MikaStars39/nano-eval huggingface dataset:
Solve the integral:
$$
2 \cdot \int \sin(-2 \cdot x)^5 \cdot \cos(2 \cdot x)^2 \, dx
$$

Options:
A. C + \frac{1}{3} \cdot (\cos(2 \cdot x))^3 + \frac{1}{7} \cdot (\cos(2 \cdot x))^7 - \frac{1}{4} \cdot (\cos(2 \cdot x))^5
B. C + \frac{1}{3} \cdot (\cos(2 \cdot x))^3 + \frac{1}{7} \cdot (\cos(2 \cdot x))^7 - \frac{1}{5} \cdot (\cos(2 \cdot x))^5
C. C + \frac{1}{3} \cdot (\cos(2 \cdot x))^3 + \frac{1}{7} \cdot (\cos(2 \cdot x))^7 - \frac{3}{5} \cdot (\cos(2 \cdot x))^5
D. C + \frac{1}{3} \cdot (\cos(2 \cdot x))^3 + \frac{1}{7} \cdot (\cos(2 \cdot x))^7 - \frac{1}{6} \cdot (\cos(2 \cdot x))^5
E. C + \frac{1}{3} \cdot (\cos(2 \cdot x))^3 + \frac{1}{7} \cdot (\cos(2 \cdot x))^7 - \frac{1}{2} \cdot (\cos(2 \cdot x))^5
F. C + \frac{1}{3} \cdot (\cos(2 \cdot x))^3 + \frac{1}{7} \cdot (\cos(2 \cdot x))^7 - \frac{1}{8} \cdot (\cos(2 \cdot x))^5
G. C + \frac{1}{3} \cdot (\cos(2 \cdot x))^3 + \frac{1}{7} \cdot (\cos(2 \cdot x))^7 - \frac{4}{5} \cdot (\cos(2 \cdot x))^5
H. C + \frac{1}{3} \cdot (\cos(2 \cdot x))^3 + \frac{1}{7} \cdot (\cos(2 \cdot x))^7 - \frac{2}{5} \cdot (\cos(2 \cdot x))^5
I. C + \frac{1}{3} \cdot (\cos(2 \cdot x))^3 + \frac{1}{7} \cdot (\cos(2 \cdot x))^7 - \frac{1}{3} \cdot (\cos(2 \cdot x))^5
J. C + \frac{1}{3} \cdot (\cos(2 \cdot x))^3 + \frac{1}{7} \cdot (\cos(2 \cdot x))^7 - \frac{1}{10} \cdot (\cos(2 \cdot x))^5
Answer: H
Compute the integral:
$$
\int \frac{ 1 }{ (x-3) \cdot \sqrt{10 \cdot x-24-x^2} } \, dx
$$

Options:
A. -\frac{2}{\sqrt{3}}\cdot\arctan\left(\frac{\sqrt{-x^2+10\cdot x-24}}{\sqrt{3}\cdot x-6\cdot\sqrt{3}}\right)+C
B. -\frac{2}{\sqrt{3}}\cdot\arctan\left(\frac{\sqrt{-x^2+10\cdot x-24}}{\sqrt{3}\cdot x-1\cdot\sqrt{3}}\right)+C
C. -\frac{2}{\sqrt{3}}\cdot\arctan\left(\frac{\sqrt{-x^2+10\cdot x-24}}{\sqrt{3}\cdot x-7\cdot\sqrt{3}}\right)+C
D. -\frac{2}{\sqrt{3}}\cdot\arctan\left(\frac{\sqrt{-x^2+10\cdot x-24}}{\sqrt{3}\cdot x-9\cdot\sqrt{3}}\right)+C
E. -\frac{2}{\sqrt{3}}\cdot\arctan\left(\frac{\sqrt{-x^2+10\cdot x-24}}{\sqrt{3}\cdot x-8\cdot\sqrt{3}}\right)+C
F. -\frac{2}{\sqrt{3}}\cdot\arctan\left(\frac{\sqrt{-x^2+10\cdot x-24}}{\sqrt{3}\cdot x+4\cdot\sqrt{3}}\right)+C
G. -\frac{2}{\sqrt{3}}\cdot\arctan\left(\frac{\sqrt{-x^2+10\cdot x-24}}{\sqrt{3}\cdot x-5\cdot\sqrt{3}}\right)+C
H. -\frac{2}{\sqrt{3}}\cdot\arctan\left(\frac{\sqrt{-x^2+10\cdot x-24}}{\sqrt{3}\cdot x-4\cdot\sqrt{3}}\right)+C
I. -\frac{2}{\sqrt{3}}\cdot\arctan\left(\frac{\sqrt{-x^2+10\cdot x-24}}{\sqrt{3}\cdot x-2\cdot\sqrt{3}}\right)+C
J. -\frac{2}{\sqrt{3}}\cdot\arctan\left(\frac{\sqrt{-x^2+10\cdot x-24}}{\sqrt{3}\cdot x-3\cdot\sqrt{3}}\right)+C
Answer: H
Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{0}^\pi \sin\left(\frac{ 1 }{ 2 } \cdot x\right) \cdot \cos\left(\frac{ 1 }{ 4 } \cdot x\right) \, dx
$$

Options:
A. 8-2*sqrt(2)/9
B. 8-2*sqrt(2)/12
C. 8-2*sqrt(2)/6
D. 8-2*sqrt(2)/8
E. 8-2*sqrt(2)/4
F. 8-2*sqrt(2)/3
G. 8-2*sqrt(2)/7
H. 8-2*sqrt(2)/10
I. 8-2*sqrt(2)/5
J. 8-2*sqrt(2)/2
Answer: F
Evaluate the integral by choosing the order of integration:
$$
\int_{0}^1 \int_{1}^2 \left(\frac{ y }{ x+y^2 }\right) \, dy \, dx
$$

Options:
A. ln(25*sqrt(5)/128)
B. ln(25*sqrt(5)/32)
C. ln(25*sqrt(5)/160)
D. ln(25*sqrt(5)/48)
E. ln(25*sqrt(5)/16)
F. ln(25*sqrt(5)/80)
G. ln(25*sqrt(5)/96)
H. ln(25*sqrt(5)/24)
I. ln(25*sqrt(5)/40)
J. ln(25*sqrt(5)/64)
Answer: B
Electrical power $P$ is given by $P = \frac{ V^2 }{ R }$, where $V$ is the voltage and $R$ is the resistance. Approximate the maximum percentage error in calculating power if $120$ V is applied to a $2000$-ohm resistor and the possible percent errors in measuring $V$ and $R$ are $3$% and $4$% respectively.

Options:
A. 0.1
B. 0.14
C. 0.15
D. 0.18
E. 0.13
F. 0.12
G. 0.16
H. 0.17
I. 0.11
J. 0.19
Answer: A
Determine the count of positive integers $J$ that fulfill all of these criteria: \begin{{enumerate}} \item[(i)] $J$ is a multiple of 404000. \item[(ii)] The number of decimal digits in $J$ does not exceed 404000. \item[(iii)] When written in decimal notation, $J$ consists of a sequence of ones immediately followed by a sequence of zeros (either sequence may be empty). \end{{enumerate}}

Options:
A. 20401293001
B. 20401293000
C. 20401293005
D. 20401293006
E. 20401293009
F. 20401293007
G. 20401293003
H. 20401293004
I. 20401293002
J. 20401293008
Answer: D
Compute the integral:
$$
\int \frac{ -1 }{ 3 \cdot \sin\left(\frac{ x }{ 3 }\right)^6 } \, dx
$$

Options:
A. frac{cosleft(frac{x}{3}right)}{5cdot sinleft(frac{x}{3}right)^5}-frac{4}{15}cdotleft(frac{cosleft(frac{x}{3}right)}{sinleft(frac{x}{3}right)^3}+2cdot cotleft(frac{x}{3}right)right)-C
B. frac{cosleft(frac{x}{3}right)}{5cdot sinleft(frac{x}{3}right)^5}+frac{4}{15}cdotleft(frac{cosleft(frac{x}{3}right)}{sinleft(frac{x}{3}right)^3}-2cdot cotleft(frac{x}{3}right)right)+C
C. frac{cosleft(frac{x}{3}right)}{5cdot sinleft(frac{x}{3}right)^5}+frac{4}{15}cdotleft(frac{cosleft(frac{x}{3}right)}{sinleft(frac{x}{3}right)^3}+2cdot cotleft(frac{x}{3}right)right)+C
D. frac{cosleft(frac{x}{3}right)}{5cdot sinleft(frac{x}{3}right)^5}+frac{4}{15}cdotleft(-frac{cosleft(frac{x}{3}right)}{sinleft(frac{x}{3}right)^3}+2cdot cotleft(frac{x}{3}right)right)+C
E. frac{cosleft(frac{x}{3}right)}{5cdot sinleft(frac{x}{3}right)^5}-frac{4}{15}cdotleft(frac{cosleft(frac{x}{3}right)}{sinleft(frac{x}{3}right)^3}-2cdot cotleft(frac{x}{3}right)right)-C
F. frac{cosleft(frac{x}{3}right)}{5cdot sinleft(frac{x}{3}right)^5}+frac{4}{15}cdotleft(frac{cosleft(frac{x}{3}right)}{sinleft(frac{x}{3}right)^3}-2cdot cotleft(frac{x}{3}right)right)-C
G. frac{cosleft(frac{x}{3}right)}{5cdot sinleft(frac{x}{3}right)^5}-frac{4}{15}cdotleft(frac{cosleft(frac{x}{3}right)}{sinleft(frac{x}{3}right)^3}+2cdot cotleft(frac{x}{3}right)right)+2C
H. frac{cosleft(frac{x}{3}right)}{5cdot sinleft(frac{x}{3}right)^5}-frac{4}{15}cdotleft(-frac{cosleft(frac{x}{3}right)}{sinleft(frac{x}{3}right)^3}-2cdot cotleft(frac{x}{3}right)right)+C
I. frac{cosleft(frac{x}{3}right)}{5cdot sinleft(frac{x}{3}right)^5}-frac{4}{15}cdotleft(frac{cosleft(frac{x}{3}right)}{sinleft(frac{x}{3}right)^3}-2cdot cotleft(frac{x}{3}right)right)+C
J. frac{cosleft(frac{x}{3}right)}{5cdot sinleft(frac{x}{3}right)^5}-frac{4}{15}cdotleft(frac{cosleft(frac{x}{3}right)}{sinleft(frac{x}{3}right)^3}+2cdot cotleft(frac{x}{3}right)right)+C
Answer: H
Evaluate the integral:
$$
I = \int 3 \cdot \ln\left(\sqrt{2-x}+\sqrt{2+x}\right) \, dx
$$

Options:
A. 3\cdot x\cdot\ln\left(\sqrt{2-x}+\sqrt{2+x}\right)-\frac{3}{2}\cdot\left((C+x)-2\cdot\arcsin\left(\frac{x}{2}\right)\right)
B. 3\cdot x\cdot\ln\left(\sqrt{2-x}+\sqrt{2+x}\right)-\frac{3}{2}\cdot\left((C+x)+2\cdot\arcsin\left(\frac{x}{2}\right)\right)
C. 3\cdot x\cdot\ln\left(\sqrt{2-x}+\sqrt{2+x}\right)-\frac{3}{2}\cdot\left((C-x)+2\cdot\arcsin\left(\frac{x}{2}\right)\right)
D. 3\cdot x\cdot\ln\left(\sqrt{2-x}+\sqrt{2+x}\right)-\frac{3}{2}\cdot\left((C-x)-2\cdot\arcsin\left(\frac{x}{2}\right)\right)
Answer: A
Kelvin the Frog's favorite quadratic function is $f(x) = x^2- 42x + 1337$ . His favorite quartic function is $f(f(x))$ , and in general his favorite function of degree $2^n$ is $f^n(x)$ . Compute the sum of the roots of Kelvin's favorite $16$ th degree function.

Options:
A. 1344
B. 840
C. 336
D. 168
E. 672
Answer: C
Compute the integral:
$$
3 \cdot \int x^{-8} \cdot \left(9+x^2\right)^{\frac{ 1 }{ 2 }} \, dx
$$

Options:
A. C+\frac{2}{1215}\cdot\left(1+\frac{9}{x^2}\right)^2\cdot\sqrt{1+\frac{9}{x^2}}-\frac{1}{729}\cdot\left(1+\frac{9}{x^2}\right)^2\cdot\sqrt{1+\frac{9}{x^2}}-\frac{1}{1701}\cdot\left(1+\frac{9}{x^2}\right)^4\cdot\sqrt{1+\frac{9}{x^2}}
B. C+\frac{2}{1215}\cdot\left(1+\frac{9}{x^2}\right)^2\cdot\sqrt{1+\frac{9}{x^2}}-\frac{1}{729}\cdot\left(1+\frac{9}{x^2}\right)\cdot\sqrt{1+\frac{9}{x^2}}-\frac{1}{1701}\cdot\left(1+\frac{9}{x^2}\right)^3\cdot\sqrt{1+\frac{9}{x^2}}
C. C+\frac{2}{1215}\cdot\left(1+\frac{9}{x^2}\right)^2\cdot\sqrt{1+\frac{9}{x^2}}-\frac{1}{729}\cdot\left(1+\frac{9}{x^2}\right)^2\cdot\sqrt{1+\frac{9}{x^2}}-\frac{1}{1701}\cdot\left(1+\frac{9}{x^2}\right)^3\cdot\sqrt{1+\frac{9}{x^2}}
D. C+\frac{2}{1215}\cdot\left(1+\frac{9}{x^2}\right)^2\cdot\sqrt{1+\frac{9}{x^2}}-\frac{1}{729}\cdot\left(1+\frac{9}{x^2}\right)^2\cdot\sqrt{1+\frac{9}{x^2}}-\frac{1}{1701}\cdot\left(1+\frac{9}{x^2}\right)^2\cdot\sqrt{1+\frac{9}{x^2}}
E. C+\frac{2}{1215}\cdot\left(1+\frac{9}{x^2}\right)^2\cdot\sqrt{1+\frac{9}{x^2}}-\frac{1}{729}\cdot\left(1+\frac{9}{x^2}\right)\cdot\sqrt{1+\frac{9}{x^2}}-\frac{1}{1701}\cdot\left(1+\frac{9}{x^2}\right)^2\cdot\sqrt{1+\frac{9}{x^2}}
Answer: B
Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that \[ f(x) + f\left( 1 - \frac{1}{x} \right) = \arctan x \] for all real $x \neq 0$. (As usual, $y = \arctan x$ means $-\pi/2 < y < \pi/2$ and $\tan y = x$.) Find \[ \int_0^1 f(x)\,dx. \]

Options:
A. \frac{5\pi}{12}
B. \frac{3\pi}{8}
C. \frac{5\pi}{16}
D. \frac{7\pi}{16}
E. \frac{\pi}{8}
F. \frac{\pi}{4}
Answer: B
Find the zeros of the function $f(x) = x^3 - (3 + \sqrt{3}) \cdot x + 3$.

Options:
A. x_1=-\sqrt{3}, x_2=\frac{\sqrt{3}-\sqrt{3+4\cdot\sqrt{3}}}{2}, x_3=\frac{\sqrt{3}+\sqrt{3+4\cdot\sqrt{3}}}{2}
B. x_1=-\sqrt{3}, x_2=\frac{\sqrt{3}-\sqrt{3-4\cdot\sqrt{3}}}{2}, x_3=\frac{\sqrt{3}+\sqrt{3-4\cdot\sqrt{3}}}{2}
C. x_1=-\sqrt{3}, x_2=\frac{\sqrt{3}+\sqrt{3-4\cdot\sqrt{3}}}{2}, x_3=\frac{\sqrt{3}-\sqrt{3-4\cdot\sqrt{3}}}{2}
D. x_1=\sqrt{3}, x_2=\frac{-\sqrt{3}+\sqrt{3-4\cdot\sqrt{3}}}{2}, x_3=\frac{-\sqrt{3}-\sqrt{3-4\cdot\sqrt{3}}}{2}
E. x_1=\sqrt{3}, x_2=\frac{-\sqrt{3}-\sqrt{3+4\cdot\sqrt{3}}}{2}, x_3=\frac{-\sqrt{3}+\sqrt{3+4\cdot\sqrt{3}}}{2}
F. x_1=-\sqrt{3}, x_2=\frac{\sqrt{3}+\sqrt{3+4\cdot\sqrt{3}}}{2}, x_3=\frac{\sqrt{3}-\sqrt{3+4\cdot\sqrt{3}}}{2}
Answer: E
$A$ , $B$ , and $C$ are chosen randomly on a unit circle. Let $H$ be the orthocenter of triangle $ABC$ , and let region $R$ be the locus of $H$ . The area of region $R$ can be expressed as $n\pi$ , for some integer $n$ . Compute $n$ .

Options:
A. 6
B. 11
C. 7
D. 12
E. 9
F. 8
G. 10
Answer: E
Find the Fourier series of the periodic function $f(x) = x^2$ in the interval $-2 \cdot \pi \leq x < 2 \cdot \pi$ if $f(x) = f(x + 4 \cdot \pi)$.

Options:
A. 4\cdot\pi^2/3 + \sum_{n=1}^\infty\left(\frac{16\cdot(-1)^{n+4}}{n^2}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)
B. 4\cdot\pi^2/3 + \sum_{n=1}^\infty\left(\frac{16\cdot(-1)^{n-1}}{n^2}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)
C. 4\cdot\pi^2/3 + \sum_{n=1}^\infty\left(\frac{16\cdot(-1)^{n+7}}{n^2}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)
D. 4\cdot\pi^2/3 + \sum_{n=1}^\infty\left(\frac{16\cdot(-1)^n}{n^2}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)
E. 4\cdot\pi^2/3 + \sum_{n=1}^\infty\left(\frac{16\cdot(-1)^{n+1}}{n^2}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)
F. 4\cdot\pi^2/3 + \sum_{n=1}^\infty\left(\frac{16\cdot(-1)^{n+3}}{n^2}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)
G. 4\cdot\pi^2/3 + \sum_{n=1}^\infty\left(\frac{16\cdot(-1)^{n+2}}{n^2}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)
H. 4\cdot\pi^2/3 + \sum_{n=1}^\infty\left(\frac{16\cdot(-1)^{n+5}}{n^2}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)
I. 4\cdot\pi^2/3 + \sum_{n=1}^\infty\left(\frac{16\cdot(-1)^{n+6}}{n^2}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)
J. 4\cdot\pi^2/3 + \sum_{n=1}^\infty\left(\frac{16\cdot(-1)^{n+8}}{n^2}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)
Answer: D
Two $5$ -digit numbers are called "responsible" if they are:
$$\begin{align*} &\text {i. In form of abcde and fghij such that fghij = 2(abcde)}\\ &\text {ii. all ten digits, a through j are all distinct.}\\ &\text {iii.} a + b + c + d + e + f + g + h + i + j = 45\end{align*}$$
If two "responsible" numbers are small as possible, what is the sum of the three middle digits of $\text {abcde}$ and last two digits on the $\text {fghij}$ ? That is, $b + c + d + i + j$ .

Options:
A. 22
B. 21
C. 20
D. 23
E. 24
F. 26
G. 25
H. 27
Answer: A
Simplify the radical expression:
$$
\frac{ 4 \cdot \sqrt{2 \cdot n} }{ \sqrt{16 \cdot n^4} }
$$

Options:
A. \frac{\sqrt{2}}{n^{\frac{1}{4}}}
B. \frac{\sqrt{2}}{n^{\frac{1}{3}}}
C. \frac{\sqrt{2}}{n^{\frac{4}{2}}}
D. \frac{\sqrt{2}}{n^{\frac{7}{2}}}
E. \frac{\sqrt{2}}{n^{\frac{5}{2}}}
F. \frac{\sqrt{2}}{n^{\frac{2}{2}}}
G. \frac{\sqrt{2}}{n^{\frac{3}{2}}}
H. \frac{\sqrt{2}}{n^{\frac{1}{2}}}
Answer: G
Compute the integral:
$$
\int \frac{ 1 }{ \sin(x)^6 } \, dx
$$

Options:
A. -\frac{\cos(x)}{5\cdot\sin(x)^5}+\frac{4}{5}\cdot\left(-\frac{\cos(x)}{3\cdot\sin(x)^3}-\frac{1}{3}\cdot\cot(x)\right)
B. -\frac{\cos(x)}{5\cdot\sin(x)^5}+\frac{4}{5}\cdot\left(-\frac{\cos(x)}{3\cdot\sin(x)^3}+\frac{1}{3}\cdot\cot(x)\right)
C. -\frac{\cos(x)}{5\cdot\sin(x)^5}+\frac{4}{5}\cdot\left(-\frac{\cos(x)}{3\cdot\sin(x)^3}+\frac{2}{3}\cdot\cot(x)\right)
D. -\frac{\cos(x)}{5\cdot\sin(x)^5}+\frac{4}{5}\cdot\left(-\frac{\cos(x)}{3\cdot\sin(x)^3}-\frac{2}{3}\cdot\cot(x)\right)
E. -\frac{\cos(x)}{5\cdot\sin(x)^5}+\frac{4}{5}\cdot\left(\frac{\cos(x)}{3\cdot\sin(x)^3}+\frac{2}{3}\cdot\cot(x)\right)
F. -\frac{\cos(x)}{5\cdot\sin(x)^5}+\frac{4}{5}\cdot\left(-\frac{\cos(x)}{3\cdot\sin(x)^3}-\frac{3}{3}\cdot\cot(x)\right)
G. -\frac{\cos(x)}{5\cdot\sin(x)^5}+\frac{4}{5}\cdot\left(\frac{\cos(x)}{3\cdot\sin(x)^3}-\frac{2}{3}\cdot\cot(x)\right)
H. -\frac{\cos(x)}{5\cdot\sin(x)^5}+\frac{4}{5}\cdot\left(-\frac{\cos(x)}{3\cdot\sin(x)^3}-\frac{1}{2}\cdot\cot(x)\right)
Answer: D
Find the remainder when $11^{2005}$ is divided by $1000$ .

Options:
A. 151
B. 111
C. 351
D. 251
E. 451
F. 211
G. 51
H. 101
I. 161
Answer: G
Let $C_1$ and $C_2$ be circles whose centers are 10 units apart, and whose radii are 1 and 3. Find the absolute difference between the inner and outer radius of the locus of all points $M$ for which there exists points $X$ on $C_1$ and $Y$ on $C_2$ such that $M$ is the midpoint of the line segment $XY$.

Options:
A. 2
B. 3
C. 10
D. 6
E. 1
F. 8
G. 7
H. 5
I. 4
Answer: A
Leaves of deciduous trees fall in an exponential rate during autumn. A large maple has 10,000 leaves and the wind starts blowing. If there are 8,000 leaves left after 3 hours, how many will be left after 4 hours?

Options:
A. 7200
B. 7600
C. 7400
D. 7350
E. 7426
F. 7475
G. 7300
H. 7450
I. 7500
Answer: E
Define the function $\omega(n) = \sum^k_{i=1} \left (a_i - a_ia_{i+1} \right)$ for all positive integers $n,$ given that $n$ can be written as $a_1a_2a_3 \dots a_k$ in base $2$ where $a_1 \ne 0,$ and $a_{k+1}$ is defined as $a_{1}.$ For example,
$$$\omega(26) = \omega(11010_2) = (1-1\cdot1) + (1-1\cdot 0)+(0-0\cdot 1) + (1-1\cdot 0)+(0-0\cdot 1)=2.$$$
Find the last three digits of the sum
$$$\sum^{4100}_{i=1} \omega(i).$$$

Options:
A. 277
B. 276
C. 271
D. 270
E. 272
F. 273
G. 278
H. 275
I. 279
J. 274
Answer: C
For positive integers $n$ , let $f(n)$ denote the number of integers $1 \leq a \leq 130$ for which there exists some integer $b$ such that $a^b-n$ is divisible by $131$ , and let $g(n)$ denote the sum of all such $a$ . Find the remainder when
$$$\sum_{n = 1}^{130} [f(n) \cdot g(n)]$$$
is divided by $131$ .

Options:
A. 56
B. 58
C. 52
D. 55
E. 50
F. 53
G. 59
H. 54
I. 57
J. 51
Answer: H
Let $x$ and $y$ be real numbers which differ in absolute value and satisfy the equations:
$$$x^3=13x+3y$$$
$$$y^3=3x+13y$$$
Find $(x^2-y^2)^2$

Options:
A. 131
B. 132
C. 153
D. 144
E. 136
F. 145
G. 134
H. 133
I. 135
J. 143
Answer: H
Find the remainder when $3^{3^{3^3}}$ is divided by $1000$ .

Options:
A. 380
B. 388
C. 387
D. 386
E. 385
F. 382
G. 383
H. 381
I. 389
J. 384
Answer: C
Find the minimum value of the product of $xyz$ , where $x,y,z$ are positive integers such that
$$x^2-12y^2-35z^2=6$$

Options:
A. 780
B. 783
C. 784
D. 795
E. 794
F. 798
G. 786
H. 790
I. 789
J. 792
Answer: J
$P$ is the probability that if you flip a fair coin, $20$ heads will occur before $19$ tails. If $P=\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find the remainder when $m+n$ is divided by $1000$ .

Options:
A. 149
B. 159
C. 148
D. 150
E. 151
F. 153
G. 155
H. 147
I. 152
J. 157
Answer: A
The function $f : N \to Z$ is defined by $f(1) = 1$ and
$$$f(n) = \sum_{d|\phi (n)}f(d).$$$
Find $f(2024)$ . (Note that $ \phi (n)$ is the number of positive integers less than or equal to $n$ that are relative prime to it).

Options:
A. 874
B. 876
C. 875
D. 877
E. 882
F. 883
G. 880
H. 881
I. 878
J. 879
Answer: G
Complex numbers $x$ , $y$ , and $z$ are such that:
$$ \begin{cases} \vspace{0.1cm} xyz - 4(x + y + z) = 5. \\ \vspace{0.15cm} xy + yz + zx = 7. \\ (x^2 + 2)(y^2 + 2)(z^2 + 2) = 246. \end{cases} $$
Find the sum of all possible values of $(x^2 + 6)(y^2 + 6)(z^2 + 6)$ .

Options:
A. 604
B. 603
C. 612
D. 610
E. 605
F. 602
G. 608
H. 606
I. 600
J. 616
Answer: A
An infinite set of balls numbered $1,2,3, \dots $ are placed in an infinite set of boxes numbered $1,2,3, \dots $ by repeating the following:
1. Flip a fair coin repeatedly until it lands on heads.
2. If the head was on flip $k,$ place the smallest-numbered ball not yet in a box into the $k$ th smallest-numbered box not yet with a ball.
The expected value of box $6$ 's ball is $\dfrac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a+b.$

Options:
A. 152
B. 143
C. 147
D. 157
E. 145
F. 159
G. 153
H. 151
I. 149
J. 155
Answer: I
A bug is on a vertex of a regular icosahedron (a polyhedron with 20 triangular faces.) Every second, it can either move to one of the adjacent vertices, or teleport to the opposite vertex (i.e. the unique vertex of the icosahedron such that the distance traveled is greatest.) However, he can teleport at most twice before exhausting himself. If $M$ is the amount of ways he can move, such that he is at the original vertex after exactly $7$ seconds, compute the last $3$ nonzero digits of $M$ (Your answer should not contain any 0s.)

Options:
A. 268
B. 261
C. 253
D. 262
E. 259
F. 256
G. 267
H. 258
I. 265
J. 264
Answer: I
Suppose $S_n$ is the set of positive divisors of $n$ , and denote $|X|$ as the number of elements in a set $X$ . Let $\xi$ be the set of positive integers $n$ where $|S_n| = 2m$ is even, and $S_n$ can be partitioned evenly into pairs $\{a_i, b_i\}$ for integers $1 \le i \le m$ such that the following conditions hold:
$\bullet$ $a_i$ and $b_i$ are relatively prime for all integers $1 \le i \le m$
$\bullet$ There exists a $j$ where $6$ divides $a^2_j+ b^2_j+ 1$
$\bullet$ $|S_n| \ge 20.$
Determine the number of positive divisors $d | 24!$ such that $d \in \xi$ .

Options:
A. 64
B. 63
C. 61
D. 68
E. 69
F. 60
G. 67
H. 62
I. 66
J. 65
Answer: A
Find the number of positive integers $n$ such that $n<10^6$ , $n$ only contains the digits $9, 1,$ and $0$ , and $n$ is divisible by $7$ .

Options:
A. 103
B. 108
C. 106
D. 101
E. 109
F. 100
G. 105
H. 102
I. 104
J. 107
Answer: I
In Zuminglish, all words consist only of the letters $M$ , $O$ , and $P$ . As in English, $O$ is said to be a vowel and $M$ and $P$ are consonants. A string of $M$ 's, $O$ 's, and $P$ 's is a word in Zuminglish if and only if between any two $O$ 's there appear at least two consonants. Let $N$ denote the number of $10$ -letter Zuminglish words. Determine the remainder obtained when $N$ is divided by $1000$ .

Options:
A. 939
B. 934
C. 937
D. 936
E. 931
F. 935
G. 938
H. 930
I. 932
J. 933
Answer: D
At the Outdian Institute of Technology, Maths is so important that there are $511$ maths buildings, called Maths Building 1, Maths Building 2, etc. There exists pathways between these buildings such that for any positive integer $k \leq 255$ , there exists a pathway between Maths Building $k$ and Maths Building $2k$ and there exists a pathway between Maths Building $k$ and Maths Building $2k + 1$ . One day, Brahmagupta takes a stroll starting from Maths Building $1$ . Every minute, Brahmagupta walks along one of the pathways connected to his building at random to another maths building, and stops when he reaches Maths Building $1$ again. What is the expected amount of time of Brahmagupta's stroll, in minutes?

Options:
A. 255
B. 512
C. 256
D. 510
E. 509
F. 515
G. 514
H. 511
I. 254
J. 513
Answer: D
Iggy has a garden with length 234 feet and width 123 feet. Iggy decides to create a sidewalk of constant width around the garden such that the area of the sidewalk is equal to the area of the garden. What is the largest possible distance, in feet, between any two points on the sidewalk?

Options:
A. 354
B. 358
C. 353
D. 356
E. 355
F. 351
G. 350
H. 359
I. 352
J. 357
Answer: J
Find the largest value of $x$ less than $1000$ such that the remainder when $x^2$ is divided by $11^3$ is $48.$
Note the problem text has been slightly modified since test release. It does not affect the solution or the answer.

Options:
A. 971
B. 969
C. 979
D. 980
E. 970
F. 981
G. 968
H. 978
I. 972
J. 973
Answer: F
A right circular cone is completely contained in a sphere. Let $p$ be the maximum possible ratio of the volume of the cone to the volume of the sphere and let $q$ be the maximum possible ratio of the lateral surface area of the cone to the surface area of the sphere. What is the value of $\dfrac{p^7}{q^{16}}$ ? Note: the lateral surface area of a cone is the surface area of the cone excluding the base, and the values of $p$ and $q$ are not necessarily achieved with the same cone.

Options:
A. 861
B. 867
C. 862
D. 864
E. 869
F. 866
G. 865
H. 868
I. 863
J. 860
Answer: D
For all positive integers $n$ , let $f(n)=\underbrace{111...11}_{n~\text{times}}.$ Find the remainder when $\sum\limits_{n=1}^{2021}(nf(n))$ is divided by 1000.

Options:
A. 336
B. 337
C. 335
D. 332
E. 339
F. 331
G. 333
H. 338
I. 330
J. 334
Answer: F
Consider the equation
$$$\left \lfloor \frac{3n}{5} \right \rfloor^2 + \left \lfloor \frac{4n}{5} \right \rfloor^2 = n^2 - k$$$
where $n, k$ are positive integers. How many distinct values can $k$ take on if $k \leq 1200$ ?

Options:
A. 655
B. 662
C. 661
D. 659
E. 658
F. 657
G. 663
H. 660
I. 656
J. 654
Answer: H
Let $S = \{5^ k | k \in Z, 0 \le k \le 2004\}$ . Given that $5^{2004} = 5443 ...0625$ has $1401$ digits, how many elements of $S$ begin with the digit $1$ ?

Options:
A. 605
B. 600
C. 604
D. 607
E. 609
F. 606
G. 602
H. 603
I. 601
J. 608
Answer: C
$ABC$ is a scalene triangle. The circle with diameter $\overline{AB}$ intersects $\overline{BC}$ at $D$ , and $E$ is the foot of the altitude from $C$ . $P$ is the intersection of $\overline{AD}$ and $\overline{CE}$ . Given that $AP = 136$ , $BP = 80$ , and $CP = 26$ , determine the circumradius of $ABC$ .

Options:
A. 84.25
B. 84.75
C. 85
D. 84.2
E. 84
F. 83
G. 84.3
H. 86
I. 87
J. 84.5
Answer: C
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\left[ X \right] = 1$, $E\left[ X^2 \right] = 2$, and $E \left[ X^3 \right] = 5$. (Here $E\left[ y \right]$ denotes the expectation of the random variable $Y$.) Determine the smallest possible value of the probability of the event $X=0$.

Options:
A. \frac{1}{2}
B. \frac{1}{6}
C. \frac{1}{8}
D. \frac{1}{9}
E. \frac{1}{3}
F. \frac{1}{5}
G. \frac{1}{10}
H. \frac{2}{5}
I. \frac{1}{7}
J. \frac{1}{4}
Answer: E
Identify the smallest natural number (positive integer) $w$ such that for every integer $s$ with $0 < s < 1979$, there exists an integer $x$ for which satisfies the expression \[ \frac{s}{1979} < \frac{x}{w} < \frac{m+1}{1980}. \]

Options:
A. 3958
B. 3961
C. 3952
D. 3959
E. 3960
F. 3953
G. 3954
H. 3956
I. 3955
J. 3957
Answer: D
For any positive integer $n$, let $\langle n\rangle$ denote the closest integer to $\sqrt{n}$. Evaluate \[\sum_{n=1}^\infty \frac{2^{\langle n\rangle}+2^{-\langle n\rangle}} {2^n}.\]

Options:
A. 2.90625
B. 2.96875
C. 2.75
D. 2.875
E. 2.8125
F. 2.625
G. 2.5
H. 2.9375
I. 2.953125
J. 3
Answer: J
Consider the set $A$ containing all integers $h$ between 1 and 2021 (inclusive) that are coprime to 2021. Define a function $S(v)$ for non-negative integers $v$ as follows: \[ S(v) = \sum_{h \in A} h^v. \] What is the largest value of $v$ below 2021 for which $S(v)$ is divisible by 2021?

Options:
A. 2013
B. 2016
C. 2018
D. 2015
E. 2017
F. 2019
G. 2014
H. 2011
I. 2012
J. 2020
Answer: J
Consider a sequence of independent random variables $K_1, K_2, \dots$ defined for $0 \leq p \leq 1/2$, where each $K_i$ follows the distribution: \[ K_i = \begin{cases} 1 & \text{with probability $p$,} \\ -1 & \text{with probability $p$,} \\ 0 & \text{with probability $1-2p$.} \end{cases} \] Let $q$ be a positive integer and $v, w_1, \dots, w_q$ be integers. Define $P(v, w_1, \dots, w_q)$ as the probability of the event $w_1 K_1 + \cdots + w_q K_q = v$. What is the largest value of $p$ that ensures \[ P(0, w_1, \dots, w_q) \geq P(v, w_1, \dots, w_q) \] holds true for any choice of positive integer $q$ and integers $v, w_1, \dots, w_q$?

Options:
A. \frac{1}{7}
B. \frac{1}{12}
C. \frac{1}{8}
D. \frac{1}{6}
E. \frac{1}{10}
F. \frac{1}{9}
G. \frac{1}{2}
H. \frac{1}{5}
I. \frac{1}{3}
J. \frac{1}{4}
Answer: J
Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \geq 0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2 + y^2 = 2^k$ for some integer $k \leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square.

Options:
A. 5n + 6
B. 5n + 1
C. 5n + 10
D. 5n + 4
E. 5n + 7
F. 5n + 5
G. 5n + 9
H. 5n + 2
I. 5n + 8
J. 5n + 3
Answer: B
Evaluate \begin{gather*} \int_0^\infty \left(x-\frac{x^3}{2}+\frac{x^5}{2\cdot 4}-\frac{x^7}{2\cdot 4\cdot 6}+\cdots\right) \\ \left(1+\frac{x^2}{2^2}+ \frac{x^4}{2^2\cdot 4^2}+\frac{x^6}{2^2\cdot 4^2 \cdot 6^2}+\cdots\right)\,dx. \end{gather*}

Options:
A. e^{2/3}
B. e^{1/5}
C. e^{1/6}
D. e^{1/3}
E. e^{1/4}
F. e^{1/8}
G. e^{1/2}
H. \sqrt{e}
I. e^{3/4}
J. e^{3/8}
Answer: H
Determine the maximum value of the sum \[ S = \sum_{n=1}^\infty \frac{n}{2^n} (a_1 a_2 \cdots a_n)^{1/n} \] over all sequences $a_1, a_2, a_3, \cdots$ of nonnegative real numbers satisfying \[ \sum_{k=1}^\infty a_k = 1. \]

Options:
A. \frac{2}{3}
B. \frac{1}{3}
C. \frac{1}{2}
D. \frac{7}{12}
E. \frac{3}{5}
F. \frac{3}{4}
G. \frac{1}{4}
H. \frac{5}{8}
I. \frac{5}{6}
J. \frac{4}{5}
Answer: A
Integrate the approximation $\sin(t) \approx t - \frac{ t^3 }{ 6 } + \frac{ t^5 }{ 120 } - \frac{ t^7 }{ 5040 } + \cdots$ evaluated at $\pi \cdot t$ to approximate $\int_{0}^1{\frac{ \sin(\pi \cdot t) }{ \pi \cdot t } d t}$

Options:
A. 0.58678689
B. 0.58678680
C. 0.58678681
D. 0.58678687
E. 0.58678686
F. 0.58678682
G. 0.58678685
H. 0.58678683
I. 0.58678684
J. 0.58678688
Answer: D
Compute the integral:
$$
\int \frac{ -\sin(2 \cdot x)^4 }{ \cos(2 \cdot x) } \, dx
$$

Options:
A. C + (1/3) * (sin(2 * x))^3 - sin(2 * x) + (1/2) * ln(|1 - sin(2 * x)| / |sin(2 * x) + 1|)
B. C + (1/3) * (sin(2 * x))^3 + sin(2 * x) + (1/2) * ln(|1 + sin(2 * x)| / |sin(2 * x) - 1|)
C. C + (1/3) * (sin(2 * x))^3 + sin(2 * x) + (1/2) * ln(|1 - sin(2 * x)| / |sin(2 * x) + 1|)
D. C + (1/3) * (sin(2 * x))^3 - sin(2 * x) - (1/2) * ln(|1 - sin(2 * x)| / |sin(2 * x) - 1|)
E. C + (1/3) * (sin(2 * x))^3 + sin(2 * x) - (1/2) * ln(|1 + sin(2 * x)| / |sin(2 * x) - 1|)
F. C + (1/3) * (sin(2 * x))^3 - sin(2 * x) + (1/2) * ln(|1 + sin(2 * x)| / |sin(2 * x) - 1|)
G. C + (1/3) * (sin(2 * x))^3 - sin(2 * x) - (1/2) * ln(|1 - sin(2 * x)| / |sin(2 * x) + 1|)
H. C + (1/3) * (sin(2 * x))^3 + sin(2 * x) - (1/2) * ln(|1 - sin(2 * x)| / |sin(2 * x) + 1|)
I. C + (1/3) * (sin(2 * x))^3 - sin(2 * x) - (1/2) * ln(|1 + sin(2 * x)| / |sin(2 * x) - 1|)
J. C + (1/3) * (sin(2 * x))^3 - sin(2 * x) - (1/2) * ln(|1 + sin(2 * x)| / |sin(2 * x) + 1|)
Answer: E
Divide the rational expressions:
$$
\frac{ q^2-16 }{ q^2+8 \cdot q+16 } \div \frac{ q^2+q-20 }{ q^2-q-20 }
$$

Options:
A. q+4/q-5
B. q-5/q+5
C. q-5/q-4
D. q+4/q-4
E. q-4/q+4
F. q+5/q-5
G. q+5/q+5
H. q-5/q-5
I. q-4/q-5
J. q+5/q-4
Answer: B
Compute the integral:
$$
8 \cdot \int \cot(-4 \cdot x)^5 \cdot \csc(4 \cdot x)^4 \, dx
$$

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

Options:
A. 3/4 * (C - sqrt(5 + 1/x^2 - 2/x) - ln(abs(1/x + sqrt(5 + 1/x^2 - 2/x) - 3)))
B. 3/4 * (C - sqrt(5 + 1/x^2 - 2/x) - ln(abs(1/x + sqrt(5 + 1/x^2 - 2/x) - 4)))
C. 3/4 * (C - sqrt(5 + 1/x^2 - 2/x) - ln(abs(1/x + sqrt(5 + 1/x^2 - 2/x) - 5)))
D. 3/4 * (C - sqrt(5 + 1/x^2 - 2/x) - ln(abs(1/x - sqrt(5 + 1/x^2 - 2/x) - 2)))
E. 3/4 * (C - sqrt(5 + 1/x^2 - 2/x) - ln(abs(1/x + sqrt(5 + 1/x^2 - 2/x) + 2)))
F. 3/4 * (C - sqrt(5 + 1/x^2 - 2/x) - ln(abs(1/x - sqrt(5 + 1/x^2 - 2/x) + 1)))
G. 3/4 * (C - sqrt(5 + 1/x^2 - 2/x) - ln(abs(1/x + sqrt(5 + 1/x^2 - 2/x) + 1)))
H. 3/4 * (C - sqrt(5 + 1/x^2 - 2/x) - ln(abs(1/x + sqrt(5 + 1/x^2 - 2/x) + 3)))
I. 3/4 * (C - sqrt(5 + 1/x^2 - 2/x) - ln(abs(1/x + sqrt(5 + 1/x^2 - 2/x) - 1)))
J. 3/4 * (C - sqrt(5 + 1/x^2 - 2/x) - ln(abs(1/x - sqrt(5 + 1/x^2 - 2/x) - 1)))
Answer: I
Evaluate $L=\lim_{P(x,y) \to P\left(1,\frac{ 1 }{ 2 }\right)}\left(f(x,y)\right)$ given $f(x,y) = \frac{ x^2 - 2 \cdot x^3 \cdot y - 2 \cdot x \cdot y^3 + y^2 }{ 1 + x + y - 2 \cdot x \cdot y - 2 \cdot x^2 \cdot y - 2 \cdot x \cdot y^2 }$.

Options:
A. 0.375
B. 0.46875
C. 0.484375
D. 0.25
E. 0.5
F. 0.45
G. 0.475
H. 0.75
I. 0.625
J. 0.4375
Answer: E
Consider a bank investment: The initial investment is $\$10000$. After $25$ years, the investment has tripled to $\$30000$. Use Newton's method to determine the interest rate if the interest was compounded annually.

Options:
A. 4.442
B. 4.392
C. 4.592
D. 4.492
E. 4.692
F. 4.482
G. 4.502
H. 4.582
I. 4.452
J. 4.542
Answer: D
Evaluate $I=\int_{0}^4 \int_{0}^x \int_{0}^{x+y} \left(3 \cdot e^x \cdot (y+2 \cdot z)\right) dz dy dx$.

Options:
A. 19*(17*e^4+2)
B. 19*(17*e^4+6)
C. 19*(17*e^4-3)
D. 19*(17*e^4+1)
E. 19*(17*e^4-4)
F. 19*(17*e^4+3)
G. 19*(17*e^4-5)
H. 19*(17*e^4-1)
I. 19*(17*e^4+4)
J. 19*(17*e^4+5)
Answer: F
Evaluate $\int\int\int_{E}{(2 \cdot x+5 \cdot y+7 \cdot z) d V}$, where $E$ is the region defined by:
$$
E = \left\{(x,y,z) | 0 \le x \le 1, 0 \le y \le -x+1, 1 \le z \le 2\right\}
$$

Options:
A. 74/12
B. 70/12
C. 77/12
D. 72/12
E. 75/12
F. 71/12
G. 79/12
H. 78/12
I. 73/12
J. 76/12
Answer: C
Find the second derivative $\frac{d ^2y}{ d x^2}$ of the function $x = \left(2 \cdot \sin(3 \cdot t)\right)^2$, $y = \sin(2 \cdot t)$.

Options:
A. \frac{\left(288\cdot\left(\sin(3\cdot t)\right)^2-144\right)\cdot\cos(2\cdot t)-96\cdot\cos(3\cdot t)\cdot\sin(2\cdot t)\cdot\sin(3\cdot t)}{13824\cdot\left(\cos(3\cdot t)\right)^2\cdot\left(\sin(3\cdot t)\right)^5}
B. \frac{\left(288\cdot\left(\sin(3\cdot t)\right)^2-144\right)\cdot\cos(2\cdot t)-96\cdot\cos(3\cdot t)\cdot\sin(2\cdot t)\cdot\sin(3\cdot t)}{13824\cdot\left(\cos(3\cdot t)\right)^3\cdot\left(\sin(3\cdot t)\right)^3}
C. \frac{\left(288\cdot\left(\sin(3\cdot t)\right)^2-144\right)\cdot\cos(2\cdot t)-96\cdot\cos(3\cdot t)\cdot\sin(2\cdot t)\cdot\sin(3\cdot t)}{13824\cdot\left(\cos(3\cdot t)\right)^3\cdot\left(\sin(3\cdot t)\right)^5}
D. \frac{\left(288\cdot\left(\sin(3\cdot t)\right)^2-144\right)\cdot\cos(2\cdot t)-96\cdot\cos(3\cdot t)\cdot\sin(2\cdot t)\cdot\sin(3\cdot t)}{13824\cdot\left(\cos(3\cdot t)\right)^4\cdot\left(\sin(3\cdot t)\right)^2}
E. \frac{\left(288\cdot\left(\sin(3\cdot t)\right)^2-144\right)\cdot\cos(2\cdot t)-96\cdot\cos(3\cdot t)\cdot\sin(2\cdot t)\cdot\sin(3\cdot t)}{13824\cdot\left(\cos(3\cdot t)\right)^3\cdot\left(\sin(3\cdot t)\right)^6}
F. \frac{\left(288\cdot\left(\sin(3\cdot t)\right)^2-144\right)\cdot\cos(2\cdot t)-96\cdot\cos(3\cdot t)\cdot\sin(2\cdot t)\cdot\sin(3\cdot t)}{13824\cdot\left(\cos(3\cdot t)\right)^3\cdot\left(\sin(3\cdot t)\right)^2}
G. \frac{\left(288\cdot\left(\sin(3\cdot t)\right)^2-144\right)\cdot\cos(2\cdot t)-96\cdot\cos(3\cdot t)\cdot\sin(2\cdot t)\cdot\sin(3\cdot t)}{13824\cdot\left(\cos(3\cdot t)\right)^5\cdot\left(\sin(3\cdot t)\right)}
H. \frac{\left(288\cdot\left(\sin(3\cdot t)\right)^2-144\right)\cdot\cos(2\cdot t)-96\cdot\cos(3\cdot t)\cdot\sin(2\cdot t)\cdot\sin(3\cdot t)}{13824\cdot\left(\cos(3\cdot t)\right)^2\cdot\left(\sin(3\cdot t)\right)^4}
I. \frac{\left(288\cdot\left(\sin(3\cdot t)\right)^2-144\right)\cdot\cos(2\cdot t)-96\cdot\cos(3\cdot t)\cdot\sin(2\cdot t)\cdot\sin(3\cdot t)}{13824\cdot\left(\cos(3\cdot t)\right)^3\cdot\left(\sin(3\cdot t)\right)^4}
J. \frac{\left(288\cdot\left(\sin(3\cdot t)\right)^2-144\right)\cdot\cos(2\cdot t)-96\cdot\cos(3\cdot t)\cdot\sin(2\cdot t)\cdot\sin(3\cdot t)}{13824\cdot\left(\cos(3\cdot t)\right)^4\cdot\left(\sin(3\cdot t)\right)^3}
Answer: B
Compute the integral:
$$
\int \frac{ -3 }{ e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}} } \, dx
$$

Options:
A. C - \frac{1}{4} \cdot \left( \frac{3}{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}}} + 3 \cdot \ln\left( \frac{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}} - 1}{1 + e^{4 \cdot x} - \sqrt{1 + e^{8 \cdot x}} + 3} \right) \right)
B. C - \frac{1}{4} \cdot \left( \frac{3}{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}}} + 3 \cdot \ln\left( \frac{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}} - 1}{1 + e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}}} \right) \right)
C. C - \frac{1}{4} \cdot \left( \frac{3}{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}}} + 3 \cdot \ln\left( \frac{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}} - 1}{1 + e^{4 \cdot x} - \sqrt{1 + e^{8 \cdot x}}} \right) \right)
D. C - \frac{1}{4} \cdot \left( \frac{3}{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}}} + 3 \cdot \ln\left( \frac{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}} - 1}{1 + e^{4 \cdot x} - \sqrt{1 + e^{8 \cdot x}} - 1} \right) \right)
E. C - \frac{1}{4} \cdot \left( \frac{3}{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}}} + 3 \cdot \ln\left( \frac{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}} - 1}{1 + e^{4 \cdot x} - \sqrt{1 + e^{8 \cdot x}} + 1} \right) \right)
F. C - \frac{1}{4} \cdot \left( \frac{3}{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}}} + 3 \cdot \ln\left( \frac{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}} - 1}{1 + e^{4 \cdot x} - \sqrt{1 + e^{8 \cdot x}} + 2} \right) \right)
G. C - \frac{1}{4} \cdot \left( \frac{3}{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}}} + 3 \cdot \ln\left( \frac{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}} + 1}{1 + e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}} + 1} \right) \right)
H. C - \frac{1}{4} \cdot \left( \frac{3}{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}}} + 3 \cdot \ln\left( \frac{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}} + 1}{1 + e^{4 \cdot x} - \sqrt{1 + e^{8 \cdot x}}} \right) \right)
I. C - \frac{1}{4} \cdot \left( \frac{3}{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}}} + 3 \cdot \ln\left( \frac{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}} - 1}{1 + e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}} + 2} \right) \right)
J. C - \frac{1}{4} \cdot \left( \frac{3}{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}}} + 3 \cdot \ln\left( \frac{e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}} + 1}{1 + e^{4 \cdot x} + \sqrt{1 + e^{8 \cdot x}}} \right) \right)
Answer: B
Calculate $E = \frac{ 1 }{ \sin(10) } - \frac{ \sqrt{3} }{ \cos(10) }$.

Options:
A. 3.99
B. 4
C. 3.97
D. 3.92
E. 3.95
F. 3.94
G. 3.9
H. 3.93
I. 3.8
J. 3.98
Answer: B
Find the 5th order Taylor polynomial $P_{5}$ (in powers of $x$) for the function $f(x) = e^x \cdot \sin(x)$.

Options:
A. x + x^2 + \frac{1}{3} \cdot x^3 - \frac{1}{80} \cdot x^5
B. x + x^2 + \frac{1}{3} \cdot x^3 - \frac{1}{40} \cdot x^5
C. x + x^2 + \frac{1}{3} \cdot x^3 - \frac{1}{70} \cdot x^5
D. x + x^2 + \frac{1}{3} \cdot x^3 - \frac{1}{60} \cdot x^5
E. x + x^2 + \frac{1}{3} \cdot x^3 - \frac{1}{45} \cdot x^5
F. x + x^2 + \frac{1}{3} \cdot x^3 - \frac{1}{25} \cdot x^5
G. x + x^2 + \frac{1}{3} \cdot x^3 - \frac{1}{35} \cdot x^5
H. x + x^2 + \frac{1}{3} \cdot x^3 - \frac{1}{30} \cdot x^5
I. x + x^2 + \frac{1}{3} \cdot x^3 - \frac{1}{20} \cdot x^5
J. x + x^2 + \frac{1}{3} \cdot x^3 - \frac{1}{50} \cdot x^5
Answer: H
Compute the integral $\int_{0}^1{e^{-x^2} \, dx}$ with an estimated error of $0.0001$ using its series expansion.

Options:
A. 0.74682403
B. 0.74683403
C. 0.74682603
D. 0.74684603
E. 0.74683803
F. 0.74683603
G. 0.74682803
H. 0.74685603
I. 0.74681603
J. 0.74682203
Answer: F
Find the center of symmetry of the curve of $f(x) = \frac{ x^2 + 2 \cdot x - 1 }{ 1 - x }$.

Options:
A. (1, -9)
B. (1, -1)
C. (1, -6)
D. (1, -2)
E. (1, -10)
F. (1, -5)
G. (1, -8)
H. (1, -7)
I. (1,-4)
J. (1, -3)
Answer: I
If $\tan(\theta) = \sqrt{x^2 - 1}$, find $\sec(\theta) + \tan(\theta)$.

Options:
A. x - sqrt(x^2 + 1)
B. x + sqrt(x^2 - 1)
C. x - sqrt(x^2 - 1)
D. x + sqrt(x^2 + 1)
Answer: B
If the reciprocal of sum of real roots of the following equation can be written in form of $\frac {r}{s}$ where $(r,s) = 1$ , find $r + s$ .
$$$1000x^6 - 1900x^5 - 1400x^4 - 190x^3 - 130x^2 - 38x - 30 = 0$$$

Options:
A. 33
B. 31
C. 35
D. 37
E. 29
Answer: E
Let $\mathcal{P}$ be a parabola with vertex $V$ , focus $F$ , and directrix $\ell$ . $FV$ intersects $\ell$ at $D$ . Let $A$ be a point on $\mathcal{P}$ , and suppose that the intersection of $AD$ and $\mathcal{P}$ is $B$ . Given that $AB=3$ and $BD=1$ , the value of $AF^2$ can be written as $\tfrac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Options:
A. 441
B. 442
C. 444
D. 445
E. 440
F. 443
G. 446
Answer: A
Consider a triangle $\triangle HGU$. The centroid is denoted by $B$, while $X$ represents the incenter. The angles at vertices $H$ and $G$ are labeled as $N$ and $J$ respectively. Given that the line segment $XB$ runs parallel to side $HG$, and $J$ is equal to $2 \arctan (1/3)$, determine the value of $N$.

Options:
A. \frac{\pi}{4}
B. \frac{\pi}{3}
C. \frac{3\pi}{8}
D. \frac{3\pi}{4}
E. \frac{\pi}{6}
F. \frac{\pi}{5}
G. \frac{\pi}{2}
Answer: G
Let $P(x)$ be a polynomial of degree $10$ satisfying $P(x^2) = P(x)P(x - 1)$ . Find the maximum possible sum of the coecients of $P(x)$ .

Options:
A. -2
B. 0.5
C. 1
D. 2
E. -1
F. -10
G. 10
H. 0
Answer: C
Let $x$ be the arithmetic mean of all positive integers $k<577$ such that
$$$k^4\equiv 144\pmod {577}.$$$
Find the greatest integer less than or equal to $x$

Options:
A. 285
B. 283
C. 290
D. 284
E. 287.5
F. 287
G. 288
H. 289
I. 286
Answer: G
Find zeros of $f(x) = 2 \cdot x^4 - 13 \cdot x^3 + 24 \cdot x^2 - 13 \cdot x + 2$.

Options:
A. x_1=2+\sqrt{3}, x_2=2-\sqrt{3}, x_3=1, x_4=\frac{1}{3}
B. x_1=2+\sqrt{3}, x_2=2-\sqrt{3}, x_3=\frac{1}{2}, x_4=2
C. x_1=2+\sqrt{3}, x_2=2-\sqrt{3}, x_3=2, x_4=\frac{1}{2}
D. x_1=2+\sqrt{3}, x_2=2-\sqrt{3}, x_3=1, x_4=\frac{1}{2}
E. x_1=2+\sqrt{3}, x_2=2-\sqrt{3}, x_3=\frac{1}{2}, x_4=\frac{1}{4}
F. x_1=2+\sqrt{2}, x_2=2-\sqrt{2}, x_3=2, x_4=\frac{1}{2}
G. x_1=2+\sqrt{3}, x_2=2-\sqrt{3}, x_3=1, x_4=\frac{1}{4}
H. x_1=2+\sqrt{3}, x_2=2-\sqrt{3}, x_3=1, x_4=2
I. x_1=2+\sqrt{3}, x_2=2-\sqrt{3}, x_3=\frac{1}{2}, x_4=1
Answer: C
Compute the integral:
$$
\int \frac{ \sin(x)^4 }{ \cos(x) } \, dx
$$

Options:
A. C - \frac{1}{2} \cdot \ln\left(\left|\frac{1 + \sin(x)}{1 - \sin(x)}\right|\right) - \frac{1}{3} \cdot \left(\sin(x)\right)^3 - \sin(x)
B. C - \frac{1}{2} \cdot \ln\left(\left|\frac{1 - \sin(x)}{1 + \sin(x)}\right|\right) - \frac{1}{3} \cdot \left(\sin(x)\right)^3 + \sin(x)
C. C - \frac{1}{2} \cdot \ln\left(\left|\frac{1 - \sin(x)}{1 + \sin(x)}\right|\right) + \frac{1}{3} \cdot \left(\sin(x)\right)^3 + 2\sin(x)
D. C - \frac{1}{2} \cdot \ln\left(\left|\frac{1 - \sin(x)}{1 + \sin(x)}\right|\right) + \frac{1}{3} \cdot \left(\sin(x)\right)^3 - \sin(x)
E. C - \frac{1}{2} \cdot \ln\left(\left|\frac{1 + \sin(x)}{1 - \sin(x)}\right|\right) + \frac{1}{3} \cdot \left(\sin(x)\right)^3 - \sin(x)
F. C - \frac{1}{2} \cdot \ln\left(\left|\frac{1 - \sin(x)}{1 + \sin(x)}\right|\right) + \frac{1}{3} \cdot \left(\sin(x)\right)^3 + \sin(x)
G. C - \frac{1}{2} \cdot \ln\left(\left|\frac{1 - \sin(x)}{1 + \sin(x)}\right|\right) - \frac{1}{3} \cdot \left(\sin(x)\right)^3 - 2\sin(x)
H. C - \frac{1}{2} \cdot \ln\left(\left|\frac{1 - \sin(x)}{1 + \sin(x)}\right|\right) - \frac{1}{3} \cdot \left(\sin(x)\right)^3 - \sin(x)
I. C - \frac{1}{2} \cdot \ln\left(\left|\frac{1 - \sin(x)}{1 + \sin(x)}\right|\right) - \frac{1}{3} \cdot \left(\sin(x)\right)^3 + 2\sin(x)
Answer: H
A sequence $a_1, a_2, a_3, ..., a_{10}$ of integers is said to be shiny if it satisfies the following conditions:
$a_1=1$ , $a_2>0$ , $a_3>0$ , and $a_{10}=1$ .
For $i=2,3,\ldots,8$ , $a_i$ is the average of $a_{i-1}$ and $a_{i+2}$ .
How many shiny sequences are there with $|a_{2} - a_{3}|\le 1000$ ?

Options:
A. 204
B. 202
C. 201
D. 205
E. 208
F. 206
G. 203
H. 209
I. 200
J. 207
Answer: C
Find the number of pairs of primes $(p, q)$ , with $2 \le p, q \le 31$ , for which there exist integers $x$ and $y$ satisfying the equation
$$$(x + p)^p = x^p + qy.$$$

Options:
A. 29
B. 25
C. 31
D. 34
E. 33
F. 26
G. 30
H. 28
I. 32
J. 27
Answer: A
If $A$ the area of the region enclosed by the equation $45x^2 + 20y^2 - 36xy = 2304$ , find $\lfloor A\rfloor $

Options:
A. 307
B. 303
C. 302
D. 309
E. 300
F. 305
G. 308
H. 301
I. 306
J. 304
Answer: H
Find $x^2-yz$ if $x,y,z$ are distinct positive integers such that $y^2-zx=-103$ and $z^2-xy=22$

Options:
A. 173
B. 172
C. 162
D. 178
E. 167
F. 163
G. 182
H. 187
I. 152
J. 177
Answer: B
Let $A$ be the area of the convex quadrilateral with vertices on the foci of the ellipses defined by
$$$16x^2-320x+25y^2-1500y-15900 = 0$$$
and
$$$25x^2-1500x+16y^2-320y-15900 = 0.$$$
Find $\frac{A}{2}$ .
.

Options:
A. 880
B. 895
C. 855
D. 825
E. 860
F. 870
G. 850
H. 890
I. 900
J. 875
Answer: I
Let $ABCD$ be a rectangle with $AB = CD = 6$ . Suppose that $E$ and $F$ lie on side $\overline{AB}$ such that $AE = EF = FB = 2$ . Let $G$ denote the intersection of lines $\overleftrightarrow{EC}$ and $\overleftrightarrow{BD}$ ; let $H$ denote the intersection of lines $\overleftrightarrow{FC}$ and $\overleftrightarrow{BD}$ . Suppose that $E, F, G,$ and $H$ lie on a common circle. Find $BC^2$ .

Options:
A. 40
B. 48
C. 47
D. 42
E. 46
F. 49
G. 45
H. 44
I. 41
J. 43
Answer: H
For positive integers $k$ , let $f_k(x)=\sum_{i=0}^{2022} k^{2022-i}x^i$ . Find the smallest positive integer $n$ such that
$$$\sum_{r_1 \mid f_1(r_1)=0}\sum_{r_2\mid f_2(r_2)=0}\cdots \sum_{r_n\mid f_n(r_n)=0}\prod_{i=1}^{n}(2022-r_i)^{999}$$$
is an integer multiple of 101.

Options:
A. 32
B. 33
C. 37
D. 34
E. 30
F. 39
G. 38
H. 31
I. 36
J. 35
Answer: J
On a certain planet, there are exactly 360 days in a year. Find the number of ways we can split the year into equally-sized weeks and months such that
a day is shorter than a week, which is shorter than a month, which is shorter than a year,
the number of days in a week and days in a month are integers,
and the number of months in a year is an integer.
Note that in particular, the number of weeks in a month and weeks in a year don't have to be integers.

Options:
A. 745
B. 755
C. 775
D. 735
E. 795
F. 760
G. 750
H. 780
I. 765
J. 720
Answer: I
The value of $(4!)! $ is
$$$620, 448, abc, 733, 239, 439, 360, 000$$$
where $a$ , $b$ , $c$ are digits. What is $100a + 10b + c$ ?

Options:
A. 402
B. 401
C. 406
D. 404
E. 407
F. 410
G. 405
H. 412
I. 411
J. 403
Answer: B
Denote
$$$S(n,k)=\log_{k}\frac{2}{1}+\log_{k}\frac{3}{2}+...+\log_{k}\frac{n-1}{n-2}+\log_{k}\frac{n}{n-1}.$$$
If $n$ and $k$ are positive integers greater than $1$ and less than $500$ , find the number of ordered pairs $(n,k)$ such that $S(n,k)$ is an integer.

Options:
A. 532
B. 533
C. 530
D. 535
E. 537
F. 538
G. 534
H. 539
I. 531
J. 536
Answer: B
Let $z_0 = \cos \frac{\pi}{2005} + i \sin \frac{\pi}{2005}$ and $z_{n+1} = (z_n)^{n+1}$ for $n = 0, 1, 2, . . ..$ Find the smallest $n$ such that $z_n$ is an integer.

Options:
A. 2008
B. 401
C. 2003
D. 2002
E. 2005
F. 2004
G. 2006
H. 2007
I. 2010
J. 2009
Answer: B
If $\log_a{c} = \log_{\sqrt{a}}{b}+ \log_{a}{b^{2}}+ \log_{a^{2}}{b^{3}}+\log_{a^{4}}{b^{4}}+\log_{a^{8}}{b^{5}}\ldots,$ and $a,b,c>1$ the least possible value of $\log_a(bc)+ \log_b(ac)+ \log_c(ab)$ can be expressed as $\frac{w+x\sqrt{y}}{z}$ for positive integers $w,x,y,z$ where $y$ is not divisible by the square of any prime and $w, x, z$ are relatively prime. Find $w+x+y+z.$
Author: Vismay Sharan

Options:
A. 113
B. 110
C. 119
D. 118
E. 117
F. 116
G. 115
H. 114
I. 112
J. 111
Answer: J
If $x^2 + y^2- 30x- 40y + 24^2 = 0$ , then the largest possible value of $\frac{y}{x}$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime, positive integers. Determine $m + n$ .

Options:
A. 165
B. 153
C. 157
D. 167
E. 151
F. 161
G. 163
H. 155
I. 158
J. 159
Answer: F
An integer $n$ is fixed with respect to $m$ if $n$ and $n^2$ leave the same remainder when divided by $m.$ Let $M$ be the largest integer such that $1000$ is fixed with respect to $M.$ Find the number of positive integer divisors of $M.$

Options:
A. 129
B. 130
C. 122
D. 119
E. 112
F. 121
G. 120
H. 128
I. 126
J. 124
Answer: H
Determine the smallest positive integer $n $ for which
$$$\frac{2 \sin (16^o) \cos (32^o)}{\sin (29^o)}=\sqrt2 - 2 \sin (n^o)$$$

Options:
A. 18
B. 16
C. 12
D. 19
E. 17
F. 13
G. 10
H. 15
I. 11
J. 14
Answer: F
Find the remainder when $\sum_{k = 1}^{899} (2k^3 + 3k)^{10}$ is divided by $899$ .

Options:
A. 284
B. 288
C. 283
D. 282
E. 281
F. 287
G. 280
H. 286
I. 285
J. 289
Answer: E
Let $f : N \to N$ be a function such that
$$$f(n) = \sum_{\substack{abc=n, a,b,c\in N}}ab + bc + ca$$$
For example, $f(5) = (1 \cdot 1 + 1 \cdot 5 + 5 \cdot 1) + (1\cdot 5 + 5 \cdot 1 + 1 \cdot 1) + (5 \cdot 1 + 1 \cdot 1 + 1 \cdot 5) = 33$ , where we are summing over the triples $(a, b,c) = (1,1, 5)$ , $(1,5,1)$ , and $(5,1,1)$ . Find the last three digits of $f(303)$ .

Options:
A. 161
B. 165
C. 167
D. 168
E. 169
F. 163
G. 166
H. 160
I. 162
J. 164
Answer: J
The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,\dots,30$. How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? [Note: the top matter on each exam paper included the logo of the Mathematical Association of America, which is itself an icosahedron.]

Options:
A. 61917364226
B. 61917364229
C. 61917364227
D. 61917364228
E. 61917364221
F. 61917364220
G. 61917364223
H. 61917364225
I. 61917364222
J. 61917364224
Answer: J
Let $I_m = \int_0^{2\pi} \cos(x)\cos(2x)\cdots \cos(mx)\,dx$. Find the sum of all integers $m$, $1 \leq m \leq 10$ for which $I_m \neq 0$?

Options:
A. \boxed{23}
B. \boxed{19}
C. \boxed{24}
D. \boxed{26}
E. \boxed{18}
F. \boxed{20}
G. \boxed{28}
H. \boxed{21}
I. \boxed{22}
J. \boxed{25}
Answer: I
Find the minimum value of \[ | \sin x + \cos x + \tan x + \cot x + \sec x + \csc x | \] for real numbers $x$.

Options:
A. 2 \sqrt{2} - 1.25
B. 2 \sqrt{2} - 1.75
C. 2 \sqrt{2} - 1
D. 2 \sqrt{2} - 1.1
E. 2 \sqrt{2} + 1
F. 2 \sqrt{2} - 1.5
G. 2 \sqrt{2} - 2
H. 2 \sqrt{2} - 3
I. 2 \sqrt{2} + 2
J. 2 \sqrt{2} - 4
Answer: C
Right triangle $ABC$ has right angle at $C$ and $\angle BAC =\theta$; the point $D$ is chosen on $AB$ so that $|AC|=|AD|=1$; the point $E$ is chosen on $BC$ so that $\angle CDE = \theta$. The perpendicular to $BC$ at $E$ meets $AB$ at $F$. Evaluate $\lim_{\theta\rightarrow 0} |EF|$.

Options:
A. \frac{2}{3}
B. \frac{1}{8}
C. \frac{1}{9}
D. \frac{1}{4}
E. \frac{1}{10}
F. \frac{1}{7}
G. \frac{1}{5}
H. \frac{1}{6}
I. \frac{1}{2}
J. \frac{1}{3}
Answer: J
Evaluate \[ \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} \sum_{n=0}^\infty \frac{1}{k2^n + 1}. \]

Options:
A. \frac{3}{2}
B. \frac{3}{4}
C. \frac{11}{8}
D. \frac{13}{8}
E. \frac{9}{8}
F. 1
G. \frac{7}{8}
H. \frac{15}{16}
I. \frac{5}{4}
J. \frac{1}{2}
Answer: F
For each continuous function $f: [0,1] \to \mathbb{R}$, let $I(f) = \int_0^1 x^2 f(x)\,dx$ and $J(x) = \int_0^1 x \left(f(x)\right)^2\,dx$. Find the maximum value of $I(f) - J(f)$ over all such functions $f$.

Options:
A. \frac{1}{18}
B. \frac{1}{15}
C. \frac{1}{19}
D. \frac{1}{17}
E. \frac{1}{13}
F. \frac{1}{12}
G. \frac{1}{16}
H. \frac{1}{14}
I. \frac{1}{20}
J. \frac{1}{24}
Answer: G
For every positive real number $x$, let \[ g(x) = \lim_{r \to 0} ((x+1)^{r+1} - x^{r+1})^{\frac{1}{r}}. \] Find $\lim_{x \to \infty} \frac{g(x)}{x}$.

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