n \cdot \pi
C. x_{1}=-\frac{ \pi }{ 3 }+\pi \cdot n, x_{2}=\frac{ \pi }{ 3 }+n \cdot \pi
D. x_{1}=-\frac{ \pi }{ 2 }+\pi \cdot n, x_{2}=\frac{ \pi }{ 2 }+n \cdot \pi
Answer: C
In triangle $ABC,$ $BC=9.$ Points $P$ and $Q$ are located on $BC$ such that $BP=PQ=2,$ $QC=5.$ The circumcircle of $APQ$ cuts $AB,AC$ at $D,E$ respectively. If $BD=CE,$ then the ratio $\frac{AB}{AC}$ can be expressed in the form $\frac{m}{n},$ where $m,n$ are relatively prime positive integers. Find $m+n.$

Options:
A. 49
B. 45
C. 43
D. 47
E. 41
Answer: C
Solve the integral:
$$
\int \frac{ 1 }{ \sin(8 \cdot x)^5 } \, dx
$$

Options:
A. C+\frac{1}{128}\cdot\left(2\cdot\left(\tan(4\cdot x)\right)^2+6\cdot\ln\left(\left|\tan(4\cdot x)\right|\right)+\frac{1}{4}\cdot\left(\tan(4\cdot x)\right)^4-\frac{2}{\left(\tan(4\cdot x)\right)^2}+\frac{1}{4\cdot\left(\tan(4\cdot x)\right)^4}\right)
B. C+\frac{1}{128}\cdot\left(2\cdot\left(\tan(4\cdot x)\right)^2+6\cdot\ln\left(\left|\tan(4\cdot x)\right|\right)+\frac{1}{4}\cdot\left(\tan(4\cdot x)\right)^4-\frac{2}{\left(\tan(4\cdot x)\right)^2}+\frac{1}{2\cdot\left(\tan(4\cdot x)\right)^4}\right)
C. C+\frac{1}{128}\cdot\left(2\cdot\left(\tan(4\cdot x)\right)^2+6\cdot\ln\left(\left|\tan(4\cdot x)\right|\right)+\frac{1}{4}\cdot\left(\tan(4\cdot x)\right)^4-\frac{2}{\left(\tan(4\cdot x)\right)^2}-\frac{1}{2\cdot\left(\tan(4\cdot x)\right)^4}\right)
D. C+\frac{1}{128}\cdot\left(2\cdot\left(\tan(4\cdot x)\right)^2+6\cdot\ln\left(\left|\tan(4\cdot x)\right|\right)+\frac{1}{4}\cdot\left(\tan(4\cdot x)\right)^4-\frac{2}{\left(\tan(4\cdot x)\right)^2}+\frac{1}{8\cdot\left(\tan(4\cdot x)\right)^4}\right)
E. C+\frac{1}{128}\cdot\left(2\cdot\left(\tan(4\cdot x)\right)^2+6\cdot\ln\left(\left|\tan(4\cdot x)\right|\right)+\frac{1}{4}\cdot\left(\tan(4\cdot x)\right)^4-\frac{2}{\left(\tan(4\cdot x)\right)^2}-\frac{1}{4\cdot\left(\tan(4\cdot x)\right)^4}\right)
Answer: E
Find the Taylor series of $f'(x)$ about $a=0$ if $f(x) = \frac{ x - \ln(1 + x) }{ x^2 }$. Use sigma notation in the final answer.

Options:
A. ∑_{k=0}^∞((-1)^{k+1}·\frac{(k+1)}{(k+2)}·x^k)
B. ∑_{k=0}^∞((-1)^{k+1}·\frac{(k+1)}{(k+5)}·x^k)
C. ∑_{k=0}^∞((-1)^{k+1}·\frac{(k+3)}{(k+3)}·x^k)
D. ∑_{k=0}^∞((-1)^{k+1}·\frac{(k+2)}{(k+2)}·x^k)
E. ∑_{k=0}^∞((-1)^{k+1}·\frac{(k+2)}{(k+5)}·x^k)
F. ∑_{k=0}^∞((-1)^{k+1}·\frac{(k+1)}{(k+4)}·x^k)
G. ∑_{k=0}^∞((-1)^{k+1}·\frac{(k+2)}{(k+3)}·x^k)
H. ∑_{k=0}^∞((-1)^{k+1}·\frac{(k+3)}{(k+4)}·x^k)
I. ∑_{k=0}^∞((-1)^{k+1}·\frac{(k+2)}{(k+4)}·x^k)
J. ∑_{k=0}^∞((-1)^{k+1}·\frac{(k+1)}{(k+3)}·x^k)
Answer: J
Consider a sequence of independent random variables $M_1, M_2, \dots$ defined for $0 \leq p \leq 1/2$, where each $M_i$ follows the distribution: \[ M_i = \begin{cases} 1 & \text{with probability $p$,} \\ -1 & \text{with probability $p$,} \\ 0 & \text{with probability $1-2p$.} \end{cases} \] Let $n$ be a positive integer and $b, k_1, \dots, k_n$ be integers. Define $P(b, k_1, \dots, k_n)$ as the probability of the event $k_1 M_1 + \cdots + k_n M_n = b$. What is the largest value of $p$ that ensures \[ P(0, k_1, \dots, k_n) \geq P(b, k_1, \dots, k_n) \] holds true for any choice of positive integer $n$ and integers $b, k_1, \dots, k_n$?

Options:
A. \frac{1}{4}
B. \frac{1}{3}
C. \frac{1}{7}
D. \frac{1}{5}
E. \frac{1}{8}
F. \frac{1}{2}
G. \frac{1}{6}
Answer: A
Consider the following function $g(x)$ defined as
$$(x^{2^{2008}-1}-1)g(x) = (x+1)(x^2+1)(x^4+1)\cdots (x^{2^{2007}}+1) - 1$$
Find $g(2)$ .

Options:
A. 2.5
B. 1.75
C. 1.5
D. 1
E. 1.25
F. 3
G. 4
H. 2
Answer: H
Consider a function $e(s)$ defined for all positive real numbers $s$ as follows: \[ e(s) = \lim_{z \to 0} \left((s+1)^{z+1} - s^{z+1}\right)^{{1}/{z}}. \] Determine the value of $\lim_{s \to \infty} \frac{e(s)}{s}$.

Options:
A. e
B. e^{1/2}
C. e^2
D. e^{3/2}
E. e^{3/4}
F. e^{1/4}
G. e^{2/3}
H. e^{1/3}
Answer: A
Compute the sum of the three smallest positive integers $n$ such that $n^4+4$ is a product of $2$ distinct primes.

Options:
A. 26
B. 23
C. 21
D. 24
E. 22
F. 27
G. 29
H. 25
I. 28
Answer: B
Find the minimum positive real value $s$ for which there exist two differentiable functions $y, d: \mathbb{R} \to \mathbb{R}$ that satisfy all of the following conditions:
\begin{enumerate}
\item[(a)] The value of $y$ at zero is positive: $y(0) > 0$
\item[(b)] The value of $d$ at zero is zero: $d(0) = 0$
\item[(c)] For every real number $x$, the absolute value of $y$'s derivative at $x$ is at most the absolute value of $d$ at $x$: $|y'(x)| \leq |d(x)|$
\item[(d)] For every real number $x$, the absolute value of $d$'s derivative at $x$ is at most the absolute value of $y$ at $x$: $|d'(x)| \leq |y(x)|$
\item[(e)] The value of $y$ at $s$ is zero: $y(s) = 0$
\end{enumerate}

Options:
A. \frac{\pi}{3}
B. \frac{\pi}{6}
C. \frac{2\pi}{3}
D. \frac{3\pi}{8}
E. \frac{\pi}{4}
F. \frac{5\pi}{8}
G. \frac{3\pi}{4}
H. \frac{\pi}{2}
I. \frac{5\pi}{6}
Answer: H
Which combination correctly identifies the social dance that originated in the Dominican Republic in the 1960s? Consider the following dances:
I. Bachata
II. Salsa
III. Tango
IV. Merengue
Select the option that correctly describes the dance.

Options:
A. III
B. I and III
C. III and IV
D. II and III
E. IV
F. I
G. I and IV
H. I and II
I. II and IV
J. II
Answer: F
In the following sentence 'Les ailes du moulin pour tourner, doivent recevoir obliquement l'impression du vent', which stylistic figure(s) can be identified? Consider the following stylistic figures:
I. Hypallage
II. Catachresis
III. Zeugma
IV. Emphasis
Select the correct combination of stylistic figures present in the sentence.

Options:
A. IV
B. II and III
C. II
D. I
E. I and II
F. I and III
G. III
H. I and IV
I. II and IV
J. III and IV
Answer: D
Among the following methods for predicting urban population size, which one can be independently applied and used as the primary prediction result? Consider the following methods:
I. Comprehensive Balancing Method
II. Environmental Capacity Method
III. Regional Population Allocation Method
IV. Analogy Method
Select the method that can be independently applied and used as the primary prediction result.

Options:
A. III and IV
B. II and IV
C. I and III
D. I and IV
E. I
F. IV
G. II and III
H. II
I. I and II
J. III
Answer: E
An initial transportation solution appears in the table. C D Factory Capacity A 10 0 10 B 15 25 40 Warehouse Demand 25 25 50 Can ths solution be improved if it costs $5 per unit to ship from A to C; $7 per unit to ship from A to D; $8 to ship from B to C; and $9 to ship from B to D?

Options:
A. Yes, this solution can be improved by $100.
B. Yes, this solution can be improved by $40.
C. Yes, this solution can be improved by $300.
D. Yes, this solution can be improved by $50.
E. none of the above
F. No, this solution is optimal.
G. No, we can not know.
H. Yes, this solution can be improved by $60.
I. Yes, this solution can be improved by $200.
J. Yes, the initial solution can be improved by $10.
Answer: F
A company intends to invest 5 million yuan in its three subsidiaries A, B and C (to distribute investment in multiples of one million yuan), and the income of each subsidiary after obtaining part of the investment is shown in the following table (in millions of yuan). The total proceeds of the Company's investments are at most ( ) million dollars.\0 1 2 3 4 5\A 0 1.2 1.8 2.5 3 3.5\B 0 0.8 1.5 3 4 4.5\C 0 1 1.2 3.5 4.2 4.8

Options:
A. 6
B. 5.2
C. 5.5
D. 5
E. 5.6
F. 5.3
G. 5.4
H. 4.8
I. 5.8
Answer: C
Hongdou Garment Factory has launched a new fashion, and based on experience and market research, it is predicted that it will be a fashion in the next 6 months
The demand is: January - 3000 pieces; February-3,600; March - 4,000 pieces; April-4,600;
May-4,800; June - 5,000 pieces. The production of each piece needs to be skilled for 4 hours, and the raw materials consumed are 150 yuan.
The selling price is 240 yuan / piece. At the beginning of January, the factory had 80 skilled workers, each of whom worked 160h per month. In order to meet the needs of production,
The factory can recruit new workers for training, but the training of a new worker needs to occupy 50 hours of skilled workers for guiding the operation, the training period is one month, and you can take up the post after the end. The monthly salary of skilled L people is 2,000 yuan, and the new worker will be given a living allowance during the training period
800 yuan, after the regularization of wages and production efficiency to ask skilled workers. and skilled workers (including new workers one month after the regularization) each
At the beginning of the month, 2% left their jobs for various reasons. It is known that the factory has added 400 pieces of this fashion as inventory at the beginning of the year, requested
At the end of June, 1,000 pieces were in storage. If the fashion produced every month is not delivered in the same month, the inventory fee is 10 yuan per piece per month
Try to design a labor arrangement for the factory to meet the inventory requirements of each month and the end of June, and make the total income from January to June the largest labor arrangement program, the maximum profit of the program is:

Options:
A. 836548
B. 845478
C. 794568
D. 869742
E. 845697
F. 902136
G. 784564
H. 864178
I. 875122
J. 874569
Answer: I
Hongdou Garment Factory uses three types of special equipment to produce shirts, short-sleeved shirts and casual wear. It is known that the amount of labor, material consumption, sales price and variable cost of each of the above three products are shown in the table.
Product Name Single-piece Labor Single-piece Material Sales Price Variable Expenses
Shirt 3 4 120 60
Short-sleeved shirt 2 3 80 40
Casual wear 6 6 180 80
It is known that the factory has a weekly usable labor capacity of 150 units, a usable material capacity of 160 units, and a weekly fixed cost of 2,000 yuan, 1,500 yuan and 1,000 yuan for the production of three special equipment for shirts, short-sleeved shirts and casual wear, respectively. It was required to design a one-week production plan for the plant to maximize its profits. The maximum profit is:

Options:
A. 1400
B. 2700
C. 1000
D. 1900
E. 4500
F. 2400
G. 1800
H. 3200
I. 1500
J. 2000
Answer: I
A museum has 4 exhibition halls of the same size. The number of visitors to the museum is subject to the Puazon distribution, with an average of 96 people/h. Visitors were roughly evenly dispersed in each exhibition hall, and the stay time in each exhibition hall obeyed a negative exponential distribution of 1/u=15min, and they left after visiting 4 exhibition halls. Q: The museum's capacity per exhibition hall (the number of visitors it can hold) is designed to be a minimum so that the probability of overcrowding at any one time is less than 5%?

Options:
A. 11
B. 8
C. 10
D. 4
E. 6
F. 9
G. 7
H. 13
I. 12
J. 5
Answer: C
The statistical results of the number of poplar individuals of different ages in a natural forest showed that the number of seedlings, mature individuals and senescent individuals increased sequentially, indicating that the age structure of the poplar population was as follows

Options:
A. Mature type
B. Growth
C. Recession
D. I type
E. Late type
F. Hybrid
G. Seedling type
H. II type
I. Stable
J. III type
Answer: C
The main factors influencing the distribution of plants are:

Options:
A. Light and climate
B. Light and moisture
C. Soil and moisture
D. Light and soil
E. Soil and Climate
F. Soil and temperature
G. Temperature and climate
H. Moisture and Climate
I. Light and temperature
J. Temperature and moisture
Answer: J
The particle size range of soil physical clay is:

Options:
A. >0.04mm
B. >0.07mm
C. >0.01cm
D. >0.05mm
E. <0.01mm
F. >0.02mm
G. >0.03mm
H. >0.01mm
I. <0.01cm
J. >0.06mm
Answer: E
It is () that imposes individual international criminal responsibility on war criminals.

Options:
A. Charter of the United Nations
B. Washington Naval Treaty
C. Statute of the International Criminal Court
D. St. Petersburg Declaration
E. Charter of the International Military Tribunal at Nuremberg
F. Hague Treaty IV
G. Treaty of the Battle of London
H. Hague Regulations on the Laws and Customs of Land Operations
I. Geneva Conventions
J. International human rights conventions
Answer: E
Which of the following options is true about Law and Freedom?

Options:
A. Freedom is free from any law, because freedom is inherently inviolable
B. Freedom can completely override order before the law
C. The sole purpose of the law is to preserve absolute liberty and brook no restrictions
D. On the scales of law, order must give complete way to freedom
E. Freedom is sacred, but there is also a limit, and this limit should be governed by the law
The law is stipulated
F. Freedom is sacred, but there is also a limit, and this limit should be determined by law
G. From an empirical point of view, all laws are free laws
H. Any law that restricts the freedom of the individual is necessarily unjust
I. The function of legal norms is only to eliminate all unfreedoms without the need to guarantee order
J. Freedom is supreme and sacred, but not by laws that restrict it
Real law
Answer: F
In 1999, there was an attempted military coup d'état in country A, and the coup leader Langman fled to country B. The court of country A sentenced Longman in absentia to 10 years in prison. There is no special agreement between the two countries. Which of the following options is correct under the relevant rules of international law?

Options:
A. State B may grant asylum to Longman
B. State B is obliged to grant Longman asylum
C. After the judgment of the court of country A takes effect, country B is obliged to arrest and hand over Longman to country A
D. State A may unilaterally declare its laws valid in the territory of State B
E. State A may directly send agents into State B to apprehend suspects without their consent
F. State B must determine whether or not a suspect is extradited in accordance with the law of State A, without regard to international law or domestic law
G. State B does not need any legal basis to refuse a request for extradition
H. State A has enforcement powers over the citizens and legal persons of State B on a global scale
I. All fugitive suspects have an obligation to provide asylum to all fugitives
J. After the judgment of the court of country A takes effect, country A may send troops into country B to arrest Langman and enforce the judgment
Answer: A
Which of the following options is considered a medical accident? Consider the following scenarios:
I. Amniotic fluid embolism
II. Drug allergy
III. Organ adhesion at surgical site
IV. Hospital-acquired infection
Select the correct combination.

Options:
A. I and II
B. II and IV
C. I and III
D. I and IV
E. IV
F. II
G. II and III
H. III
I. III and IV
J. I
Answer: J
A 38-year-old woman comes to the physician for the evaluation of progressive weakness and numbness for 3 months. The symptoms started in her lower legs and gradually moved to her upper legs and arms. During the last 5 days, she has also had bilateral facial weakness with headaches. She has hepatitis B and Graves' disease. She is sexually active with one male partner and they use condoms inconsistently. Her current medications include methimazole and a multivitamin. Vital signs are within normal limits. She is alert and fully oriented. She has bilateral upper and lower facial paralysis. There is generalized weakness of the muscles. Sensation to light touch is decreased throughout and is absent in her fingertips and toes. Deep tendon reflexes are 1+ bilaterally. Further evaluation of this patient is most likely to show which of the following findings?

Options:
A. Elevated TSH and decreased FT4 levels
B. Rapid plasma reagin titer of 1:128
C. Positive serum botulinum neurotoxin
D. Elevated HbA1c level
E. Negative GM1 ganglioside autoantibodies
F. Positive Lyme ELISA test
G. Positive Campylobacter stool culture
H. Positive GM1 ganglioside autoantibodies
I. Positive anti-acetylcholine receptor antibodies
J. Low vitamin B12 level
Answer: H