MikaStars39/nano-eval huggingface dataset:
If $y=100$ at $t=4$ and $y=10$ at $t=8$, when does $y=1$? Use $y=y_{0} \cdot e^{k \cdot t}$, where $y_{0}$ is the beginning amount, $y$ is the ending amount, $k$ is the growth or decay rate, and $t$ is time.

Options:
A. 16
B. 12.2
C. 14
D. 10
E. 12
F. 13.5
G. 15
H. 13
I. 12.5
J. 11
Answer: E
Compute the integral:
$$
2 \cdot \int \left(\cos\left(\frac{ x }{ 4 }\right)\right)^6 \, dx
$$

Options:
A. C + 2⋅sin(x/2) + 3/8⋅sin(x) + 5/8⋅x - 1/6⋅(sin(x/2))^3
B. C + 2⋅sin(x/2) + 3/8⋅sin(x) + 5/8⋅x - 1/6⋅(sin(x/2))^6
C. C + 2⋅sin(x/2) + 3/8⋅sin(x) + 5/8⋅x - 1/6⋅(sin(x/2))^2
D. C + 2⋅sin(x/2) + 3/8⋅sin(x) + 5/8⋅x - 1/6⋅(sin(x/2))^9
E. C + 2⋅sin(x/2) + 3/8⋅sin(x) + 5/8⋅x - 1/6⋅(sin(x/2))^7
F. C + 2⋅sin(x/2) + 3/8⋅sin(x) + 5/8⋅x - 1/6⋅(sin(x/2))^4
G. C + 2⋅sin(x/2) + 3/8⋅sin(x) + 5/8⋅x - 1/6⋅(sin(x/2))^1
H. C + 2⋅sin(x/2) + 3/8⋅sin(x) + 5/8⋅x - 1/6⋅(sin(x/2))^5
I. C + 2⋅sin(x/2) + 3/8⋅sin(x) + 5/8⋅x - 1/6⋅(sin(x/2))^8
J. C + 2⋅sin(x/2) + 3/8⋅sin(x) + 5/8⋅x - 1/6⋅(sin(x/4))^3
Answer: A
Compute the limit:
$$
\lim_{x \to 3} \left( \frac{ x }{ x-3 } - \frac{ 1 }{ \ln\left( \frac{ x }{ 3 } \right) } \right)
$$

Options:
A. 0.625
B. 0.5208
C. 0.6
D. 0.5833
E. 0.6667
F. 0.5
G. 0.5417
H. 0.5556
I. 0.75
J. 0.5714
Answer: F
Solve the integral:
$$
\int 2 \cdot \tan(-10 \cdot x)^4 \, dx
$$

Options:
A. C + \frac{1}{5} \cdot \left(\frac{1}{3} \cdot \left(\tan(10 \cdot x)\right)^3 + \arctan\left(\tan(10 \cdot x)\right) - \frac{1}{3} \cdot \tan(10 \cdot x)\right)
B. C + \frac{1}{5} \cdot \left(\frac{1}{3} \cdot \left(\tan(10 \cdot x)\right)^3 - \arctan\left(\tan(10 \cdot x)\right) + 2 \cdot \tan(10 \cdot x)\right)
C. C + \frac{1}{5} \cdot \left(\frac{1}{3} \cdot \left(\tan(10 \cdot x)\right)^3 + \arctan\left(\tan(10 \cdot x)\right) - \frac{1}{2} \cdot \tan(10 \cdot x)\right)
D. C + \frac{1}{5} \cdot \left(\frac{1}{3} \cdot \left(\tan(10 \cdot x)\right)^3 + \arctan\left(\tan(10 \cdot x)\right) - \tan(10 \cdot x)\right)
E. C + \frac{1}{5} \cdot \left(\frac{1}{3} \cdot \left(\tan(10 \cdot x)\right)^3 + \arctan\left(\tan(10 \cdot x)\right) - \frac{2}{3} \cdot \tan(10 \cdot x)\right)
F. C + \frac{1}{5} \cdot \left(\frac{1}{3} \cdot \left(\tan(10 \cdot x)\right)^3 + \arctan\left(\tan(10 \cdot x)\right) - \frac{3}{2} \cdot \tan(10 \cdot x)\right)
G. C + \frac{1}{5} \cdot \left(\frac{1}{3} \cdot \left(\tan(10 \cdot x)\right)^3 + \arctan\left(\tan(10 \cdot x)\right) + \tan(10 \cdot x)\right)
H. C + \frac{1}{5} \cdot \left(\frac{1}{3} \cdot \left(\tan(10 \cdot x)\right)^3 - \arctan\left(\tan(10 \cdot x)\right) + \tan(10 \cdot x)\right)
I. C + \frac{1}{5} \cdot \left(\frac{1}{3} \cdot \left(\tan(10 \cdot x)\right)^3 - \arctan\left(\tan(10 \cdot x)\right) - \tan(10 \cdot x)\right)
J. C + \frac{1}{5} \cdot \left(\frac{1}{3} \cdot \left(\tan(10 \cdot x)\right)^3 + \arctan\left(\tan(10 \cdot x)\right) - 2 \cdot \tan(10 \cdot x)\right)
Answer: D
Find zeros of $f(x) = \sin(x) + \sin(2 \cdot x) + 2 \cdot \sin(x) \cdot \sin(2 \cdot x) - 2 \cdot \cos(x) - \cos(2 \cdot x)$.

Options:
A. x_1=-\frac{\pi}{2}+2\cdot\pi\cdot n, x_2=-\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_3=\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_4=(-1)^n\cdot\frac{\pi}{9}+\pi\cdot n
B. x_1=-\frac{\pi}{2}+2\cdot\pi\cdot n, x_2=-\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_3=\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_4=(-1)^n\cdot\frac{\pi}{7}+\pi\cdot n
C. x_1=-\frac{\pi}{2}+2\cdot\pi\cdot n, x_2=-\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_3=\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_4=(-1)^n\cdot\frac{\pi}{8}+\pi\cdot n
D. x_1=-\frac{\pi}{2}+2\cdot\pi\cdot n, x_2=-\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_3=\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_4=(-1)^n\cdot\frac{\pi}{10}+\pi\cdot n
E. x_1=-\frac{\pi}{2}+2\cdot\pi\cdot n, x_2=-\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_3=\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_4=(-1)^n\cdot\frac{\pi}{11}+\pi\cdot n
F. x_1=-\frac{\pi}{2}+2\cdot\pi\cdot n, x_2=-\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_3=\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_4=(-1)^n\cdot\frac{\pi}{6}+\pi\cdot n
G. x_1=-\frac{\pi}{2}+2\cdot\pi\cdot n, x_2=-\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_3=\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_4=(-1)^n\cdot\frac{\pi}{12}+\pi\cdot n
H. x_1=-\frac{\pi}{2}+2\cdot\pi\cdot n, x_2=-\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_3=\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_4=(-1)^n\cdot\frac{\pi}{5}+\pi\cdot n
I. x_1=-\frac{\pi}{2}+2\cdot\pi\cdot n, x_2=-\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_3=\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_4=(-1)^n\cdot\frac{\pi}{4}+\pi\cdot n
J. x_1=-\frac{\pi}{2}+2\cdot\pi\cdot n, x_2=-\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_3=\frac{2\cdot\pi}{3}+2\cdot\pi\cdot n, x_4=(-1)^n\cdot\frac{\pi}{3}+\pi\cdot n
Answer: F
Solve the integral:
$$
\int \frac{ \sin(x)^5 }{ \cos(x)^4 } \, dx
$$

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

Options:
A. -\frac{2\cdot\cos(3\cdot x)}{5\cdot\sin(3\cdot x)^5}+\frac{24}{5}\cdot\left(-\frac{\cos(3\cdot x)}{9\cdot\sin(3\cdot x)^3}+\frac{1}{12}\cdot\cot(3\cdot x)\right)+C
B. -\frac{2\cdot\cos(3\cdot x)}{5\cdot\sin(3\cdot x)^5}+\frac{24}{5}\cdot\left(-\frac{\cos(3\cdot x)}{9\cdot\sin(3\cdot x)^3}-\frac{2}{9}\cdot\cot(3\cdot x)\right)+C
C. -\frac{2\cdot\cos(3\cdot x)}{5\cdot\sin(3\cdot x)^5}+\frac{24}{5}\cdot\left(-\frac{\cos(3\cdot x)}{9\cdot\sin(3\cdot x)^3}+\frac{1}{6}\cdot\cot(3\cdot x)\right)+C
D. -\frac{2\cdot\cos(3\cdot x)}{5\cdot\sin(3\cdot x)^5}+\frac{24}{5}\cdot\left(-\frac{\cos(3\cdot x)}{9\cdot\sin(3\cdot x)^3}-\frac{1}{3}\cdot\cot(3\cdot x)\right)+C
E. -\frac{2\cdot\cos(3\cdot x)}{5\cdot\sin(3\cdot x)^5}+\frac{24}{5}\cdot\left(-\frac{\cos(3\cdot x)}{9\cdot\sin(3\cdot x)^3}+\frac{2}{9}\cdot\cot(3\cdot x)\right)+C
F. -\frac{2\cdot\cos(3\cdot x)}{5\cdot\sin(3\cdot x)^5}+\frac{24}{5}\cdot\left(-\frac{\cos(3\cdot x)}{9\cdot\sin(3\cdot x)^3}+\frac{1}{18}\cdot\cot(3\cdot x)\right)+C
G. -\frac{2\cdot\cos(3\cdot x)}{5\cdot\sin(3\cdot x)^5}+\frac{24}{5}\cdot\left(-\frac{\cos(3\cdot x)}{9\cdot\sin(3\cdot x)^3}-\frac{1}{9}\cdot\cot(3\cdot x)\right)+C
H. -\frac{2\cdot\cos(3\cdot x)}{5\cdot\sin(3\cdot x)^5}+\frac{24}{5}\cdot\left(-\frac{\cos(3\cdot x)}{9\cdot\sin(3\cdot x)^3}+\frac{1}{9}\cdot\cot(3\cdot x)\right)+C
I. -\frac{2\cdot\cos(3\cdot x)}{5\cdot\sin(3\cdot x)^5}+\frac{24}{5}\cdot\left(-\frac{\cos(3\cdot x)}{9\cdot\sin(3\cdot x)^3}-\frac{1}{4}\cdot\cot(3\cdot x)\right)+C
J. -\frac{2\cdot\cos(3\cdot x)}{5\cdot\sin(3\cdot x)^5}+\frac{24}{5}\cdot\left(-\frac{\cos(3\cdot x)}{9\cdot\sin(3\cdot x)^3}+\frac{1}{3}\cdot\cot(3\cdot x)\right)+C
Answer: B
Solve the integral:
$$
\int 5 \cdot \cos(3 \cdot x)^6 \, dx
$$

Options:
A. \frac{5}{8}\cdot\left(\frac{5}{2}\cdot x+\frac{2\cdot\sin(6\cdot x)}{3}+\frac{\sin(12\cdot x)}{8}-\frac{\sin(6\cdot x)^3}{18}\right)+C
B. \frac{5}{8}\cdot\left(\frac{5}{2}\cdot x+\frac{2\cdot\sin(6\cdot x)}{3}+\frac{\sin(12\cdot x)}{8}-\frac{\sin(6\cdot x)^3}{15}\right)+C
C. \frac{5}{8}\cdot\left(\frac{5}{2}\cdot x+\frac{2\cdot\sin(6\cdot x)}{3}+\frac{\sin(12\cdot x)}{8}-\frac{\sin(6\cdot x)^3}{22}\right)+C
D. \frac{5}{8}\cdot\left(\frac{5}{2}\cdot x+\frac{2\cdot\sin(6\cdot x)}{3}+\frac{\sin(12\cdot x)}{8}-\frac{\sin(6\cdot x)^3}{20}\right)+C
E. \frac{5}{8}\cdot\left(\frac{5}{2}\cdot x+\frac{2\cdot\sin(6\cdot x)}{3}+\frac{\sin(12\cdot x)}{8}-\frac{\sin(6\cdot x)^3}{21}\right)+C
F. \frac{5}{8}\cdot\left(\frac{5}{2}\cdot x+\frac{2\cdot\sin(6\cdot x)}{3}+\frac{\sin(12\cdot x)}{8}-\frac{\sin(6\cdot x)^3}{17}\right)+C
G. \frac{5}{8}\cdot\left(\frac{5}{2}\cdot x+\frac{2\cdot\sin(6\cdot x)}{3}+\frac{\sin(12\cdot x)}{8}-\frac{\sin(6\cdot x)^3}{23}\right)+C
H. \frac{5}{8}\cdot\left(\frac{5}{2}\cdot x+\frac{2\cdot\sin(6\cdot x)}{3}+\frac{\sin(12\cdot x)}{8}-\frac{\sin(6\cdot x)^3}{16}\right)+C
I. \frac{5}{8}\cdot\left(\frac{5}{2}\cdot x+\frac{2\cdot\sin(6\cdot x)}{3}+\frac{\sin(12\cdot x)}{8}-\frac{\sin(6\cdot x)^3}{19}\right)+C
J. \frac{5}{8}\cdot\left(\frac{5}{2}\cdot x+\frac{2\cdot\sin(6\cdot x)}{3}+\frac{\sin(12\cdot x)}{8}-\frac{\sin(6\cdot x)^3}{24}\right)+C
Answer: A
Use the double-angle formulas to evaluate the integral:
$$
\int \sin(x)^2 \cdot \cos(x)^2 \, dx
$$

Options:
A. 1/8 * x - 1/40 * sin(4 * x) + C
B. 1/8 * x - 1/48 * sin(4 * x) + C
C. 1/8 * x - 1/24 * sin(4 * x) + C
D. 1/8 * x - 1/20 * sin(4 * x) + C
E. 1/8 * x - 1/36 * sin(4 * x) + C
F. 1/8 * x - 1/28 * sin(4 * x) + C
G. 1/8 * x - 1/16 * sin(4 * x) + C
H. 1/8 * x - 1/32 * sin(4 * x) + C
I. 1/8 * x - 1/44 * sin(4 * x) + C
J. 1/8 * x - 1/64 * sin(4 * x) + C
Answer: H
Evaluate $\lim_{x \to 0^{+}} \left( \left( \frac{ \tan(4 \cdot x) }{ 4 \cdot x } \right)^{\frac{ 1 }{ 2 \cdot x^2 }} \right)$ using L'Hopital's Rule.

Options:
A. e^(16/3)
B. e^(12/3)
C. e^(8/3)
D. e^(2/3)
E. e^(7/3)
F. e^(10/3)
G. e^(4/3)
H. e^(14/3)
I. e^(11/3)
J. e^(1/3)
Answer: C
Compute the integral:
$$
\int \frac{ 6 \cdot x^3-7 \cdot x^2+3 \cdot x-1 }{ 2 \cdot x-3 \cdot x^2 } \, dx
$$

Options:
A. -x^2 + x - \frac{1}{3} \cdot \ln\left(\left|x - \frac{2}{3}\right|\right) + \frac{1}{2} \cdot \ln\left(\left|1 - \frac{2}{3 \cdot x^4}\right|\right) + C
B. -x^2 + x - \frac{1}{3} \cdot \ln\left(\left|x - \frac{2}{3}\right|\right) - \frac{1}{2} \cdot \ln\left(\left|1 + \frac{2}{3 \cdot x}\right|\right) + C
C. -x^2 + x - \frac{1}{3} \cdot \ln\left(\left|x - \frac{2}{3}\right|\right) + \frac{1}{2} \cdot \ln\left(\left|1 - \frac{2}{3 \cdot x}\right|\right) + C
D. -x^2 + x - \frac{1}{3} \cdot \ln\left(\left|x - \frac{2}{3}\right|\right) - \frac{1}{2} \cdot \ln\left(\left|1 - \frac{2}{3 \cdot x}\right|\right) + C
E. -x^2 + x - \frac{1}{3} \cdot \ln\left(\left|x - \frac{2}{3}\right|\right) + \frac{1}{2} \cdot \ln\left(\left|1 - \frac{2}{3 \cdot x^3}\right|\right) + C
F. -x^2 + x - \frac{1}{3} \cdot \ln\left(\left|x - \frac{2}{3}\right|\right) + \frac{1}{2} \cdot \ln\left(\left|1 - \frac{2}{3 \cdot x^5}\right|\right) + C
G. -x^2 + x - \frac{1}{3} \cdot \ln\left(\left|x - \frac{2}{3}\right|\right) + \frac{1}{2} \cdot \ln\left(\left|1 - \frac{2}{3 \cdot x^7}\right|\right) + C
H. -x^2 + x - \frac{1}{3} \cdot \ln\left(\left|x - \frac{2}{3}\right|\right) + \frac{1}{2} \cdot \ln\left(\left|1 - \frac{2}{3 \cdot x^2}\right|\right) + C
I. -x^2 + x - \frac{1}{3} \cdot \ln\left(\left|x - \frac{2}{3}\right|\right) + \frac{1}{2} \cdot \ln\left(\left|1 + \frac{2}{3 \cdot x}\right|\right) + C
J. -x^2 + x - \frac{1}{3} \cdot \ln\left(\left|x - \frac{2}{3}\right|\right) + \frac{1}{2} \cdot \ln\left(\left|1 - \frac{2}{3 \cdot x^6}\right|\right) + C
Answer: C
Find the derivative of the function $y = \frac{ 3 \cdot \csc(x) - 2 \cdot \sin(x) }{ 5 \cdot \left(\cos(x)\right)^5 } - \frac{ 16 }{ 5 } \cdot \cot(2 \cdot x)$.

Options:
A. 1 / (sin(x)^2 * cos(x)^5)
B. 1 / (sin(x)^2 * cos(x)^5 * cos(x)^2)
C. 1 / (sin(x)^2 * cos(x)^5 * sin(2x))
D. 1 / (sin(x)^2 * cos(x)^4)
E. 1 / (sin(x)^2 * cos(x)^5 * cos(x))
F. 1 / (sin(x)^2 * cos(x)^6)
G. 1 / (sin(x)^2 * cos(x)^5 * sin(x))
H. 1 / (sin(x)^2 * cos(x)^7)
I. 1 / (sin(x)^2 * cos(x)^5 * cos(2x))
J. 1 / (sin(x)^2 * cos(x)^8)
Answer: F
Find the integral:
$$
\int \frac{ 1 }{ \sqrt[3]{\left(\sin(x)\right)^{11} \cdot \cos(x)} } \, dx
$$

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

Options:
A. C + \frac{1}{3} \cdot \left(\frac{1}{x^2} + 1\right) \cdot \sqrt{\frac{1}{x^2} + 1} - \frac{1}{10} \cdot \left(\frac{1}{x^2} + 1\right)^2 \cdot \sqrt{\frac{1}{x^2} + 1}
B. C + \frac{1}{3} \cdot \left(\frac{1}{x^2} + 1\right) \cdot \sqrt{\frac{1}{x^2} + 1} - \frac{1}{12} \cdot \left(\frac{1}{x^2} + 1\right)^2 \cdot \sqrt{\frac{1}{x^2} + 1}
C. C + \frac{1}{3} \cdot \left(\frac{1}{x^2} + 1\right) \cdot \sqrt{\frac{1}{x^2} + 1} - \frac{1}{7} \cdot \left(\frac{1}{x^2} + 1\right)^2 \cdot \sqrt{\frac{1}{x^2} + 1}
D. C + \frac{1}{3} \cdot \left(\frac{1}{x^2} + 1\right) \cdot \sqrt{\frac{1}{x^2} + 1} - \frac{1}{5} \cdot \left(\frac{1}{x^2} + 1\right)^2 \cdot \sqrt{\frac{1}{x^2} + 1}
E. C + \frac{1}{3} \cdot \left(\frac{1}{x^2} + 1\right) \cdot \sqrt{\frac{1}{x^2} + 1} - \frac{1}{9} \cdot \left(\frac{1}{x^2} + 1\right)^2 \cdot \sqrt{\frac{1}{x^2} + 1}
F. C + \frac{1}{3} \cdot \left(\frac{1}{x^2} + 1\right) \cdot \sqrt{\frac{1}{x^2} + 1} - \frac{1}{4} \cdot \left(\frac{1}{x^2} + 1\right)^2 \cdot \sqrt{\frac{1}{x^2} + 1}
G. C + \frac{1}{3} \cdot \left(\frac{1}{x^2} + 1\right) \cdot \sqrt{\frac{1}{x^2} + 1} - \frac{1}{11} \cdot \left(\frac{1}{x^2} + 1\right)^2 \cdot \sqrt{\frac{1}{x^2} + 1}
H. C + \frac{1}{3} \cdot \left(\frac{1}{x^2} + 1\right) \cdot \sqrt{\frac{1}{x^2} + 1} - \frac{1}{8} \cdot \left(\frac{1}{x^2} + 1\right)^2 \cdot \sqrt{\frac{1}{x^2} + 1}
I. C + \frac{1}{3} \cdot \left(\frac{1}{x^2} + 1\right) \cdot \sqrt{\frac{1}{x^2} + 1} - \frac{1}{5} \cdot \left(\frac{1}{x^2} + 1\right)^{3/2} \cdot \sqrt{\frac{1}{x^2} + 1}
J. C + \frac{1}{3} \cdot \left(\frac{1}{x^2} + 1\right) \cdot \sqrt{\frac{1}{x^2} + 1} - \frac{1}{6} \cdot \left(\frac{1}{x^2} + 1\right)^2 \cdot \sqrt{\frac{1}{x^2} + 1}
Answer: D
Determine the Taylor series for $y = \left(\sin(x)\right)^2$, centered at $x_{0} = \frac{ \pi }{ 2 }$. Write out the sum of the first three non-zero terms, followed by dots.

Options:
A. 1 - \frac{2}{2!} \cdot \left(x - \frac{\pi}{2}\right)^2 + \frac{2^3}{3!} \cdot \left(x - \frac{\pi}{2}\right)^4 + \cdots
B. 1 - \frac{2}{2!} \cdot \left(x - \frac{\pi}{2}\right)^2 + \frac{2^5}{4!} \cdot \left(x - \frac{\pi}{2}\right)^4 + \cdots
C. 1 - \frac{2}{2!} \cdot \left(x - \frac{\pi}{2}\right)^2 + \frac{2^1}{4!} \cdot \left(x - \frac{\pi}{2}\right)^4 + \cdots
D. 1 - \frac{2}{2!} \cdot \left(x - \frac{\pi}{2}\right)^2 + \frac{2^3}{6!} \cdot \left(x - \frac{\pi}{2}\right)^4 + \cdots
E. 1 - \frac{2}{2!} \cdot \left(x - \frac{\pi}{2}\right)^2 + \frac{2^3}{4!} \cdot \left(x - \frac{\pi}{2}\right)^4 + \cdots
F. 1 - \frac{2}{2!} \cdot \left(x - \frac{\pi}{2}\right)^2 + \frac{2^2}{4!} \cdot \left(x - \frac{\pi}{2}\right)^4 + \cdots
G. 1 - \frac{2}{2!} \cdot \left(x - \frac{\pi}{2}\right)^2 + \frac{2^3}{5!} \cdot \left(x - \frac{\pi}{2}\right)^4 + \cdots
H. 1 - \frac{2}{2!} \cdot \left(x - \frac{\pi}{2}\right)^2 + \frac{2^6}{4!} \cdot \left(x - \frac{\pi}{2}\right)^4 + \cdots
I. 1 - \frac{2}{2!} \cdot \left(x - \frac{\pi}{2}\right)^2 + \frac{2^4}{4!} \cdot \left(x - \frac{\pi}{2}\right)^4 + \cdots
J. 1 - \frac{2}{2!} \cdot \left(x - \frac{\pi}{2}\right)^2 + \frac{2^2}{3!} \cdot \left(x - \frac{\pi}{2}\right)^4 + \cdots
Answer: E
Consider the quartic polynomial $P(x) = x^4 - Ax^3 + Bx^2 - Ax + 1$ , for some constants $A$ and $B$ . There exist constants $m$ and $n$ such that:
$$\{2m+n, \ 3m+2n, \ 2m-n, \ 3m-2n\}$$
are the four, distinct roots of $P(x)$ . If the sum of all possible values of $(mn)^2$ may be expressed as $\frac{p}{q}$ , for relatively prime positive integers $p$ and $q$ , find $p+q$ .

Options:
A. 124
B. 128
C. 122
D. 120
E. 64
F. 121
G. 129
H. 125
I. 123
J. 126
Answer: E
Find the derivative of $f(x) = \frac{ 1 }{ 15 } \cdot \left(\cos(x)\right)^3 \cdot \left(\left(\cos(x)\right)^2-5\right)$.

Options:
A. -\frac{\left(\cos(x)\right)^2}{15}\cdot\left(3\cdot\sin(x)\cdot\left(\left(\cos(x)\right)^2-5\right)-2\cdot\cos(x)\cdot\sin(x)\right)
B. -\frac{\left(\cos(x)\right)^2}{15}\cdot\left(3\cdot\sin(x)\cdot\left(\left(\cos(x)\right)^2-5\right)+2\cdot\cos(x)\cdot\sin(x)\right)
C. -\frac{\left(\cos(x)\right)^2}{15}\cdot\left(3\cdot\sin(x)\cdot\left(\left(\cos(x)\right)^2-5\right)-\cos(x)\cdot\sin(2\cdot x)\right)
D. -\frac{\left(\cos(x)\right)^2}{15}\cdot\left(3\cdot\sin(x)\cdot\left(\left(\cos(x)\right)^2-5\right)+\cos(x)\cdot\sin(2\cdot x)\right)
Answer: D
Solve the integral:
$$
\int \left(\frac{ x+2 }{ x-2 } \right)^{\frac{ 3 }{ 2 }} \, dx
$$

Options:
A. C + sqrt((x+2)/(x-2)) * (x-8) - 6 * ln(abs((sqrt(x-2) - sqrt(x+2)) / (sqrt(x-2) + sqrt(x+2))))
B. C + sqrt((x+2)/(x-2)) * (x-10) - 6 * ln(abs((sqrt(x-2) - sqrt(x+2)) / (sqrt(x-2) + sqrt(x+2))))
C. C + sqrt((x+2)/(x-2)) * (x-11) - 6 * ln(abs((sqrt(x-2) - sqrt(x+2)) / (sqrt(x-2) + sqrt(x+2))))
D. C + sqrt((x+2)/(x-2)) * (x-9) - 6 * ln(abs((sqrt(x-2) - sqrt(x+2)) / (sqrt(x-2) + sqrt(x+2))))
E. C + sqrt((x+2)/(x-2)) * (x-12) - 6 * ln(abs((sqrt(x-2) - sqrt(x+2)) / (sqrt(x-2) + sqrt(x+2))))
Answer: B
Consider the sequence defined by $f_0=1$, $f_1=2$, and $f_a=4f_{a-1}-f_{a-2}$ for $a\geq 2$. Determine the smallest odd prime factor of $f_{40}$.

Options:
A. 173
B. 179
C. 193
D. 181
E. 199
F. 191
G. 197
Answer: D
Let $A$ and $B$ be $2 \times 2$ matrices with integer entries such that $A, A+B, A+2B, A+3B$, and $A+4B$ are all invertible matrices whose inverses have integer entries. Find the maximum possible value of $\det(A + 5B)$.

Options:
A. -3
B. 2
C. 3
D. 1
E. -1
F. 0
G. -2
Answer: D
Find the radius of convergence of the series:
$$
\sum_{n=1}^\infty \left( \frac{ \prod_{i=1}^n(2 \cdot n) }{ (2 \cdot n)! } \cdot x^n \right)
$$

Options:
A. 2
B. 3
C. ∞
D. 1/2
E. 1/4
F. 1
G. 4
H. 1/3
Answer: C
The $\textit{digital root}$ of a number is defined as the result obtained by repeatedly adding the digits of the number until a single digit remains. For example, the $\textit{digital root}$ of $237$ is $3$ ( $2+3+7=12, 1+2=3$ ). Find the $\textit{digital root}$ of $2012^{2012^{2012}}$ .

Options:
A. 1
B. 7
C. 5
D. 6
E. 9
F. 4
G. 2
H. 8
I. 3
Answer: F
The following vectors are given:
$u = 2 \cdot \mathbf{i} - 3 \cdot \mathbf{j} + 2 \cdot \mathbf{k}$
$v = 3 \cdot \mathbf{i} + 2 \cdot \mathbf{j} - 3 \cdot \mathbf{k}$
Find the following component:
Project of $u$ onto $v$

Options:
A. w=\frac{9}{22}\cdot i+\frac{6}{22}\cdot j-\frac{9}{22}\cdot k
B. w=\frac{9}{22}\cdot i+\frac{6}{22}\cdot j-\frac{9}{11}\cdot k
C. w=\frac{9}{22}\cdot i+\frac{6}{11}\cdot j-\frac{9}{22}\cdot k
D. w=\frac{9}{11}\cdot i+\frac{6}{11}\cdot j-\frac{9}{11}\cdot k
E. w=\frac{9}{22}\cdot i+\frac{6}{22}\cdot j-\frac{18}{11}\cdot k
F. w=\frac{18}{11}\cdot i+\frac{12}{11}\cdot j-\frac{18}{11}\cdot k
G. w=\frac{9}{22}\cdot i+\frac{6}{11}\cdot j-\frac{9}{11}\cdot k
H. w=\frac{9}{22}\cdot i+\frac{6}{22}\cdot j-\frac{18}{22}\cdot k
I. w=\frac{18}{22}\cdot i+\frac{12}{22}\cdot j-\frac{18}{22}\cdot k
Answer: D
The positive integers $x_1, x_2, ... , x_7$ satisfy $x_6 = 144$ and $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n = 1, 2, 3, 4$ . Find the last three digits of $x_7$ .

Options:
A. 455
B. 453
C. 454
D. 452
E. 457
F. 456
G. 450
H. 451
I. 458
J. 459
Answer: F
Consider the set $S=\{1, 2, 3, 4, 5, 6, 7, 8\}$ . We say a subset $s \in S$ , with $|s| \geq 2$ , is prime-like if every two-element subset of $s$ will always contain at least one prime number. Find the number of prime-like subsets.

Options:
A. 72
B. 70
C. 71
D. 66
E. 69
F. 74
G. 68
H. 65
I. 73
J. 67
Answer: C
If $k$ is the unique real number such that $\sqrt[7]{k+\sqrt[7]{k+\sqrt[7]{k+2}}}=2$ , find $\lfloor k \rfloor$ .
( $\lfloor k \rfloor$ denotes the greatest integer less than or equal to $k$ .)

Options:
A. 127
B. 125
C. 124
D. 128
E. 122
F. 120
G. 123
H. 129
I. 121
J. 126
Answer: J
A positive integer $n$ is called spammy if the base ten repeating decimal for $\frac{1}{n}$ has period $1$ . Find the number of spammy positive integers less than or equal to $1000$ .

Options:
A. 35
B. 31
C. 39
D. 33
E. 38
F. 37
G. 32
H. 34
I. 36
J. 30
Answer: F
A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are colored red. An arrangement of the cubes is $\mathfrak{Intriguing}$ if there is exactly $1$ red unit cube in every $1\times1\times4$ rectangular box composed of $4$ unit cubes. Determine the number of $\mathfrak{Intriguing}$ colorings.

Options:
A. 579
B. 571
C. 578
D. 575
E. 574
F. 573
G. 580
H. 572
I. 576
J. 577
Answer: I
Let $k$ be the value of
$$$\lfloor log_2 1 \rfloor + \lfloor log_2 2 \rfloor + \lfloor log_2 3 \rfloor + \dots + \lfloor log_2 2017 \rfloor+ \lfloor log_2 2018 \rfloor$$$
where $\lfloor m \rfloor$ denotes the integer part of $m$ (the biggest integer which is $\le m$ ).
What is the remainder when $k$ is divided by $1000$ ?
Problem

Options:
A. 142
B. 146
C. 141
D. 148
E. 144
F. 145
G. 143
H. 149
I. 147
J. 140
Answer: E
Sequence $S = x_0, x_1, x_2, x_3, ...$ satisfies that for all $i > 2$ , $x_i = x_{i-3} + x_{i-2}$ (mod $x_{i-1}$ ). If $x_{i-1} = 0$ , the sequence ends. What is the smallest value of $x_2$ such that we can choose values of $x_0$ and $x_1$ that make the length of $S$ at least $7$ ? Anything modulo $1$ is $0$ and the last number in the sequence is $0$ .

Options:
A. 7
B. 10
C. 12
D. 14
E. 11
F. 9
G. 8
H. 15
I. 16
J. 13
Answer: F
At the Amalgamated Institute of Math Excellence, there are $10$ clubs and a number of students. For any two different students, there is some club such that exactly one of the two belongs to that club. For any three students, there is some club such that either one of the three belongs to that club, or all three belong to that club. What is the largest possible number of students in the Institute?

Options:
A. 518
B. 514
C. 510
D. 517
E. 519
F. 515
G. 511
H. 512
I. 516
J. 513
Answer: J
Let $P(x) = x^4 + 2x^3 + 3x^2 + 4x + 5$ have roots $r_1, r_2, r_3, r_4$ . Given that $Q(x)$ is a monic polynomial of degree $4$ with roots $r^2_1 - 1$ , $r^2_2 - 1$ , $ r^2_3 - 1$ , $r^2_4 - 1$ ,find the remainder when $Q(8)$ is divided by $1000$ .

Options:
A. 12
B. 14
C. 16
D. 23
E. 18
F. 17
G. 13
H. 19
I. 15
J. 11
Answer: G
$\Delta ABC$ has integer side lengths for which some ordering of the lengths of all three of its medians form a right triangle. Find the minimum possible perimeter of $\Delta ABC$ .

Options:
A. 50
B. 56
C. 54
D. 52
E. 51
F. 59
G. 57
H. 53
I. 55
J. 58
Answer: C
Find the number of four digit multiples of $11$ that have the property that it is possible to remove a digit such that the remaining three-digit number is still a multiple of $11$ .

Options:
A. 201
B. 198
C. 195
D. 203
E. 194
F. 199
G. 200
H. 196
I. 202
J. 197
Answer: A
Define a normal series to be an expression $a_{1} \pm a_{2} \pm...\pm a_{n}$ , where each $\pm$ is either a $+$ or a $-$ , and the operations are performed in the normal order from left to right. Define a reverse series to be an expression $R: a_{1} \pm 1_{2} \pm a_{3} \pm...\pm a_{n}$ , where again each $\pm$ is either a $+$ or a $-$ , but now the operations are grouped from left to right. For instance,
$$$R: 3-4-6+5=3-(4-(6+5))=3-4+6+5=10$$$
As just illustrated, each reverse series is equivalent to a normal series with the same terms in the same order, but with some of the signs changed. Conversely, each normal series has an equivalent reversed series. If the normal series $a_{1} \pm a_{2} \pm a_{3} \pm...\pm a_{1000}$ has $600$ minuses among its $999$ signs, at most how many minuses does the equivalent reverse series have?

Options:
A. 796
B. 790
C. 791
D. 798
E. 795
F. 797
G. 794
H. 793
I. 792
J. 799
Answer: J
Let $a$ , $b$ , and $c$ be positive real numbers such that $a^2+b^2=c^2+ab$ and $(a+b+c)(a+b-c)(a-b+c)(-a+b+c)=2020$ . Then $ab$ can be expressed as $\frac{p}{q}\sqrt{r}$ for positive integers $p$ , $q$ , and $r$ such that $p$ and $q$ are relatively prime and $r$ is as small as possible. Compute the remainder when $p+q+r$ is divided by $1000$ .

Options:
A. 511
B. 516
C. 517
D. 514
E. 520
F. 513
G. 515
H. 519
I. 512
J. 518
Answer: E
Triangle $ABC$ has side lengths $AB = 9$ and $AC = 15.$ Points $P$ and $Q$ are on sides $\overline{AC}$ and $\overline{AB}$ respectively, such that $PB = AB$ and $QC = AC.$ If the circumcircle of $\triangle APQ$ passes through the midpoint of $\overline{BC}$ , find $BC^2.$

Options:
A. 216
B. 208
C. 205
D. 204
E. 202
F. 200
G. 209
H. 212
I. 206
J. 210
Answer: D
Let $f(x) =\frac{2}{x^2 -1}$ . Find the largest positive integer $n$ such that
$$$f(2) + f(3) + ... + f(n) \ge \frac{2006}{1337} - \frac{101}{n} + \frac{99}{n + 1}.$$$

Options:
A. 516
B. 512
C. 513
D. 517
E. 518
F. 511
G. 510
H. 515
I. 514
J. 519
Answer: A
A point $D$ is on side $BC$ of triangle $ABC$ such that $BD=20$ and $DC=14$ . A circle $\Gamma$ centered at $D$ is tangent to rays $\overrightarrow{AB}$ and $\overrightarrow{AC}$ . Line $l$ is a tangent of $\Gamma$ that is parallel to $BC$ , and let $l$ intersect $\overrightarrow{AB}$ at $X$ and $\overrightarrow{AC}$ at $Y$ . How many different integer lengths can $XY$ take on?

Options:
A. 54
B. 56
C. 58
D. 51
E. 57
F. 50
G. 53
H. 59
I. 55
J. 52
Answer: A
Let $x_1, . . . ,x_{2024}$ be complex numbers such that for all $1 \le i \le 2024$
$$$\sum_{1\le j_1<j_2<···<j_i\le 2024} x_{j1}x_{j2}... x_{ji} = 1$$$
Calculate $\left|\sum^{2024}_{i=1} x^{2024}_i \right|$ .
.

Options:
A. 2024^2
B. 2023
C. 2024
D. 2024 - 3
E. 2024 - 2
F. 1
G. 2022
H. 2024 + 1
I. 2025
J. 2024 - 1
Answer: F
Find the remainder when
$$\sum_{a=0}^8 \sum_{b=0}^8 \sum_{c=0}^8 \sum_{d=0}^8 \max\left( |a+b-c-d|, |a-b-c+d| \right)$$
is divided by $1000$ .

Options:
A. 874
B. 875
C. 873
D. 876
E. 872
F. 878
G. 877
H. 871
I. 879
J. 880
Answer: J
Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$?

Options:
A. \boxed{6}
B. \boxed{1}
C. \boxed{3}
D. \boxed{4}
E. \boxed{5}
F. \boxed{7}
G. \boxed{2}
H. \boxed{10}
I. \boxed{9}
J. \boxed{8}
Answer: C
Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers \[ N = a + (a+1) +(a+2) + \cdots + (a+k-1) \] for $k=2017$ but for no other values of $k>1$. Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of these expressions?

Options:
A. 15.5
B. 17
C. 15.25
D. 13
E. 14
F. 19
G. 18
H. 15
I. 15.75
J. 16
Answer: J
Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is, \[N=1111\cdots 11.\] Find the 1000th digit after the decimal point of $\sqrt N$.

Options:
A. 8
B. 0
C. 1
D. 2
E. 5
F. 9
G. 3
H. 4
I. 7
J. 6
Answer: C
Find the positive number $m$ such that the area in the first quadrant bounded by the line $y = \frac{1}{88} x$, the $x$-axis, and the ellipse $\frac{1}{8649} x^2 + y^2 = 1$ is equal to the area of the region in the first quadrant bounded by the line $y = mx$, the $y$-axis, and the ellipse $\frac{1}{8649} x^2 + y^2 = 1$.

Options:
A. \frac{88}{8650}
B. \frac{88}{8656}
C. \frac{88}{8651}
D. \frac{88}{8647}
E. \frac{88}{8655}
F. \frac{88}{8648}
G. \frac{88}{8654}
H. \frac{88}{8652}
I. \frac{88}{8649}
J. \frac{88}{8653}
Answer: I
Let $R$ be the region consisting of the points $(x,y)$ of the cartesian plane satisfying both $|x|-|y| \leq 1$ and $|y| \leq 1$. Find the area of $R$.

Options:
A. 4
B. 5.8
C. 5.9
D. 6
E. 7
F. 5.5
G. 5
H. 8
I. 5.7
J. 5.2
Answer: D
How many positive integers $N$ satisfy all of the following three conditions? \begin{enumerate} \item[(i)] $N$ is divisible by 2020. \item[(ii)] $N$ has at most 2020 decimal digits. \item[(iii)] The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros. \end{enumerate}

Options:
A. 508530
B. 508531
C. 508533
D. 508538
E. 508536
F. 508537
G. 508539
H. 508532
I. 508534
J. 508535
Answer: E
Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k$, the integer \[ p^{(j)}(k) = \left. \frac{d^j}{dx^j} p(x) \right|_{x=k} \] (the $j$-th derivative of $p(x)$ at $k$) is divisible by 2016.

Options:
A. 7
B. 9
C. 8
D. 12
E. 14
F. 11
G. 16
H. 10
I. 15
J. 6
Answer: C
Find the area of a triangle bounded by the x-axis, the line $f(x) = 20 - \frac{ 1 }{ 3 } \cdot x$, and the line perpendicular to $f(x)$ that passes through the origin.

Options:
A. 540 / 3
B. 180
C. 540 / 5
D. 540 / 6
E. 540 / 4
F. 270
G. 540
H. 540 / 7
I. 540 / 2
J. 540 / 8
Answer: G
Evaluate the double integral:
$$
\int \int \rho^2 \, d \varphi \, d \rho
$$
where the integration area $Q$ is bounded by the curve $\rho = a \cdot \sin(2 \cdot \varphi)$.

Options:
A. 16/9 * a^3
B. 12/9 * a^3
C. 10/12 * a^3
D. 10/9 * a^3
E. 18/9 * a^3
F. 8/9 * a^3
G. 6/9 * a^3
H. 14/9 * a^3
I. 2/3 * a^3
J. 4/9 * a^3
Answer: F
Compute the area of the figure bounded by curves $y = 2 \cdot x^2$, $y = 1 + x^2$, lines $x = 3$, $x = -2$, and the $x$-axis.

Options:
A. 47/3
B. 41/3
C. 43/3
D. 44/3
E. 42/3
F. 46/3
G. 48/3
H. 40/3
I. 49/3
J. 45/3
Answer: F
Solve the integral:
$$
\int \frac{ \cos(x)^3 }{ \sin(x)^9 } \, dx
$$

Options:
A. C - (1/3 * (cot(x))^6 + 1/4 * (cot(x))^4 + 1/6 * (cot(x))^8)
B. C - (1/3 * (cot(x))^6 + 1/4 * (cot(x))^4 + 1/14 * (cot(x))^8)
C. C - (1/3 * (cot(x))^6 + 1/4 * (cot(x))^4 + 1/8 * (cot(x))^8)
D. C - (1/3 * (cot(x))^6 + 1/4 * (cot(x))^4 + 1/5 * (cot(x))^8)
E. C - (1/3 * (cot(x))^6 + 1/4 * (cot(x))^4 + 1/10 * (cot(x))^8)
F. C - (1/3 * (cot(x))^6 + 1/4 * (cot(x))^4 + 1/16 * (cot(x))^8)
G. C - (1/3 * (cot(x))^6 + 1/4 * (cot(x))^4 + 1/12 * (cot(x))^8)
H. C - (1/3 * (cot(x))^6 + 1/4 * (cot(x))^4 + 1/9 * (cot(x))^8)
I. C - (1/3 * (cot(x))^6 + 1/4 * (cot(x))^4 + 1/7 * (cot(x))^8)
J. C - (1/3 * (cot(x))^6 + 1/4 * (cot(x))^4 + 1/11 * (cot(x))^8)
Answer: C
The velocity of a bullet from a rifle can be approximated by $v(t) = 6400 \cdot t^2 - 6505 \cdot t + 2686$ where $t$ is seconds after the shot and $v$ is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: $0 \le t \le 0.5$. What is the total distance the bullet travels in $0.5$ sec?

Options:
A. 796.54166670
B. 796.54166666
C. 796.54166662
D. 796.54166667
E. 796.54166668
F. 796.54166661
G. 796.54166664
H. 796.54166665
I. 796.54166663
J. 796.54166669
Answer: D
Evaluate the integral:
$$
I = \int 3 \cdot x \cdot \ln\left(4 + \frac{ 1 }{ x } \right) \, dx
$$

Options:
A. 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/2 * x^2 + C
B. 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/6 * x^2 + C
C. 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/10 * x^2 + C
D. 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/9 * x^2 + C
E. 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/7 * x^2 + C
F. 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/4 * x^2 + C
G. 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/16 * x^2 + C
H. 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/8 * x^2 + C
I. 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/5 * x^2 + C
J. 3/2 * x^2 * ln(4 * x + 1) - 3/4 * x^2 + 3/8 * x - 3/32 * ln(x + 1/4) - 3/2 * x^2 * ln(x) + 3/3 * x^2 + C
Answer: F
Evaluate $\tan\left(\arcsin\left(\frac{ 3 }{ 5 }\right)+\arccos\left(\frac{ 5 }{ 13 }\right)\right)$.

Options:
A. -63/19
B. -63/17
C. -63/13
D. -63/12
E. -63/16
F. -63/18
G. -63/15
H. -63/14
I. -63/11
J. -63/20
Answer: E
Find the period of the function $f(x) = \left(\sin(x)\right)^4 + \left(\cos(x)\right)^4$.

Options:
A. T=\pi/10
B. T=\pi/8
C. T=\pi/16
D. T=\pi/4
E. T=\pi/6
F. T=\pi/12
G. T=\pi/3
H. T=\pi
I. T=\pi/5
J. T=\frac{\pi}{2}
Answer: J
Find $\lim_{x \to 0}\frac{ 2 \cdot \left(\tan(x)-\sin(x)\right)-x^3 }{ x^5 }$, using the expansion of the function in the Maclaurin series.

Options:
A. 1/2
B. 1/4
C. 1/15
D. 1/10
E. 1/12
F. 1/6
G. 1/5
H. 1/3
I. 1/7
J. 1/8
Answer: B
Calculate the integral:
$$
\int_{-\sqrt{5}}^{\sqrt{5}} \frac{ 4 \cdot x^7+6 \cdot x^6-12 \cdot x^5-14 \cdot x^3-24 \cdot x^2+2 \cdot x+4 }{ x^2+2 } \, dx
$$

Options:
A. 20*sqrt(5) + 4*sqrt(2)*arctan(sqrt(5)/sqrt(10))
B. 20*sqrt(5) + 4*sqrt(2)*arctan(sqrt(5)/sqrt(7))
C. 20*sqrt(5) + 4*sqrt(2)*arctan(2*sqrt(5)/sqrt(2))
D. 20*sqrt(5) + 4*sqrt(2)*arctan(sqrt(5)/2)
E. 20*sqrt(5) + 4*sqrt(2)*arctan(sqrt(5)/sqrt(4))
F. 20*sqrt(5) + 4*sqrt(2)*arctan(sqrt(5)/sqrt(6))
G. 20*sqrt(5) + 4*sqrt(2)*arctan(sqrt(5)/sqrt(8))
H. 20*sqrt(5) + 4*sqrt(2)*arctan(sqrt(5)/sqrt(3))
I. 20*sqrt(5) + 4*sqrt(2)*arctan(sqrt(5)/sqrt(2))
J. 20*sqrt(5) + 4*sqrt(2)*arctan(sqrt(5)/sqrt(9))
Answer: I
Calculate the derivative of the function $r = 2 \cdot \ln\left(\sqrt[6]{\frac{ 1+\tan(3 \cdot \varphi) }{ 1-\tan(3 \cdot \varphi) }}\right)$.

Options:
A. 2 * sec(3 * φ) * tan(6 * φ)
B. 2 * sec(4 * φ)
C. 2 * sec(5 * φ)
D. 2 * sec(3 * φ) / 2
E. 2 * sec(3 * φ) * tan(3 * φ) / 2
F. 2 * sec(6 * φ)
G. 2 * sec(6 * φ) * tan(3 * φ)
H. 2 * sec(3 * φ)
I. 2 * sec(2 * φ)
J. 2 * sec(3 * φ) * tan(3 * φ)
Answer: F
What are the points of inflection of the graph of $f(x) = \frac{ x+1 }{ x^2+1 }$?

Options:
A. x_1=1, x_2=-2+\sqrt{3}, x_3=-2-\sqrt{3}
B. x_1=1, x_2=-2+\sqrt{8}, x_3=-2-\sqrt{8}
C. x_1=1, x_2=-2+\sqrt{10}, x_3=-2-\sqrt{10}
D. x_1=1, x_2=-1+\sqrt{3}, x_3=-1-\sqrt{3}
E. x_1=1, x_2=-2+\sqrt{4}, x_3=-2-\sqrt{4}
F. x_1=1, x_2=-2+\sqrt{6}, x_3=-2-\sqrt{6}
G. x_1=1, x_2=-2+\sqrt{7}, x_3=-2-\sqrt{7}
H. x_1=1, x_2=-2+\sqrt{2}, x_3=-2-\sqrt{2}
I. x_1=1, x_2=-2+\sqrt{5}, x_3=-2-\sqrt{5}
J. x_1=1, x_2=-2+\sqrt{1}, x_3=-2-\sqrt{1}
Answer: A
Compute $\sqrt[3]{130}$ with accuracy $0.0001$.

Options:
A. 5.0653
B. 5.0651
C. 5.0652
D. 5.0657
E. 5.0658
F. 5.0654
G. 5.0660
H. 5.0659
I. 5.0655
J. 5.0656
Answer: E
Suppose that $\Omega_1$ and $\Omega_2$ are two circles with radii $9$ and $16$ , respectively, and that they are externally tangent at $P$ . A common external tangent $\ell$ meets $\Omega_1$ at $X$ and $\Omega_2$ at $Y$ . $\overleftrightarrow{XP}$ meets $\Omega_2$ again at $W$ , and $\overleftrightarrow{YP}$ meets $\Omega_1$ again at $Z$ . If $WZ$ can be expressed as $a \sqrt b$ for positive integers $a$ and $b$ with $b$ squarefree, compute $a+ b$ .

Options:
A. 194
B. 195
C. 190
D. 198
E. 196
F. 191
G. 192
H. 199
I. 193
J. 197
Answer: B
Evaluate $\lim_{x \to 0^{+}} \left( \left( \frac{ \tan(2 \cdot x) }{ 2 \cdot x } \right)^{\frac{ 1 }{ 3 \cdot x^2 }} \right)$ using L'Hopital's Rule.

Options:
A. e^(2/3)
B. e^(4/9)
C. e^(8/27)
D. e^(1/18)
E. e^(1/9)
F. e^(1/3)
G. e^(1/27)
H. e^(8/9)
I. e^(2/9)
J. e^(1/6)
Answer: B
Compute the integral:
$$
\int \frac{ \sqrt{1+x^2} }{ x } \, dx
$$

Options:
A. sqrt{x^2+1} + 1/2 * ln(|(sqrt{x^2+1} + 1) / (sqrt{x^2+1} - 1)|) + 3C
B. sqrt{x^2+1} + 1/2 * ln(|(sqrt{x^2+1} + 1) / (sqrt{x^2+1} - 1)|) - C
C. sqrt{x^2+1} + 1/2 * ln(|(sqrt{x^2+1} + 1) / (sqrt{x^2+1} - 1)|) + C/2
D. sqrt{x^2+1} - 1/2 * ln(|(sqrt{x^2+1} + 1) / (sqrt{x^2+1} - 1)|) + C
E. sqrt{x^2+1} - 1/2 * ln(|(sqrt{x^2+1} - 1) / (sqrt{x^2+1} + 1)|) + C
F. sqrt{x^2+1} - 1/2 * ln(|(sqrt{x^2+1} + 1) / (sqrt{x^2+1} - 1)|) - C
G. sqrt{x^2+1} - 1/2 * ln(|(sqrt{x^2+1} - 1) / (sqrt{x^2+1} + 1)|) + 2C
H. sqrt{x^2+1} + 1/2 * ln(|(sqrt{x^2+1} + 1) / (sqrt{x^2+1} - 1)|) + 2C
I. sqrt{x^2+1} + 1/2 * ln(|(sqrt{x^2+1} + 1) / (sqrt{x^2+1} - 1)|) + C
J. sqrt{x^2+1} + 1/2 * ln(|(sqrt{x^2+1} - 1) / (sqrt{x^2+1} + 1)|) + C
Answer: J
Solve the integral:
$$
\int \frac{ 6 }{ \cos(-4 \cdot x)^7 \cdot \sin(4 \cdot x) } \, dx
$$

Options:
A. C + \frac{3}{2} \cdot \left( \frac{3}{2 \cdot \left( \cot(4 \cdot x) \right)^2} + \frac{3}{4 \cdot \left( \cot(4 \cdot x) \right)^4} + \frac{1}{6 \cdot \left( \cot(4 \cdot x) \right)^6} + \ln\left( \left| \cot(4 \cdot x) \right| \right) \right)
B. C + \frac{3}{2} \cdot \left( \frac{3}{2 \cdot \left( \cot(4 \cdot x) \right)^2} + \frac{3}{4 \cdot \left( \cot(4 \cdot x) \right)^4} - \frac{1}{6 \cdot \left( \cot(4 \cdot x) \right)^6} + \ln\left( \left| \cot(4 \cdot x) \right| \right) \right)
C. C + \frac{3}{2} \cdot \left( \frac{3}{2 \cdot \left( \cot(4 \cdot x) \right)^2} + \frac{3}{4 \cdot \left( \cot(4 \cdot x) \right)^4} + \frac{1}{6 \cdot \left( \cot(4 \cdot x) \right)^6} - \ln\left( \left| \cot(4 \cdot x) \right| \right) \right)
D. C + \frac{3}{2} \cdot \left( \frac{3}{2 \cdot \left( \cot(4 \cdot x) \right)^2} + \frac{3}{4 \cdot \left( \cot(4 \cdot x) \right)^4} - \frac{1}{6 \cdot \left( \cot(4 \cdot x) \right)^6} - \ln\left( \left| \cot(4 \cdot x) \right| \right) \right)
Answer: C
Kelvin the Frog's current home is a regular hexagon. One of the vertices of his hexagon lies at the origin, a vertex adjacent to this lies on the line $y=-2$ , and a vertex adjacent to that one lies on the line $y=-1$ . Let $S_x$ denote the sum of the $x$ -coordinates of the vertices of Kelvin the Frog's home, and $S_y$ denote the sum of the $y$ -coordinates. Then $|S_x| + |S_y|$ can be expressed in the form $a\sqrt{b} + c$ , where $a$ , $b$ , $c$ are positive integers and $b$ is not divisible by the square of any prime. Compute $a + b + c$ .

Options:
A. 20
B. 22
C. 19
D. 21
E. 23
Answer: C
Find the zeros of the polynomial function with complex roots: $f(x) = 2 \cdot x^3 - 2 \cdot x^2 + 5 \cdot x - 5$.

Options:
A. 1, i·√(10/3), -i·√(10/3)
B. 1, i·√(10/2), -i·√(10/2)
C. 1, i·√(15/2), -i·√(15/2)
D. 1, i·√(5/2), -i·√(5/2)
E. 1, i·√(10/4), -i·√(10/4)
Answer: D
Find the formula for an exponential function that passes through the two points: $P(-2,6)$ and $P(3,1)$.

Options:
A. 6^{\frac{2}{5}}\cdot6^{\frac{x}{5}}
B. 6^{\frac{4}{5}}\cdot6^{-\frac{x}{5}}
C. 6^{\frac{3}{5}}\cdot6^{-\frac{x}{5}}
D. 6^{\frac{1}{5}}\cdot6^{\frac{x}{5}}
E. 6^{\frac{1}{5}}\cdot6^{-\frac{x}{5}}
F. 6^{\frac{2}{5}}\cdot6^{-\frac{x}{5}}
Answer: C
Compute the sum of all $n$ less than $450$ such that
$$$2^n-187n$$$
leaves a remainder of $69$ when divided by $420$ .

Options:
A. 175
B. 235
C. 119
D. 295
E. 115
F. 55
G. 179
Answer: F
The expansion of $(x+1)^n$ has $3$ consecutive terms with coefficients in the ratio $1:2:3$ that can be written in the form
$$${n\choose k} : {n\choose k+1} : {n \choose k+2}$$$
Find the sum of all possible values of $n+k$ .

Options:
A. 14
B. 19
C. 17
D. 15
E. 18
F. 20
G. 16
Answer: E
Compute the length of the arc $y = 2 \cdot \ln(3 \cdot x)$ between the points $x = \sqrt{5}$ and $x = 2 \cdot \sqrt{3}$.

Options:
A. 1+ln(15/3)
B. 1+ln(3/5)
C. 1+ln(3/15)
D. 1+ln(9/5)
E. 1+ln(15/9)
F. 1+ln(15/5)
G. 1+ln(5/3)
Answer: G
What is the coefficient of $x^3y^{13}$ in $\left(\frac 12x + y\right)^{17}$ ?

Options:
A. 34
B. 136
C. 17
D. 170
E. 68
F. 340
G. 0
H. 85
Answer: G
Consider a triangle $\triangle OXM$. The centroid is denoted by $D$, while $C$ represents the incenter. The angles at vertices $O$ and $X$ are labeled as $G$ and $z$ respectively. Given that the line segment $CD$ runs parallel to side $OX$, and $z$ is equal to $2 \arctan (1/3)$, determine the value of $G$.

Options:
A. \frac{3\pi}{8}
B. \frac{5\pi}{12}
C. \frac{\pi}{5}
D. \frac{\pi}{4}
E. \frac{\pi}{6}
F. \frac{3\pi}{4}
G. \frac{\pi}{2}
H. \frac{\pi}{3}
Answer: G
A sequence of positive reals defined by $a_0=x$ , $a_1=y$ , and $a_n\cdot a_{n+2}=a_{n+1}$ for all integers $n\ge 0$ . Given that $a_{2007}+a_{2008}=3$ and $a_{2007}\cdot a_{2008}=\frac 13$ , find $x^3+y^3$ .

Options:
A. 652
B. 650
C. 647
D. 645
E. 648
F. 651
G. 649
H. 646
I. 653
Answer: E
Qin Shi Huang wants to count the number of warriors he has to invade China. He puts his warriors into lines with the most people such that they have even length. The people left over are the remainder. He makes $2$ lines, with a remainder of $1$ , $3$ lines with a remainder of $2$ , $4$ lines with a remainder of $3$ , $5$ lines with a remainder of $4$ , and $6$ lines with a remainder of $5$ . Find the minimum number of warriors that he has.

Options:
A. 115
B. 118
C. 117
D. 59
E. 114
F. 119
G. 116
H. 179
I. 113
Answer: D
Iggy and Twiggy are creating problem proposals for their Mock AIME. Iggy's question asks for the remainder when a perfect square is divided by 1000, and Twiggy's question asks for the remainder when a perfect fourth power is divided by 1000. Let $S_i$ be the set of all possible answers to Iggy's question and let $S_t$ be the set of all possible answers to Twiggy's question. Find the number of elements that are contained in $S_i$ but are not contained in $S_t$ .

Options:
A. 106
B. 101
C. 105
D. 103
E. 108
F. 104
G. 100
H. 109
I. 107
J. 102
Answer: I
In a round robin tournament, occasionally a cyclic $3-$ set occurs, that is, a set of $[a,b,c]$ of teams where $a$ beats $b$ , $b$ beats $c$ , and $c$ beats $a$ . If $23$ teams play a round robin (without any ties), what is the greatest number of cyclic $3-$ sets that can occur?

Options:
A. 484
B. 535
C. 510
D. 515
E. 506
F. 465
G. 495
H. 490
I. 529
J. 475
Answer: E
A circle $\omega_1$ of radius $6\sqrt2$ is internally tangent to a larger circle $\omega_2$ of radius $12\sqrt2$ such that the center of $\omega_2$ lies on $\omega_1$ . A diameter $AB$ of $\omega_2$ is drawn tangent to $\omega_1$ . A second line $\ell$ is drawn from $B$ tangent to $\omega_1$ . Let the line tangent to $\omega_2$ at $A$ intersect $\ell$ at $C$ . Find the area of $\vartriangle ABC$ .

Options:
A. 784
B. 760
C. 768
D. 744
E. 776
F. 720
G. 756
H. 736
I. 752
J. 740
Answer: C
Let $n$ be the smallest positive integer such that the last $2015$ digits of $21^n$ are $000\ldots0001$ , where there are $2014$ $0$ 's followed by one $1$ . Compute the remainder when the number of positive factors of $n$ is divided by $1000$ .

Options:
A. 213
B. 215
C. 212
D. 211
E. 214
F. 210
G. 216
H. 218
I. 219
J. 217
Answer: F
Let $k(n)$ be the $n^\text{th}$ smallest positive integer relatively prime to $n$ . Find the sum of all primes $p \le 100$ such that the equation
$$ k(p + 1) + k(p) = k(2p)$$
is satisfied.

Options:
A. 228
B. 236
C. 235
D. 229
E. 233
F. 227
G. 231
H. 230
I. 232
J. 234
Answer: D
A $5$ -digit number is leet if and only if the sum of the first $2$ digits, the sum of the last $2$ digits and the middle digit are equal. How many $5$ -digit leet numbers exist?

Options:
A. 370
B. 300
C. 310
D. 350
E. 330
F. 320
G. 390
H. 360
I. 340
J. 380
Answer: E
Let $x$ be an integer with $10\leq x\leq99$ . $M$ is a positive integer such that for every pair of even positive integers $(i,j)$ , the difference $x^i-x^j$ is a multiple of $M$ . Find the remainder when the maximum possible value of $M$ is divided by $1000$ .

Options:
A. 798
B. 788
C. 784
D. 800
E. 795
F. 799
G. 792
H. 794
I. 796
J. 790
Answer: D
For real numbers $a,b,$ and $c,$ suppose that $\tfrac{a+b}{c}$ and $\tfrac{b+c}{a}$ are the roots of the polynomial $x^2 - 574 x + 17.$ Find $\tfrac{a+c}{b}$ .

Options:
A. 35
B. 38
C. 36
D. 35.7
E. 34
F. 35.8
G. 35.2
H. 37
I. 35.9
J. 35.5
Answer: C
Let $a$ and $b$ be real numbers such that
$$$\left(8^a+2^{b+7}\right)\left(2^{a+3}+8^{b-2}\right)=4^{a+b+2}.$$$
The value of the product $ab$ can be written as $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Options:
A. 278
B. 272
C. 276
D. 275
E. 277
F. 274
G. 271
H. 279
I. 270
J. 273
Answer: G
Find the smallest positive integer $n$ such that ${2n \choose n}$ is divisible by all the primes between $10$ and $30$ .

Options:
A. 137
B. 138
C. 133
D. 131
E. 135
F. 134
G. 130
H. 139
I. 132
J. 136
Answer: D
Let
$$$K =\sum^8_{a_1=0}\sum^{a_1}_{a_2=0}...\sum^{a_{99}}_{a_{100}=0} {8 \choose a_1}{a_1 \choose a_2}...{a_{99} \choose a_{100}}.$$$
Find the sum of digits of $K$ in base- $100$ .

Options:
A. 258
B. 256
C. 255
D. 259
E. 250
F. 252
G. 253
H. 254
I. 257
J. 251
Answer: B
Circle $\omega_1$ is defined by the equation $(x-7)^2+(y-1)^2=k,$ where $k$ is a positive real number. Circle $\omega_2$ passes through the center of $\omega_1$ and its center lies on the line $7x+y=28.$ Suppose that one of the tangent lines from the origin to circles $\omega_1$ and $\omega_2$ meets $\omega_1$ and $\omega_2$ at $A_1,A_2$ respectively, that $OA_1=OA_2,$ where $O$ is the origin, and that the radius of $\omega_2$ is $\frac{2011} {211}$ . What is $k$ ?

Options:
A. 42
B. 44
C. 47
D. 48
E. 46
F. 43
G. 45
H. 49
I. 40
J. 41
Answer: B
A card deck consists of $32$ cards. On each card, a different subset of
$$$S = \{algebra, \,\,\, combinatorics, \,\,\, geometry, \,\,\, number \,\,\, theory, \,\,\, sequences\}$$$
is written (thus, one of the cards is empty). Let $N$ be the number of ways to burn $5$ cards from this deck so that each element of $S$ appears an odd number of times among the remaining cards. Find the remainder when $N$ is divided by $1000$ .

Options:
A. 394
B. 293
C. 396
D. 393
E. 397
F. 391
G. 395
H. 399
I. 398
J. 392
Answer: B
Compute the largest integer $n$ such that $2005^{2^{100}} - 2003^{2^{100}}$ is divisible by $2^n$ .

Options:
A. 106
B. 105
C. 109
D. 108
E. 100
F. 107
G. 103
H. 101
I. 102
J. 104
Answer: G
Each of $11$ friends begin with $10^{11}$ coins. A friend may $\textit{share}$ their coins with the rest of the group by distributing all of their coins evenly with the other $10$ friends. Each friend $\textit{shares}$ exactly once, one at a time starting with Alice. In the end, Alice has $N$ coins. Find the $\underline{\text{first}}$ three digits of $N$ .

Options:
A. 174
B. 172
C. 179
D. 170
E. 171
F. 175
G. 177
H. 173
I. 178
J. 176
Answer: F
Find the number of functions $f$ from $\{1,2, 3, 4,5\}$ to itself such that $f(f(f(x))) = f(f(x))$ for all $x \in \{1,2, 3, 4,5\}$ .

Options:
A. 735
B. 750
C. 700
D. 760
E. 744
F. 756
G. 720
H. 752
I. 740
J. 764
Answer: F
Let $S$ be the sum of all positive integers $n$ such that $n^2+12n-2007$ is a perfect square. Find the remainder when $S$ is divided by $1000.$

Options:
A. 460
B. 465
C. 462
D. 467
E. 468
F. 463
G. 461
H. 466
I. 464
J. 469
Answer: I
Let $F(x)$ be a polynomial such that $F(6) = 15$ and
$$\frac{F(3x)}{F(x+3)} = 9-\frac{48x+54}{x^2+5x+6}$$
for $x \in \mathbb{R}$ such that both sides are defined. Find $F(12)$ .

Options:
A. 78
B. 63
C. 60
D. 64
E. 66
F. 72
G. 54
H. 69
I. 75
J. 65
Answer: E
Find the number of sequences of positive integers $a_1, a_2, \dots , a_{14}$ such that $a_1 = 1,$ $11$ divides $a_{14},$ and for all integers $1 \le n \le 13,$ $a_{n+1}$ may be constructed by starting with either the integer $a_n$ or the integer $a_n+1,$ concatenating the digit $0$ at the right, followed by an instance of the integer $a_n$ at the right. (For example, $a_2$ may either be $101$ or $201.$ )

Options:
A. 747
B. 745
C. 741
D. 748
E. 749
F. 746
G. 740
H. 744
I. 743
J. 742
Answer: B
Identify the smallest natural number (positive integer) $p$ such that for every integer $e$ with $0 < e < 1110$, there exists an integer $b$ for which satisfies the expression \[ \frac{e}{1110} < \frac{b}{p} < \frac{m+1}{1111}. \]

Options:
A. \boxed{2220}
B. \boxed{2219}
C. \boxed{2226}
D. \boxed{2221}
E. \boxed{2227}
F. \boxed{2224}
G. \boxed{2223}
H. \boxed{2225}
I. \boxed{2222}
J. \boxed{2228}
Answer: D
Given positive integers $y$, define the sequence $z(y)$ by the following rules: $z(1) = 1$, $z(2y) = z(y)$, and $z(2y+1) = (-1)^y z(y)$. Find the value of \[ \sum_{y=1}^{4727} z(y) z(y+2). \]

Options:
A. -5
B. -4
C. -3
D. 0
E. 2
F. -1
G. -7
H. -2
I. -6
J. 1
Answer: F
Given a positive integer $k$, define the function $h_k(y) = \prod_{{k=1}}^k \cos(ky)$.Determine the smallest $k$ such that the absolute value of the second derivative at zero satisfies $|h_k''(0)| > 86707$.

Options:
A. 60
B. 67
C. 61
D. 68
E. 62
F. 66
G. 69
H. 63
I. 65
J. 64
Answer: J
Consider a function $t(y)$ defined for all positive real numbers $y$ as follows: \[ t(y) = \lim_{q \to 0} \left((y+1)^{q+1} - y^{q+1}\right)^{{1}/{q}}. \] Determine the value of $\lim_{y \to \infty} \frac{t(y)}{y}$.

Options:
A. e^{2/3}
B. e
C. e^{1/2}
D. e^{3/4}
E. e^{4/3}
F. e^{3/2}
G. e^{5/4}
H. e^2
I. e^{1/3}
J. e^{1/4}
Answer: B