MikaStars39/nano-eval huggingface dataset:
Find $\frac{d^3}{dx^3}f(x)$, given $f(x) = \ln\left(\frac{ x+7 }{ x-7 }\right)$.

Options:
A. -\frac{84\cdot x^2-1372}{\left(x^2+49\right)^3}
B. -\frac{84\cdot x^2+1372}{\left(x^2-49\right)^2}
C. -\frac{84\cdot x^2-1372}{\left(x^2-49\right)^2}
D. -\frac{84\cdot x^2+1372}{\left(x^2-49\right)^3}
E. -\frac{84\cdot x^2-1372}{\left(x^2-49\right)^3}
F. -\frac{84\cdot x^2+1372}{\left(x^2+49\right)^3}
G. -\frac{84\cdot x^2+1372}{\left(x^2-49\right)^4}
Answer: D
Find the solution to the following inequality and express it in interval notation:
$$-4 (x-5) (x+4) (x-1) \le 0$$

Options:
A. (-∞, -4] ∪ (1, 5]
B. [-4, 1] ∪ [5, ∞)
C. (-∞, -4] ∪ (1, 5)
D. [-4, 1) ∪ (5, ∞)
E. (-∞, -4) ∪ [1, 5]
F. (-∞, -4] ∪ [1, 5)
G. [-4, 1) ∪ [5, ∞)
H. (-∞, -4] ∪ [1, 5]
Answer: B
What is the units (i.e., rightmost) digit of \[ \left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ? \] %Here $\lfloor x \rfloor$ is the greatest integer less than or equal to %$x$.

Options:
A. 5
B. 1
C. 7
D. 6
E. 4
F. 2
G. 9
H. 8
I. 3
Answer: I
Compute the integral:
$$
\int x \cdot \arctan(2 \cdot x)^2 \, dx
$$

Options:
A. \frac{1}{16}\cdot\left(2\cdot\left(\arctan(2\cdot x)\right)^2+2\cdot\ln\left(4\cdot x^2+1\right)+8\cdot x^2\cdot\left(\arctan(2\cdot x)\right)^2-12\cdot x\cdot\arctan(2\cdot x)\right)+C
B. \frac{1}{16}\cdot\left(2\cdot\left(\arctan(2\cdot x)\right)^2+2\cdot\ln\left(4\cdot x^2+1\right)+8\cdot x^2\cdot\left(\arctan(2\cdot x)\right)^2-4\cdot x\cdot\arctan(2\cdot x)\right)+C
C. \frac{1}{16}\cdot\left(2\cdot\left(\arctan(2\cdot x)\right)^2+2\cdot\ln\left(4\cdot x^2+1\right)+8\cdot x^2\cdot\left(\arctan(2\cdot x)\right)^2-18\cdot x\cdot\arctan(2\cdot x)\right)+C
D. \frac{1}{16}\cdot\left(2\cdot\left(\arctan(2\cdot x)\right)^2+2\cdot\ln\left(4\cdot x^2+1\right)+8\cdot x^2\cdot\left(\arctan(2\cdot x)\right)^2-14\cdot x\cdot\arctan(2\cdot x)\right)+C
E. \frac{1}{16}\cdot\left(2\cdot\left(\arctan(2\cdot x)\right)^2+2\cdot\ln\left(4\cdot x^2+1\right)+8\cdot x^2\cdot\left(\arctan(2\cdot x)\right)^2-10\cdot x\cdot\arctan(2\cdot x)\right)+C
F. \frac{1}{16}\cdot\left(2\cdot\left(\arctan(2\cdot x)\right)^2+2\cdot\ln\left(4\cdot x^2+1\right)+8\cdot x^2\cdot\left(\arctan(2\cdot x)\right)^2-6\cdot x\cdot\arctan(2\cdot x)\right)+C
G. \frac{1}{16}\cdot\left(2\cdot\left(\arctan(2\cdot x)\right)^2+2\cdot\ln\left(4\cdot x^2+1\right)+8\cdot x^2\cdot\left(\arctan(2\cdot x)\right)^2-2\cdot x\cdot\arctan(2\cdot x)\right)+C
H. \frac{1}{16}\cdot\left(2\cdot\left(\arctan(2\cdot x)\right)^2+2\cdot\ln\left(4\cdot x^2+1\right)+8\cdot x^2\cdot\left(\arctan(2\cdot x)\right)^2-8\cdot x\cdot\arctan(2\cdot x)\right)+C
I. \frac{1}{16}\cdot\left(2\cdot\left(\arctan(2\cdot x)\right)^2+2\cdot\ln\left(4\cdot x^2+1\right)+8\cdot x^2\cdot\left(\arctan(2\cdot x)\right)^2-16\cdot x\cdot\arctan(2\cdot x)\right)+C
Answer: H
$\zeta_1$ ; $\zeta_2$ , and $\zeta_3$ are complex numbers such that
$$$\zeta_1 + \zeta_2 + \zeta_3 = 1$$$
$$$\zeta^2_1 + \zeta^2_2 +\zeta^2_3 = 3$$$
$$$\zeta^3_1 + \zeta^3_2 + \zeta^3_3 = 7$$$
Compute $\zeta^7_1 + \zeta^7_2 + \zeta^7_3$ .

Options:
A. 65
B. 69
C. 72
D. 67
E. 74
F. 70
G. 73
H. 68
I. 71
J. 75
Answer: I
Find the number of solutions, in degrees, to the equation $\sin^{10}x + \cos^{10}x = \frac{29}{16}\cos^4 2x,$ where $0^\circ \le x^\circ \le 2007^\circ.$

Options:
A. 44
B. 43
C. 48
D. 46
E. 40
F. 50
G. 42
H. 47
I. 45
J. 41
Answer: I
Let $m$ denote the minimum value of:
$$$a + \sqrt{a^2 - 7a - \sqrt{a^2 - 50}}$$$
over all positive real numbers $a \geq \sqrt{50}$ . Compute $\lfloor 100m \rfloor$ .

Options:
A. 724
B. 720
C. 727
D. 721
E. 728
F. 725
G. 729
H. 722
I. 726
J. 723
Answer: J
Find $|\sum_{k=0}^{2n} \dbinom{2n}{k}^{3}|$ $\mod {1000}$ where $n=5$ .

Options:
A. 260
B. 280
C. 240
D. 270
E. 290
F. 265
G. 261
H. 255
I. 250
J. 262
Answer: A
Let $n$ be the smallest positive integer such that
$$$\frac{n^{100}}{100!}$$$
is an integer. Let $d$ be the number of positive factors of $n$ . Compute the number of positive factors of $d$ .

Options:
A. 27
B. 22
C. 24
D. 29
E. 20
F. 26
G. 28
H. 23
I. 21
J. 25
Answer: F
If $S$ is the number of $4$ -digit positive integers with all distinct nonzero digits satisfy the condition that the sum of the first and last digit is greater than or equal to the sum of the middle two digits, find the remainder obtained when $S$ is divided by $1000$ .

Options:
A. 644
B. 656
C. 640
D. 636
E. 642
F. 652
G. 654
H. 632
I. 648
J. 646
Answer: I
Determine the number of integers $n$ such that $1\le n \le 1000$ and $n^{12}-1$ is divisible by $73$ .

Options:
A. 164
B. 167
C. 168
D. 165
E. 162
F. 163
G. 160
H. 161
I. 166
J. 169
Answer: A
$2006$ points are evenly spaced on a circle. Given one point, find the number of other points that are less than $1$ radius distance away from that point

Options:
A. 669
B. 670
C. 666
D. 665
E. 663
F. 668
G. 667
H. 664
I. 672
J. 671
Answer: F
Let $a$ , $b$ , and $c$ be positive real numbers such that $a+b+c=12288$ . What is the smallest value of $n$ such that the maximum possible value of
$$$\left(\sum_{i=1}^na^{\frac1i}\right)+\left(\sum_{i=1}^nb^{\frac1i}\right)+\left(\sum_{i=1}^nc^{\frac1i}\right)$$$
is not an integer?

Options:
A. 6
B. 7
C. 11
D. 5
E. 8
F. 12
G. 4
H. 3
I. 10
J. 9
Answer: D
A sequence is defined recursively by
$$$a_n-a_{n-1} \in \{-1,0,1\},$$$
for all integers $n$ chosen from $\{1,2,3,\dots,9\}$ . Find the remainder when the number of sequences with $a_1=a_9=0$ is divided by $1000$ .

Options:
A. 108
B. 101
C. 103
D. 107
E. 104
F. 109
G. 105
H. 102
I. 110
J. 106
Answer: D
Given $\triangle ABC$ and $D$ a point on side $AC$ such that $AB=DC, \angle{DBC}=(5x)^\circ, \angle{ABD}=(60-2x)^\circ$ , and $\angle{ACB}=(3x)^\circ$ find $x$ .

Options:
A. 19
B. 10
C. 11
D. 16
E. 15
F. 17
G. 18
H. 12
I. 14
J. 13
Answer: H
The walkway to the front of Euler Elementary School is a $6 \times 15$ grid. ACME Sidewalk company has been chartered to tile it with $15$ non-overlapping $2 \times 3$ concrete slabs. Compute the number of tilings ACME can choose from for this task.

Options:
A. 32
B. 28
C. 34
D. 31
E. 37
F. 29
G. 36
H. 33
I. 35
J. 30
Answer: B
For how many positive integers $n < 1000$ does there exist a regular $n$ -sided polygon such that the number of diagonals is a nonzero perfect square?

Options:
A. 11
B. 8
C. 5
D. 7
E. 12
F. 6
G. 10
H. 3
I. 9
J. 4
Answer: J
There is an equiangular hexagon $OUTDIA$ such that $OU=DI$ , $UT=IA$ , $TD=AO$ are all integer side lengths. The incircles of triangles $OUT$ and $OTD$ are tangent to each other. Given that $OU,UT,TD$ are all distinct integers, what is the smallest possible value of the sum of the sides of the hexagon?

Options:
A. 24
B. 25
C. 21
D. 28
E. 20
F. 23
G. 26
H. 27
I. 22
J. 29
Answer: I
Richard has an infinite row of empty boxes labeled $1, 2, 3, \ldots$ and an infinite supply of balls. Each minute, Richard finds the smallest positive integer $k$ such that box $k$ is empty. Then, Richard puts a ball into box $k$ , and if $k \geq 3$ , he removes one ball from each of boxes $1,2,\ldots,k-2$ . Find the smallest positive integer $n$ such that after $n$ minutes, both boxes $9$ and $10$ have at least one ball in them.

Options:
A. 90
B. 96
C. 98
D. 95
E. 97
F. 89
G. 93
H. 92
I. 91
J. 94
Answer: F
Let $R$ be a set of $13$ points in the plane, no three of which lie on the same line. At most how many ordered triples of points $(A,B,C)$ in $R$ exist such that $\angle ABC$ is obtuse?

Options:
A. 585
B. 568
C. 530
D. 528
E. 560
F. 552
G. 580
H. 558
I. 546
J. 572
Answer: J
How many non-decreasing sequences $a_0, a_1, a_2, a_3, a_4, a_5, a_6$ of non-negative integers are there satisfying $a_i\leq i$ for $i=0,2,4,6$ ?

Options:
A. 551
B. 555
C. 556
D. 550
E. 559
F. 552
G. 558
H. 553
I. 554
J. 557
Answer: I
$ABC$ is a scalene triangle. Points $D$ , $E$ , and $F$ are selected on sides $BC$ , $CA$ , and $AB$ respectively. The cevians $AD$ , $BE$ , and $CF$ concur at point $P$ . If $ [AFP] = 126$ , $[FBP] = 63$ , and $[CEP] = 24$ , determine the area of triangle $ABC$ .

Options:
A. 336
B. 357
C. 350
D. 345
E. 342
F. 348
G. 363
H. 360
I. 354
J. 351
Answer: J
For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of $2023 \cdot n$. What is the minimum value of $k(n)$?

Options:
A. 5
B. 3
C. 10
D. 6
E. 9
F. 2
G. 4
H. 1
I. 8
J. 7
Answer: B
Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches at the very least are needed to remove every point?

Options:
A. \boxed{3}
B. \boxed{9}
C. \boxed{6}
D. \boxed{8}
E. \boxed{10}
F. \boxed{5}
G. \boxed{1}
H. \boxed{2}
I. \boxed{7}
J. \boxed{4}
Answer: A
For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\dots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties: \begin{enumerate} \item[(a)] $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1,\dots,t_n$; \item[(b)] $f(t_0) = 1/2$; \item[(c)] $\lim_{t \to t_k^+} f'(t) = 0$ for $0 \leq k \leq n$; \item[(d)] For $0 \leq k \leq n-1$, we have $f''(t) = k+1$ when $t_k < t< t_{k+1}$, and $f''(t) = n+1$ when $t>t_n$. \end{enumerate} Considering all choices of $n$ and $t_0,t_1,\dots,t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f(t_0+T) = 2023$?

Options:
A. 32
B. 37
C. 29
D. 35
E. 38
F. 33
G. 30
H. 31
I. 36
J. 34
Answer: C
For a positive integer $n$, let $f_n(x) = \cos(x) \cos(2x) \cos(3x) \cdots \cos(nx)$. Find the smallest $n$ such that $|f_n''(0)| > 2023$.

Options:
A. 17
B. 13
C. 20
D. 18
E. 19
F. 21
G. 16
H. 14
I. 15
J. 12
Answer: D
On a linear board with 98567 adjacent cells, Alice and Bob engage in a strategic game. Players alternate turns, starting with Alice, placing non-overlapping dominos that cover exactly two neighboring cells. The game concludes when no more dominos can be legally placed. Alice aims to maximize the count of uncovered cells at the game's end, while Bob strives to minimize this number. Determine the maximum number of uncovered cells Alice can guarantee, regardless of Bob's strategy.

Options:
A. 14078
B. 14082
C. 14081
D. 14087
E. 14080
F. 14086
G. 14083
H. 14085
I. 14084
J. 14079
Answer: C
Consider the function $a(o)$ defined for all positive integers $o$. This function counts the number of 1's that appear when you write $2023 \cdot o$ in binary notation. What is the smallest possible output of $a(o)$?

Options:
A. 2
B. 4
C. 8
D. 6
E. 3
F. 9
G. 5
H. 7
I. 10
J. 1
Answer: E
Consider the sequence defined by $c_0=1$, $c_1=2$, and $c_f=4c_{f-1}-c_{f-2}$ for $f\geq 2$. Determine the smallest odd prime factor of $c_{265}$.

Options:
A. 183
B. 182
C. 179
D. 187
E. 178
F. 173
G. 181
H. 175
I. 177
J. 171
Answer: G
Compute the integral:
$$
\int \frac{ -4 }{ 3+\sin(4 \cdot x)+\cos(4 \cdot x) } \, dx
$$

Options:
A. C - \frac{2}{\sqrt{7}} \cdot \arctan\left(\frac{2}{\sqrt{7}} \cdot \left(\frac{1}{2} + \tan(3 \cdot x)\right)\right)
B. C - \frac{2}{\sqrt{7}} \cdot \arctan\left(\frac{2}{\sqrt{7}} \cdot \left(\frac{1}{2} + \tan(4 \cdot x)\right)\right)
C. C - \frac{2}{\sqrt{7}} \cdot \arctan\left(\frac{2}{\sqrt{7}} \cdot \left(\frac{1}{2} + \tan(2 \cdot x + \frac{\pi}{6})\right)\right)
D. C - \frac{2}{\sqrt{7}} \cdot \arctan\left(\frac{2}{\sqrt{7}} \cdot \left(\frac{1}{2} + \tan(2 \cdot x + \frac{\pi}{4})\right)\right)
E. C - \frac{2}{\sqrt{7}} \cdot \arctan\left(\frac{2}{\sqrt{7}} \cdot \left(\frac{1}{2} + \tan(2 \cdot x + \frac{\pi}{3})\right)\right)
F. C - \frac{2}{\sqrt{7}} \cdot \arctan\left(\frac{2}{\sqrt{7}} \cdot \left(\frac{1}{2} + \tan(2 \cdot x)\right)\right)
G. C - \frac{2}{\sqrt{7}} \cdot \arctan\left(\frac{2}{\sqrt{7}} \cdot \left(\frac{1}{2} + \tan(2 \cdot x + \frac{\pi}{12})\right)\right)
H. C - \frac{2}{\sqrt{7}} \cdot \arctan\left(\frac{2}{\sqrt{7}} \cdot \left(\frac{1}{2} - \tan(2 \cdot x)\right)\right)
I. C - \frac{2}{\sqrt{7}} \cdot \arctan\left(\frac{2}{\sqrt{7}} \cdot \left(\frac{1}{2} + \tan(2 \cdot x + \frac{\pi}{8})\right)\right)
J. C - \frac{2}{\sqrt{7}} \cdot \arctan\left(\frac{2}{\sqrt{7}} \cdot \left(\frac{1}{2} + \tan(2 \cdot x - \frac{\pi}{4})\right)\right)
Answer: F
Find the value $k$ that satisfies the following integral equation:
$$
\int_{1}^k \log_{2}(x) \, dx = 2 - \frac{ 1 }{ \ln(2) }
$$

Options:
A. 2.25
B. 2.1
C. 2.125
D. 2.75
E. 2
F. 2.5
G. 2.625
H. 3
I. 2.375
J. 2.05
Answer: E
Find the second derivative $\frac{d ^2y}{ d x^2}$ of the function $x = \left(4 \cdot \sin(t)\right)^3$, $y = 2 \cdot \sin(2 \cdot t)$.

Options:
A. \frac{\left(2304\cdot\left(\sin(t)\right)^3-1536\cdot\sin(t)\right)\cdot\cos(2\cdot t)-1536\cdot\left(\sin(t)\right)^2\cdot\cos(t)\cdot\sin(2\cdot t)}{7077888\cdot\left(\cos(t)\right)^3\cdot\left(\sin(t)\right)^5}
B. \frac{\left(2304\cdot\left(\sin(t)\right)^3-1536\cdot\sin(t)\right)\cdot\cos(2\cdot t)-1536\cdot\left(\sin(t)\right)^2\cdot\cos(t)\cdot\sin(2\cdot t)}{7077888\cdot\left(\cos(t)\right)^6\cdot\left(\sin(t)\right)^5}
C. \frac{\left(2304\cdot\left(\sin(t)\right)^3-1536\cdot\sin(t)\right)\cdot\cos(2\cdot t)-1536\cdot\left(\sin(t)\right)^2\cdot\cos(t)\cdot\sin(2\cdot t)}{7077888\cdot\left(\cos(t)\right)^3\cdot\left(\sin(t)\right)^6}
D. \frac{\left(2304\cdot\left(\sin(t)\right)^3-1536\cdot\sin(t)\right)\cdot\cos(2\cdot t)-1536\cdot\left(\sin(t)\right)^2\cdot\cos(t)\cdot\sin(2\cdot t)}{7077888\cdot\left(\cos(t)\right)^2\cdot\left(\sin(t)\right)^7}
E. \frac{\left(2304\cdot\left(\sin(t)\right)^3-1536\cdot\sin(t)\right)\cdot\cos(2\cdot t)-1536\cdot\left(\sin(t)\right)^2\cdot\cos(t)\cdot\sin(2\cdot t)}{7077888\cdot\left(\cos(t)\right)^4\cdot\left(\sin(t)\right)^5}
F. \frac{\left(2304\cdot\left(\sin(t)\right)^3-1536\cdot\sin(t)\right)\cdot\cos(2\cdot t)-1536\cdot\left(\sin(t)\right)^2\cdot\cos(t)\cdot\sin(2\cdot t)}{7077888\cdot\left(\cos(t)\right)^2\cdot\left(\sin(t)\right)^6}
G. \frac{\left(2304\cdot\left(\sin(t)\right)^3-1536\cdot\sin(t)\right)\cdot\cos(2\cdot t)-1536\cdot\left(\sin(t)\right)^2\cdot\cos(t)\cdot\sin(2\cdot t)}{7077888\cdot\left(\cos(t)\right)^5\cdot\left(\sin(t)\right)^4}
H. \frac{\left(2304\cdot\left(\sin(t)\right)^3-1536\cdot\sin(t)\right)\cdot\cos(2\cdot t)-1536\cdot\left(\sin(t)\right)^2\cdot\cos(t)\cdot\sin(2\cdot t)}{7077888\cdot\left(\cos(t)\right)^4\cdot\left(\sin(t)\right)^7}
I. \frac{\left(2304\cdot\left(\sin(t)\right)^3-1536\cdot\sin(t)\right)\cdot\cos(2\cdot t)-1536\cdot\left(\sin(t)\right)^2\cdot\cos(t)\cdot\sin(2\cdot t)}{7077888\cdot\left(\cos(t)\right)^4\cdot\left(\sin(t)\right)^6}
J. \frac{\left(2304\cdot\left(\sin(t)\right)^3-1536\cdot\sin(t)\right)\cdot\cos(2\cdot t)-1536\cdot\left(\sin(t)\right)^2\cdot\cos(t)\cdot\sin(2\cdot t)}{7077888\cdot\left(\cos(t)\right)^3\cdot\left(\sin(t)\right)^7}
Answer: C
Given $k$ as an integer and $a \ne \pi \cdot k$, evaluate $P=\cos(a) \cdot \cos(2 \cdot a) \cdot \cos(4 \cdot a) \ldots \cos\left(2^n \cdot a\right)$.

Options:
A. P=\frac{1}{2^{n+1}}\cdot\frac{\sin\left(2^{n+1}\cdot a\right)}{\sin(2^{n+1}a)}
B. P=\frac{1}{2^{n+2}}\cdot\frac{\sin\left(2^{n+1}\cdot a\right)}{\sin(a)}
C. P=\frac{1}{2^{n+1}}\cdot\frac{\sin\left(2^{n+1}\cdot a\right)}{\sin(2^{n+2}a)}
D. P=\frac{1}{2^{n+1}}\cdot\frac{\sin\left(2^{n+1}\cdot a\right)}{\sin(a)}
E. P=\frac{1}{2^{n}}\cdot\frac{\sin\left(2^{n}\cdot a\right)}{\sin(a)}
F. P=\frac{1}{2^{n+1}}\cdot\frac{\sin\left(2^{n+1}\cdot a\right)}{\sin(2a)}
G. P=\frac{1}{2^{n+1}}\cdot\frac{\sin\left(2^{n+1}\cdot a\right)}{\sin(2^{n}a)}
H. P=\frac{1}{2^{n+1}}\cdot\frac{\sin\left(2^{n}\cdot a\right)}{\sin(a)}
I. P=\frac{1}{2^{n+1}}\cdot\frac{\sin\left(2^{n+2}\cdot a\right)}{\sin(a)}
J. P=\frac{1}{2^n}\cdot\frac{\sin\left(2^{n+1}\cdot a\right)}{\sin(a)}
Answer: D
Change the order of integration by integrating first with respect to $z$, then $x$, then $y$:
$$
\int_{0}^1 \int_{1}^2 \int_{2}^3 \left( x^2 + \ln(y) + z \right) \, dx \, dy \, dz
$$

Options:
A. 2*ln(2)+30/6
B. 2*ln(2)+35/6
C. 2*ln(2)+36/6
D. 2*ln(2)+33/6
E. 2*ln(2)+32/6
F. 2*ln(2)+37/6
G. 2*ln(2)+39/6
H. 2*ln(2)+31/6
I. 2*ln(2)+38/6
J. 2*ln(2)+34/6
Answer: B
For the function $y = \frac{ x }{ x^2 - 1 }$ find the derivative $y^{(n)}$.

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

Options:
A. \frac{1}{2}\cdot\ln\left(\tan(x)^2+\sqrt{\tan(x)^4+2}\right)+C
B. \frac{1}{2}\cdot\ln\left(\tan(x)^2+\sqrt{\tan(x)^4+8}\right)+C
C. \frac{1}{2}\cdot\ln\left(\tan(x)^2+\sqrt{\tan(x)^4-1}\right)+C
D. \frac{1}{2}\cdot\ln\left(\tan(x)^2+\sqrt{\tan(x)^4+1}\right)+C
E. \frac{1}{2}\cdot\ln\left(\tan(x)^2+\sqrt{\tan(x)^4+7}\right)+C
F. \frac{1}{2}\cdot\ln\left(\tan(x)^2+\sqrt{\tan(x)^4+5}\right)+C
G. \frac{1}{2}\cdot\ln\left(\tan(x)^2+\sqrt{\tan(x)^4+4}\right)+C
H. \frac{1}{2}\cdot\ln\left(\tan(x)^2+\sqrt{\tan(x)^4+6}\right)+C
I. \frac{1}{2}\cdot\ln\left(\tan(x)^2+\sqrt{\tan(x)^4+3}\right)+C
J. \frac{1}{2}\cdot\ln\left(\tan(x)^2+\sqrt{\tan(x)^4+9}\right)+C
Answer: D
Find the integral:
$$
\int \frac{ 4 \cdot x^2+25 \cdot x+7 }{ \sqrt{x^2+8 \cdot x} } \, dx
$$

Options:
A. (2*x+1)*sqrt(x^2+8*x)-3*ln(|x+4-sqrt(x^2+8*x)|)+C
B. (2*x+1)*sqrt(x^2+8*x)+3*ln(|x+4+sqrt(x^2+8*x)|)+C/4
C. (2*x+1)*sqrt(x^2+8*x)+3*ln(|x+4+sqrt(x^2+8*x)|)-C
D. (2*x+1)*sqrt(x^2+8*x)+3*ln(|x+4+sqrt(x^2+8*x)|)+C/5
E. (2*x+1)*sqrt(x^2+8*x)+3*ln(|x+4-sqrt(x^2+8*x)|)+C
F. (2*x+1)*sqrt(x^2+8*x)+3*ln(|x+4+sqrt(x^2+8*x)|)+C
G. (2*x+1)*sqrt(x^2+8*x)-3*ln(|x+4+sqrt(x^2+8*x)|)+C
H. (2*x+1)*sqrt(x^2+8*x)+3*ln(|x+4+sqrt(x^2+8*x)|)+2*C
I. (2*x+1)*sqrt(x^2+8*x)+3*ln(|x+4+sqrt(x^2+8*x)|)+C/3
J. (2*x+1)*sqrt(x^2+8*x)+3*ln(|x+4+sqrt(x^2+8*x)|)+C/2
Answer: F
Use integration by substitution and/or by parts to compute the integral:
$$
\int x \cdot \ln(5+x) \, dx
$$

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

Options:
A. C + \frac{3}{2} \cdot \left(\ln\left(\left|\tan(2 \cdot x)\right|\right) - \frac{3}{2 \cdot \left(\tan(2 \cdot x)\right)^2} - \frac{3}{4 \cdot \left(\tan(2 \cdot x)\right)^4} - \frac{1}{13 \cdot \left(\tan(2 \cdot x)\right)^6}\right)
B. C + \frac{3}{2} \cdot \left(\ln\left(\left|\tan(2 \cdot x)\right|\right) - \frac{3}{2 \cdot \left(\tan(2 \cdot x)\right)^2} - \frac{3}{4 \cdot \left(\tan(2 \cdot x)\right)^4} - \frac{1}{9 \cdot \left(\tan(2 \cdot x)\right)^6}\right)
C. C + \frac{3}{2} \cdot \left(\ln\left(\left|\tan(2 \cdot x)\right|\right) - \frac{3}{2 \cdot \left(\tan(2 \cdot x)\right)^2} - \frac{3}{4 \cdot \left(\tan(2 \cdot x)\right)^4} - \frac{1}{12 \cdot \left(\tan(2 \cdot x)\right)^6}\right)
D. C + \frac{3}{2} \cdot \left(\ln\left(\left|\tan(2 \cdot x)\right|\right) - \frac{3}{2 \cdot \left(\tan(2 \cdot x)\right)^2} - \frac{3}{4 \cdot \left(\tan(2 \cdot x)\right)^4} - \frac{1}{8 \cdot \left(\tan(2 \cdot x)\right)^6}\right)
E. C + \frac{3}{2} \cdot \left(\ln\left(\left|\tan(2 \cdot x)\right|\right) - \frac{3}{2 \cdot \left(\tan(2 \cdot x)\right)^2} - \frac{3}{4 \cdot \left(\tan(2 \cdot x)\right)^4} - \frac{1}{5 \cdot \left(\tan(2 \cdot x)\right)^6}\right)
F. C + \frac{3}{2} \cdot \left(\ln\left(\left|\tan(2 \cdot x)\right|\right) - \frac{3}{2 \cdot \left(\tan(2 \cdot x)\right)^2} - \frac{3}{4 \cdot \left(\tan(2 \cdot x)\right)^4} - \frac{1}{10 \cdot \left(\tan(2 \cdot x)\right)^6}\right)
G. C + \frac{3}{2} \cdot \left(\ln\left(\left|\tan(2 \cdot x)\right|\right) - \frac{3}{2 \cdot \left(\tan(2 \cdot x)\right)^2} - \frac{3}{4 \cdot \left(\tan(2 \cdot x)\right)^4} - \frac{1}{7 \cdot \left(\tan(2 \cdot x)\right)^6}\right)
H. C + \frac{3}{2} \cdot \left(\ln\left(\left|\tan(2 \cdot x)\right|\right) - \frac{3}{2 \cdot \left(\tan(2 \cdot x)\right)^2} - \frac{3}{4 \cdot \left(\tan(2 \cdot x)\right)^4} - \frac{1}{6 \cdot \left(\tan(2 \cdot x)\right)^6}\right)
I. C + \frac{3}{2} \cdot \left(\ln\left(\left|\tan(2 \cdot x)\right|\right) - \frac{3}{2 \cdot \left(\tan(2 \cdot x)\right)^2} - \frac{3}{4 \cdot \left(\tan(2 \cdot x)\right)^4} - \frac{1}{11 \cdot \left(\tan(2 \cdot x)\right)^6}\right)
J. C + \frac{3}{2} \cdot \left(\ln\left(\left|\tan(2 \cdot x)\right|\right) - \frac{3}{2 \cdot \left(\tan(2 \cdot x)\right)^2} - \frac{3}{4 \cdot \left(\tan(2 \cdot x)\right)^4} - \frac{1}{14 \cdot \left(\tan(2 \cdot x)\right)^6}\right)
Answer: H
Find the derivative of the function: $y = -4 \cdot x^{\sqrt{5 \cdot x}}$

Options:
A. -\left(\frac{4\cdot\sqrt{5}}{\sqrt{x}}+\frac{2\cdot\sqrt{5}\cdot\ln(x)}{2\sqrt{x}}\right)\cdot x^{\sqrt{5}\cdot\sqrt{x}}
B. -\left(\frac{4\cdot\sqrt{5}}{\sqrt{x}}+\frac{2\cdot\sqrt{5}\cdot\ln(x)}{\sqrt{x}}\right)\cdot x^{\sqrt{5}\cdot\sqrt{x}}
C. -\left(\frac{4\cdot\sqrt{5}}{\sqrt{x}}+\frac{2\cdot\sqrt{5}\cdot\ln(x)}{4\sqrt{x}}\right)\cdot x^{\sqrt{5}\cdot\sqrt{x}}
D. -\left(\frac{4\cdot\sqrt{5}}{\sqrt{x}}+\frac{2\cdot\sqrt{5}\cdot\ln(x)}{3\sqrt{x}}\right)\cdot x^{\sqrt{5}\cdot\sqrt{x}}
E. -\left(\frac{4\cdot\sqrt{5}}{\sqrt{x}}+\frac{2\cdot\sqrt{5}\cdot\ln(x)}{x^2}\right)\cdot x^{\sqrt{5}\cdot\sqrt{x}}
F. -\left(\frac{4\cdot\sqrt{5}}{\sqrt{x}}+\frac{2\cdot\sqrt{5}\cdot\ln(x)}{x^{1/3}}\right)\cdot x^{\sqrt{5}\cdot\sqrt{x}}
G. -\left(\frac{4\cdot\sqrt{5}}{\sqrt{x}}+\frac{2\cdot\sqrt{5}\cdot\ln(x)}{x^{3/2}}\right)\cdot x^{\sqrt{5}\cdot\sqrt{x}}
H. -\left(\frac{4\cdot\sqrt{5}}{\sqrt{x}}+\frac{2\cdot\sqrt{5}\cdot\ln(x)}{2x}\right)\cdot x^{\sqrt{5}\cdot\sqrt{x}}
I. -\left(\frac{4\cdot\sqrt{5}}{\sqrt{x}}+\frac{2\cdot\sqrt{5}\cdot\ln(x)}{x}\right)\cdot x^{\sqrt{5}\cdot\sqrt{x}}
J. -\left(\frac{4\cdot\sqrt{5}}{\sqrt{x}}+\frac{2\cdot\sqrt{5}\cdot\ln(x)}{x^{1/2}}\right)\cdot x^{\sqrt{5}\cdot\sqrt{x}}
Answer: B
Find the sum of the series $\sum_{n=0}^\infty \left(\frac{ (-1)^n }{ (2 \cdot n+1)! }\right)$ with an estimate error of $0.01$.

Options:
A. 0.8415666666666667
B. 0.8413666666666667
C. 0.8416666666666667
D. 0.8416466666666667
E. 0.8414666666666667
F. 0.8418666666666667
G. 0.8419666666666667
H. 0.8416566666666667
I. 0.8416766666666667
J. 0.8417666666666667
Answer: C
Find the measure of the angle $\theta$ between the three-dimensional vectors $\vec{a}$ and $\vec{b}$, expressed in radians rounded to two decimal places, if it is not possible to express it exactly.
Given:
$\vec{a} = 3 \cdot \vec{i} - \vec{j} - 2 \cdot \vec{k}$
$\vec{b} = \vec{v} - \vec{w}$, where $\vec{v} = 2 \cdot \vec{i} + \vec{j} + 4 \cdot \vec{k}$ and $\vec{w} = 6 \cdot \vec{i} + \vec{j} + 2 \cdot \vec{k}$

Options:
A. 2.79
B. 2.54
C. 2.64
D. 2.49
E. 2.84
F. 2.44
G. 2.69
H. 2.94
I. 2.59
J. 2.74
Answer: E
Calculate the derivative $\frac{ d }{d x}\left(\log_{x}(a)\right)$ for $x > 0$, $a > 0$, $x \ne 1$, $a \ne 1$.

Options:
A. -ln(a) / (x * ln(x) * ln(a))
B. -ln(a) / (x * ln(x) * ln(x) * ln(x) * ln(x) * ln(x))
C. -1 / (x * ln(x))
D. -ln(a) / (x * (ln(x))^2)
E. -ln(a) / (x * ln(x) * ln(x) * ln(x))
F. -ln(a) / (x * ln(x) * ln(x))
G. -ln(a) / (x * ln(x) * ln(x) * ln(x) * ln(x))
H. -ln(a) / (x * ln(x))
I. -1 / (x * (ln(x))^2)
J. -ln(a) / (x * ln(x) * ln(x) * ln(a))
Answer: D
Two nonintersecting circles with radii 3 and 4 have internal tangent with length $x$ and external tangent with length $y,$ where $x$ and $y$ are positive integers. Find the sum of all possible values of $x+y.$

Options:
A. 43
B. 42
C. 49
D. 48
E. 46
F. 45
G. 40
H. 47
I. 44
J. 41
Answer: I
In the triangle $\triangle ABC$, let $G$ be the centroid, and let $I$ be the center of the inscribed circle. Let $\alpha$ and $\beta$ be the angles at the vertices $A$ and $B$, respectively. Suppose that the segment $IG$ is parallel to $AB$ and that $\beta = 2 \tan^{-1} (1/3)$. Find $\alpha$.

Options:
A. \frac{\pi}{4}
B. \frac{\pi}{2}
C. \frac{\pi}{6}
D. \frac{\pi}{3}
Answer: B
A circle of maximal area is inscribed in the region bounded by the graph of $y = -x^2 - 7x + 12$ and the $x$ -axis. The radius of this circle is $\frac{\sqrt{p} + q}{r}$ , where $p$ , $q$ , and $r$ are integers and $q$ and $r$ are relatively prime. What is $p + q + r$ ?

Options:
A. 7
B. 11
C. 9
D. 8
E. 10
Answer: D
Newton's Law of Cooling states that the temperature $T$ of an object at any time $t$ can be described by the equation $T = T_{s} + (T_{0} - T_{s}) \cdot e^{-k \cdot t}$, where $T_{s}$ is the temperature of the surrounding environment, $T_{0}$ is the initial temperature of the object, and $k$ is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time $t$ such that $t$ is equal to a single logarithm.

Options:
A. t = ln((T_0 - T_s) / (T - T_s))^(-1/k)
B. t = ln((T_0 - T) / (T_s - T_0))^(-1/k)
C. t = ln((T_s - T) / (T_0 - T_s))^(-1/k)
D. t = ln((T - T_s) / (T_0 - T_s))^(-1/k)
E. t = ln((T_s - T_0) / (T - T_s))^(-1/k)
Answer: D
In triangle $ABC$ , $BA = 15$ , $AC = 20$ , and $BC = 25$ . In addition, there is a point $D$ lying on segment $BC$ such that $BD = 16$ . Given that the length of the radius of the circle through $B$ and $D$ that is tangent to side $AC$ can be expressed in the form $\frac{p}{q}$ , where $p$ and $q$ are relatively prime integers, find $p + q$ .

Options:
A. 25
B. 26
C. 28
D. 29
E. 24
F. 27
Answer: C
Evaluate the triple integral $\int_{0}^2 \int_{4}^6 \int_{3 \cdot z}^{3 \cdot z+2} (5-4 \cdot y) \, dx \, dz \, dy$ by using the transformation $u=x-3 \cdot z$, $v=4 \cdot y$, and $w=z$.

Options:
A. 14
B. 10
C. 8
D. 16
E. 12
F. 6
Answer: C
There exists a not necessarily convex quadrilateral $ABCD$ such that
$$ \angle B = \angle C = 60^\circ \quad \text{and} \quad \angle A = 30^\circ .$$
Lines $AB$ and $CD$ intersect at $E$ , lines $AD$ and $BC$ intersect at $F$ , and $EF$ meets $BD$ at $P$ . If $CF=AE=1$ , then $EP^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find the value of $m + n$ .

Options:
A. 18
B. 19
C. 14
D. 17
E. 15
F. 16
G. 13
Answer: F
What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?

Options:
A. \frac{\sqrt{3}}{2}
B. \frac{\sqrt{2}}{3}
C. \frac{\sqrt{2}}{4}
D. \frac{\sqrt{2}}{2}
E. \frac{\sqrt{5}}{2}
F. \frac{\sqrt{1}}{2}
G. \frac{\sqrt{4}}{2}
Answer: D
Find the sum of the series $\sum_{n=1}^\infty \frac{ 10 }{ n \cdot 9^n }$. (Use the formula $\int_{9}^\infty \frac{ 10 }{ x^{n+1} } \, dx = \frac{ 10 }{ n \cdot 9^n }$)

Options:
A. -10*ln(9/8)
B. -10*ln(8/9)
C. -10*ln(7/9)
D. -10*ln(1/9)
E. -10*ln(10/9)
F. -10*ln(9/7)
G. -10*ln(9/10)
Answer: B
In $\vartriangle ABC$ , let $D$ and $E$ be points on segment $BC$ such that $BD =29$ , $DE = 30$ , $EC = 31$ , $BD < BE < BC$ , and $\angle BAD = \angle EAC$ . If $\frac{AB}{AD}= \frac{AE}{AC} = \frac{m}{n}$ , where $m$ , $n$ are relatively prime positive integers, find $m + n$ .

Options:
A. 119
B. 118
C. 117
D. 120
E. 113
F. 114
G. 115
H. 116
Answer: D
Compute the derivative of the implicit function:
$$
2 \cdot y^4 - \frac{ 3 \cdot x + 3 \cdot y }{ 2 \cdot x - 2 \cdot y } = 0
$$

Options:
A. -\frac{y^2}{2\cdot\left(x^2+y^2\right)+x\cdot y}
B. -\frac{y^2}{2\cdot\left(x^2-y^2\right)+3\cdot x\cdot y}
C. -\frac{y^2}{2\cdot\left(x^2-y^2\right)-2\cdot x\cdot y}
D. -\frac{y^2}{2\cdot\left(x^2-y^2\right)+x\cdot y}
E. -\frac{y^2}{2\cdot\left(x^2+y^2\right)-x\cdot y}
F. -\frac{y^2}{2\cdot\left(x^2-y^2\right)+2\cdot x\cdot y}
G. -\frac{y^2}{2\cdot\left(x^2-y^2\right)-x\cdot y}
H. -\frac{y^2}{2\cdot\left(x^2-y^2\right)-3\cdot x\cdot y}
Answer: G
The sum
$$$\sum_{k=1}^{2007} \arctan\left(\frac{1}{k^2+k+1}\right)$$$
can be written in the form $\arctan\left(\frac{m}{n}\right)$ , where $\gcd(m,n) = 1$ . Compute the remainder when $m+n$ is divided by $100$ .

Options:
A. 18
B. 14
C. 12
D. 16
E. 13
F. 11
G. 15
H. 17
I. 19
Answer: D
Let $F$ be the positive integer consisting of 1998 decimal digits, all of which are 1; that is, \[F=1111\cdots 11.\] Determine the $482$th digit after the decimal point in the decimal expansion of $\sqrt{F}$.

Options:
A. 2
B. 1
C. 3
D. 5
E. 4
F. 8
G. 9
H. 6
I. 7
Answer: C
Given that:
$$\begin{align*}\left(\frac {1}{r}\right)\left(\frac {1}{s}\right)\left(\frac {1}{t}\right) &= \frac {3}{391} \\ r + \frac {1}{s} &= \frac {35}{46} \\ s + \frac {1}{t} &= \frac {1064}{23} \\ t + \frac {1}{r} &= \frac {529}{102}.\end{align*}$$
Then what is the smallest integer that is divisible $rs$ and $12t$ ?

Options:
A. 1564
B. 1172
C. 1175
D. 1955
E. 391
F. 1174
G. 1568
H. 782
I. 1176
J. 1173
Answer: H
A sequence $(a_n)$ is defined by $a_0=\dfrac{350}{27}$ , $a_1=200$ , and $a_n=36a_{n-1}-243a_{n-2}$ for $n\ge 2$ . Then, $\sqrt[3]{a_0 + \sqrt[3]{a_1 + \sqrt[3]{a_2 + \cdots}}}$ can be expressed in the form $\dfrac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Options:
A. 14
B. 15
C. 11
D. 10
E. 19
F. 17
G. 13
H. 16
I. 18
J. 12
Answer: C
If five points in a plane are given, then considering all possible triples of these points, we can form 30 angles. Denote the least of these angles by $\alpha$ . What is the greatest possible value of $\alpha$ ? Put your answer in degrees.

Options:
A. 35
B. 40
C. 34
D. 32
E. 30
F. 37
G. 45
H. 39
I. 36
J. 38
Answer: I
Find the number of quadruples of integers $(a, b, c, d)$ satisfying
$$a^2 + b^2 + c^2 + d^2 = ab + ac + ad + 9.$$

Options:
A. 310
B. 312
C. 314
D. 316
E. 315
F. 318
G. 313
H. 317
I. 319
J. 311
Answer: B
Let $S_n$ denote the set $\{1, 2,..., n\}$ , and define $f(S)$ , where $S$ is a subset of the positive integers, to output the greatest common divisor of all elements in $S$ , unless $S$ is empty, in which case it will output $0$ . Find the last three digits of $\sum_{S \subseteq S_{10}}f(S)$ , where $S$ ranges over all subsets of $S_{10}$ .

Options:
A. 111
B. 107
C. 103
D. 102
E. 105
F. 108
G. 106
H. 104
I. 110
J. 109
Answer: F
For each integer $1\le n\le 72$ , let $f(n)$ denote the smallest positive integer $d$ such that $n^d-1$ is divisible by $73$ . Find the sum of all positive integers $n\le 72$ such that $f(f(f(n))) = f(f(f(f(n))))$ .

Options:
A. 950
B. 955
C. 957
D. 948
E. 956
F. 952
G. 951
H. 949
I. 953
J. 954
Answer: G
$3$ Elm trees, $4$ Dogwood trees, and $5$ Oak trees are to be planted in a line in front of a library such that
i) No two Elm trees are next to each other.
ii) No Dogwood tree is adjacent to an Oak tree.
iii) All of the trees are planted.
How many ways can the trees be situated in this manner?

Options:
A. 146
B. 154
C. 150
D. 156
E. 152
F. 144
G. 140
H. 158
I. 148
J. 142
Answer: E
Let $A$ and $B$ be two points on a plane, with $AB = 28$ . Denote $\omega_1$ as the circle with center $A$ and radius $20$ , and denote $\omega_2$ as the circle with center $B$ and radius $32$ . Let the intersections of $\omega_1$ and $\omega_2$ be $X$ and $Y$ . Let $l$ be the line passing through $X$ that is perpendicular to $XY$ , and let $C$ be the second intersection of $\omega_2$ and $l$ , and let $D$ be the second intersection of $\omega_1$ and $l$ . If $[YCD]$ can be expressed in the form $m\sqrt{n}$ , where $n$ is not divisible by the square of a prime, find $m+n$ .

Options:
A. 649
B. 642
C. 646
D. 647
E. 645
F. 640
G. 644
H. 648
I. 643
J. 641
Answer: I
How many positive integers $n\le 2010$ satisfy $\phi (n)|n$ , where $ \phi (n)$ is the number of integers less than or equal to $n$ relatively prime to $n$ ?

Options:
A. 41
B. 44
C. 48
D. 40
E. 39
F. 47
G. 43
H. 42
I. 46
J. 45
Answer: A
Call a positive integer $\textit{troubling}$ if its only prime divisors are $2$ and $5$ . Two $\textit{troubling}$ numbers have a sum of $135,000$ . Find the number of positive integer divisors of their product.

Options:
A. 80
B. 90
C. 85
D. 83
E. 88
F. 82
G. 84
H. 87
I. 81
J. 86
Answer: E
Define $f(n) = {2006 \choose n} \sum^{2006}_{i=1}i^n$ . Find the remainder obtained when $\sum ^{2005}_{k=0}f(k)$ is divided by $1000$ .

Options:
A. 647
B. 653
C. 649
D. 654
E. 650
F. 645
G. 651
H. 652
I. 646
J. 648
Answer: J
For how many positive integers $n > 1$ is it possible to express $2005$ as the sum of $n$ distinct positive integers?

Options:
A. 66
B. 62
C. 61
D. 64
E. 67
F. 65
G. 60
H. 63
I. 58
J. 59
Answer: C
Let
$$$ \left(1 + x^3 \right) \left(1 + 2x^{3^2} \right) ... \left(1 + kx^{3^k} \right) ... \left(1 + 1997x^{3^{1997}} \right) = 1 + a_1x^{k_1} + a_2^{k_2} + ... + a_mx^{k_m}$$$
where $a_i \ne 0$ and $k_1 < k_2 < ... < k_m$ . Determine the remainder obtained when $a_{1997}$ is divided by $1000$ .

Options:
A. 282
B. 283
C. 279
D. 277
E. 285
F. 286
G. 281
H. 278
I. 284
J. 280
Answer: J
Let $a, b, c$ be positive reals such that $abc + a + b = c$ and
$$ \frac{19}{\sqrt{a^2 + 1}} + \frac{20}{\sqrt{b^2 + 1}} = 31.$$
The maximum possible value of $c^2$ can be written in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$ .

Options:
A. 365
B. 360
C. 359
D. 356
E. 363
F. 362
G. 357
H. 361
I. 358
J. 364
Answer: H
Let $n$ be the number of distinct ordered pairs $(x,y)$ of positive integers such that $1 \le x,y \le 49$ and $\dfrac{x^2 + xy + y^2}{49}$ is an integer. Find $n$ .

Options:
A. 136
B. 130
C. 133
D. 132
E. 135
F. 119
G. 121
H. 131
I. 120
J. 134
Answer: C
Suppose $b > 1$ is a real number where $\log_5 (\log_5 b + \log_b 125) = 2$ . Find $log_5 \left(b^{\log_5 b}\right) + log_b \left(125^{\log_b 125}\right).$

Options:
A. 617
B. 623
C. 619
D. 618
E. 620
F. 624
G. 621
H. 625
I. 616
J. 622
Answer: C
Consider the functions
$$\begin{align*} f_1 (x) &= 2x \\ f_2 (x) &= 8x+3 \end{align*}$$
Find the number of positive integers $n < 131072$ which can be written in the form $f_{a_1}(f_{a_2}( \dots f_{a_k}(1) \dots ))$ for some positive integer $k$ and some sequence $(a_1, a_2, a_3 ... a_{k-1}, a_k)$ where each $a_i$ is $1$ or $2$ .

Options:
A. 871
B. 866
C. 870
D. 868
E. 869
F. 872
G. 865
H. 867
I. 874
J. 873
Answer: C
Find the remainder when the number of functions $f\colon \mathbb{Z}\to \mathbb{Z}$ that satisfy the following conditions
i)
$$f(x-14f(x)+1) = f(x)$$
for all integers $x$
ii) The smallest root of $f$ is $35$ .
iii) $f(x)\le 14$ for all $x$ .
is divided by $1000$ .

Options:
A. 774
B. 769
C. 772
D. 776
E. 775
F. 771
G. 770
H. 773
I. 768
J. 777
Answer: G
Let $S=\{1003,1002,...,1\}$ be an ordered set. Define the operation: to 3-reduce, as follows: take 4 adjacent elements, $a_{k}$ , $a_{k+1}$ , $a_{k+2}$ , and $a_{k+3}$ , and replace those 4 elements with one element, $(a_{k}+a_{k+3}+\omega* a_{k+1}+\omega^{2}* a_{k+2})$ , where $\omega=\frac{-1+i\sqrt{3}}{2}$ . So $\{1,2,3,4,5\}$ could be 3-reduced into $\{5+2*\omega+3*\omega^{2},5\}$ . If $S=\{k\}$ after being 3-reduced $334$ times, find the maximum value for the real part of $k$ .

Options:
A. 506
B. 503
C. 505
D. 500
E. 507
F. 501.5
G. 508
H. 501
I. 502
J. 504
Answer: I
Sum the series \[\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{m^2 n}{3^m(n3^m+m3^n)}.\]

Options:
A. \frac{9}{33}
B. \frac{9}{37}
C. \frac{9}{35}
D. \frac{9}{32}
E. \frac{9}{30}
F. \frac{9}{31}
G. \frac{9}{39}
H. \frac{9}{34}
I. \frac{9}{38}
J. \frac{9}{36}
Answer: D
Consider a prime number $f$ exceeding 5. Define $f(f)$ as the count of infinite sequences $q_1, q_2, q_3, \dots$ where each $q_n$ is chosen from $\{{1, 2, \dots, f-1\}}$, satisfying the congruence $q_n q_{{n+2}} \equiv 1 + q_{{n+1}} \pmod{{f}}$ for all positive integers $n$. Determine the sum of all possible residues of $f(f)$ when divided by 5.

Options:
A. 9
B. 2
C. 6
D. 1
E. 0
F. 4
G. 3
H. 8
I. 5
J. 7
Answer: B
For each positive integer $n$, let $a(n)$ be the number of zeroes in the base 3 representation of $n$. Find the upper bound of the set of real numbers $x$ for which the series \[ \sum_{n=1}^\infty \frac{x^{a(n)}}{n^3} \] converge?

Options:
A. 28
B. 29
C. 26
D. 25
E. 24
F. 27
G. 26.5
H. 23
I. 22
J. 21
Answer: D
Find the maximum value of \[ \int_0^y \sqrt{x^4 + (y-y^2)^2}\,dx \] for $0 \leq y \leq 1$.

Options:
A. \frac{1}{4}
B. \frac{1}{3}
C. \frac{1}{10}
D. \frac{1}{6}
E. \frac{1}{2}
F. \frac{1}{12}
G. \frac{1}{7}
H. \frac{1}{9}
I. \frac{1}{8}
J. \frac{1}{5}
Answer: B
Determine the greatest possible value of $\sum_{i=1}^{10} \cos(3x_i)$ for real numbers $x_1,x_2,\dots,x_{10}$ satisfying $\sum_{i=1}^{10} \cos(x_i) = 0$.

Options:
A. \frac{480}{49}
B. \frac{480}{56}
C. \frac{480}{51}
D. \frac{480}{48}
E. \frac{480}{47}
F. \frac{480}{52}
G. \frac{480}{55}
H. \frac{480}{53}
I. \frac{480}{50}
J. \frac{480}{54}
Answer: A
Given real numbers $b_0, b_1, \dots, b_{2019}$ with $b_{2019} \neq 0$, let $z_1,z_2,\dots,z_{2019}$ be the roots in the complex plane of the polynomial \[ P(z) = \sum_{k=0}^{2019} b_k z^k. \] Let $\mu = (|z_1| + \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\dots,z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu \geq M$ for all choices of $b_0,b_1,\dots, b_{2019}$ that satisfy \[ 1 \leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019. \]

Options:
A. 2019^{-1/2019}
B. 2019^{-1/2026}
C. 2019^{-1/2025}
D. 2019^{-1/2020}
E. 2019^{-1/2017}
F. 2019^{-1/2024}
G. 2019^{-1/2021}
H. 2019^{-1/2018}
I. 2019^{-1/2022}
J. 2019^{-1/2023}
Answer: A
Consider all positive integers written in base 35, are alternating 1's and 0's, beginning and ending with 1. How many of these numbers are prime?

Options:
A. \boxed{6}
B. \boxed{4}
C. \boxed{8}
D. \boxed{7}
E. \boxed{9}
F. \boxed{0}
G. \boxed{1}
H. \boxed{5}
I. \boxed{3}
J. \boxed{2}
Answer: G
Evaluate the integral by interchanging the order of integration:
$$
\int_{1}^{27} \int_{1}^2 \left(\sqrt[3]{x} + \sqrt[3]{y}\right) \, dy \, dx
$$

Options:
A. 39\cdot\sqrt[3]{2}+\frac{81}{9}
B. 39\cdot\sqrt[3]{2}+\frac{81}{11}
C. 39\cdot\sqrt[3]{2}+\frac{81}{2}
D. 39\cdot\sqrt[3]{2}+\frac{81}{4}
E. 39\cdot\sqrt[3]{2}+\frac{81}{7}
F. 39\cdot\sqrt[3]{2}+\frac{81}{10}
G. 39\cdot\sqrt[3]{2}+\frac{81}{8}
H. 39\cdot\sqrt[3]{2}+\frac{81}{6}
I. 39\cdot\sqrt[3]{2}+\frac{81}{3}
J. 39\cdot\sqrt[3]{2}+\frac{81}{5}
Answer: C
Compute the integral:
$$
-2 \cdot \int x^{-4} \cdot \left(4+x^2\right)^{\frac{ 1 }{ 2 }} \, dx
$$

Options:
A. C + \frac{1}{10} \cdot \left(\frac{4}{x^2} + 1\right) \cdot \sqrt{\frac{4}{x^2} + 1}
B. C + \frac{1}{3} \cdot \left(\frac{4}{x^2} + 1\right) \cdot \sqrt{\frac{4}{x^2} + 1}
C. C + \frac{1}{6} \cdot \left(\frac{4}{x^2} + 1\right) \cdot \sqrt{\frac{4}{x^2} + 1}
D. C + \frac{1}{8} \cdot \left(\frac{4}{x^2} + 1\right) \cdot \sqrt{\frac{4}{x^2} + 1}
E. C + \frac{1}{7} \cdot \left(\frac{4}{x^2} + 1\right) \cdot \sqrt{\frac{4}{x^2} + 1}
F. C + \frac{1}{12} \cdot \left(\frac{4}{x^2} + 1\right) \cdot \sqrt{\frac{4}{x^2} + 1}
G. C + \frac{1}{4} \cdot \left(\frac{4}{x^2} + 1\right) \cdot \sqrt{\frac{4}{x^2} + 1}
H. C + \frac{1}{9} \cdot \left(\frac{4}{x^2} + 1\right) \cdot \sqrt{\frac{4}{x^2} + 1}
I. C + \frac{1}{5} \cdot \left(\frac{4}{x^2} + 1\right) \cdot \sqrt{\frac{4}{x^2} + 1}
J. C + \frac{1}{11} \cdot \left(\frac{4}{x^2} + 1\right) \cdot \sqrt{\frac{4}{x^2} + 1}
Answer: C
Find the degree and leading coefficient for the given polynomial $f(x) = x^3 \cdot (4 \cdot x - 3)^2$.

Options:
A. 5, 24
B. 5, 8
C. 5, 36
D. 5, 18
E. 5, 4
F. 5, 20
G. 5, 32
H. 5, 12
I. 5, 16
J. 6, 16
Answer: I
Calculate the integral:
$$
\int \frac{ 3 \cdot x + 4 }{ \left( x^2 + 1 \cdot x + 7 \right)^2 } \, dx
$$

Options:
A. frac{frac{5}{27}cdot x-frac{38}{27}}{frac{27}{4}+left(x+frac{1}{2}right)^2}+frac{30cdot sqrt{3}}{729}cdot arctanleft(sqrt{frac{4}{27}}cdot left(x+frac{1}{2}right)right)+frac{1}{162}+C
B. frac{frac{5}{27}cdot x-frac{38}{27}}{frac{27}{4}+left(x+frac{1}{2}right)^2}+frac{30cdot sqrt{3}}{729}cdot arctanleft(sqrt{frac{4}{27}}cdot left(x+frac{1}{2}right)right)+frac{1}{81}+C
C. frac{frac{5}{27}cdot x-frac{38}{27}}{frac{27}{4}+left(x+frac{1}{2}right)^2}+frac{30cdot sqrt{3}}{729}cdot arctanleft(sqrt{frac{4}{27}}cdot left(x+frac{1}{2}right)right)-frac{1}{27}+C
D. frac{frac{5}{27}cdot x-frac{38}{27}}{frac{27}{4}+left(x+frac{1}{2}right)^2}+frac{30cdot sqrt{3}}{729}cdot arctanleft(sqrt{frac{4}{27}}cdot left(x+frac{1}{2}right)right)+frac{1}{54}+C
E. frac{frac{5}{27}cdot x-frac{38}{27}}{frac{27}{4}+left(x+frac{1}{2}right)^2}+frac{30cdot sqrt{3}}{729}cdot arctanleft(sqrt{frac{4}{27}}cdot left(x+frac{1}{2}right)right)+frac{1}{108}+C
F. frac{frac{5}{27}cdot x-frac{38}{27}}{frac{27}{4}+left(x+frac{1}{2}right)^2}+frac{30cdot sqrt{3}}{729}cdot arctanleft(sqrt{frac{4}{27}}cdot left(x+frac{1}{2}right)right)+frac{1}{27}+C
G. frac{frac{5}{27}cdot x-frac{38}{27}}{frac{27}{4}+left(x+frac{1}{2}right)^2}+frac{30cdot sqrt{3}}{729}cdot arctanleft(sqrt{frac{4}{27}}cdot left(x+frac{1}{2}right)right)+C
H. frac{frac{5}{27}cdot x-frac{38}{27}}{frac{27}{4}+left(x+frac{1}{2}right)^2}+frac{30cdot sqrt{3}}{729}cdot arctanleft(sqrt{frac{4}{27}}cdot left(x-frac{1}{2}right)right)+C
I. frac{frac{5}{27}cdot x-frac{38}{27}}{frac{27}{4}+left(x+frac{1}{2}right)^2}+frac{30cdot sqrt{3}}{729}cdot arctanleft(sqrt{frac{4}{27}}cdot left(x+frac{3}{2}right)right)+C
J. frac{frac{5}{27}cdot x-frac{38}{27}}{frac{27}{4}+left(x+frac{1}{2}right)^2}+frac{30cdot sqrt{3}}{729}cdot arctanleft(sqrt{frac{4}{27}}cdot left(x+frac{1}{2}right)right)+frac{1}{216}+C
Answer: G
Given positive numbers $a$, $b$, $c$, and $d$, find the minimum value of
$$
S = \frac{ a }{ b+c+d }+\frac{ b }{ a+c+d }+\frac{ c }{ a+b+d }+\frac{ d }{ a+b+c }+\frac{ b+c+d }{ a }+\frac{ a+c+d }{ b }+\frac{ a+b+d }{ c }+\frac{ a+b+c }{ d }
$$

Options:
A. 13.67
B. 14
C. 12
D. 13.33
E. 13
F. 13.5
G. 13.34
H. 13.35
I. 40/3
J. 13.32
Answer: I
If a culture of bacteria doubles in $3$ hours, how many hours does it take to multiply by $10$? 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. Leave your answer in exact form.

Options:
A. 3 * ln(10) / (ln(2) + ln(3))
B. 3 * ln(10) / (2 * ln(2))
C. 3 * ln(10) / (ln(2) + 1)
D. 3 * ln(10) / ln(2)
E. 3 * ln(10) / ln(4)
F. 3 * ln(10) / (ln(2) + ln(5))
G. 3 * ln(10) / (3 * ln(2))
H. 3 * ln(2) / ln(10)
I. 3 * ln(5) / ln(2)
J. 3 * ln(20) / ln(2)
Answer: D
Compute $\lim_{x \to 0}\left(\frac{ 3 \cdot \cos(x)+6 }{ 4 \cdot x^3 \cdot \sin(x) }-\frac{ 9 }{ 4 \cdot x^4 }\right)$. Use the expansion of the function in the Taylor series.

Options:
A. 1/80
B. 1/70
C. 1/90
D. 1/140
E. 1/160
F. 1/60
G. 1/40
H. 1/100
I. 1/120
J. 1/150
Answer: A
Calculate the integral:
$$
\int \frac{ \sqrt[5]{x}+\sqrt[5]{x^4}+x \cdot \sqrt[5]{x} }{ x \cdot \left(1+\sqrt[5]{x^2}\right) } \, dx
$$

Options:
A. C + 5 * arctan(sqrt[5]{x}) + 5/5 * sqrt[5]{x}^4
B. C + 5 * arctan(sqrt[5]{x}) + 5/4 * sqrt[5]{x}^4
C. C + 5 * arctan(sqrt[5]{x}) + 5/5 * sqrt[5]{x}^5
D. C + 5 * arctan(sqrt[5]{x}) + 5/6 * sqrt[5]{x}^6
E. C + 5 * arctan(sqrt[5]{x}) + 5/8 * sqrt[5]{x}^8
F. C + 5 * arctan(sqrt[5]{x}) + 5/3 * sqrt[5]{x}^3
G. C + 5 * arctan(sqrt[5]{x}) + 5/2 * sqrt[5]{x}^2
H. C + 5 * arctan(sqrt[5]{x}) + 5/9 * sqrt[5]{x}^9
I. C + 5 * arctan(sqrt[5]{x}) + 5/10 * sqrt[5]{x}^10
J. C + 5 * arctan(sqrt[5]{x}) + 5/7 * sqrt[5]{x}^7
Answer: B
Find the area of the surface formed by rotating the arc of the circle $x^2 + y^2 = 1$ between the points $(1,0)$ and $(0,1)$ in the first quadrant, around the line $x + y = 1$.

Options:
A. \frac{4\cdot\pi-\pi^2}{7\sqrt{2}}
B. \frac{4\cdot\pi-\pi^2}{\sqrt{2}}
C. \frac{4\cdot\pi-\pi^2}{3\sqrt{2}}
D. \frac{4\cdot\pi-\pi^2}{2\sqrt{2}}
E. \frac{4\cdot\pi-\pi^2}{10\sqrt{2}}
F. \frac{4\cdot\pi-\pi^2}{5\sqrt{2}}
G. \frac{4\cdot\pi-\pi^2}{6\sqrt{2}}
H. \frac{4\cdot\pi-\pi^2}{9\sqrt{2}}
I. \frac{4\cdot\pi-\pi^2}{8\sqrt{2}}
J. \frac{4\cdot\pi-\pi^2}{4\sqrt{2}}
Answer: B
Let $Q$ be the solid situated outside the sphere $x^2+y^2+z^2=z$ and inside the upper hemisphere $x^2+y^2+z^2=R^2$, where $R>1$. If the density of the solid is $\rho(x,y,z) = \frac{ 1 }{ \sqrt{x^2+y^2+z^2} }$, find $R$ such that the mass of the solid is $\frac{ 7 \cdot \pi }{ 2 }$.

Options:
A. R = \frac{\sqrt{138}}{6}
B. R = \frac{\sqrt{138}}{11}
C. R = \frac{\sqrt{138}}{13}
D. R = \frac{\sqrt{138}}{5}
E. R = \frac{\sqrt{138}}{9}
F. R = \frac{\sqrt{138}}{10}
G. R = \frac{\sqrt{138}}{12}
H. R = \frac{\sqrt{138}}{7}
I. R = \frac{\sqrt{138}}{4}
J. R = \frac{\sqrt{138}}{8}
Answer: A
For the curve $x = 10 \left(t - \sin(t)\right)$, $y = 10 \left(1 - \cos(t)\right)$, determine the curvature. Use $a=10$.

Options:
A. 1 / (20 * |sin(t / 2)|)
B. 1 / (10 * |sin(t / 2)|)
C. 1 / (45 * |sin(t / 2)|)
D. 1 / (50 * |sin(t / 2)|)
E. 1 / (80 * |sin(t / 2)|)
F. 1 / (40 * |sin(t / 2)|)
G. 1 / (30 * |sin(t / 2)|)
H. 1 / (60 * |sin(t / 2)|)
I. 1 / (70 * |sin(t / 2)|)
J. 1 / (90 * |sin(t / 2)|)
Answer: F
Use the Rational Zero Theorem to find the real solution(s) for the equation: $8 \cdot x^4 + 26 \cdot x^3 + 39 \cdot x^2 + 26 \cdot x + 6 = 0$

Options:
A. -1/2, -1/2
B. -1/2, -1/7
C. -1/2, -1/6
D. -1/2, -3/4
E. -1/2, -1/8
F. -1/2, -1/3
G. -1/2, -2/3
H. -1/2, -1/5
I. -1/2, -1/4
J. -1/2, -1/10
Answer: D
Find the radius of convergence of the series:
$$
\sum_{n=1}^\infty \left( \left(n^{\frac{ 1 }{ n }}-1\right)^n \cdot x^n \right)
$$

Options:
A. e^(-1)
B. e
C. e^(-1/2)
D. e^(-2)
E. 2
F. 0
G. e^(1/2)
H. e^2
I. 1
J. ∞
Answer: J
Solve the integral:
$$
\int \left(\frac{ x+6 }{ x-6 } \right)^{\frac{ 3 }{ 2 }} \, dx
$$

Options:
A. C + sqrt((x+6)/(x-6)) * (x-36) - 18 * ln(abs((sqrt(x-6) - sqrt(x+6)) / (sqrt(x-6) + sqrt(x+6))))
B. C + sqrt((x+6)/(x-6)) * (x-30) - 18 * ln(abs((sqrt(x-6) + sqrt(x+6)) / (sqrt(x-6) - sqrt(x+6))))
C. C + sqrt((x+6)/(x-6)) * (x-32) - 18 * ln(abs((sqrt(x-6) - sqrt(x+6)) / (sqrt(x-6) + sqrt(x+6))))
D. C + sqrt((x+6)/(x-6)) * (x-34) - 18 * ln(abs((sqrt(x-6) - sqrt(x+6)) / (sqrt(x-6) + sqrt(x+6))))
E. C + sqrt((x+6)/(x-6)) * (x-26) - 18 * ln(abs((sqrt(x-6) - sqrt(x+6)) / (sqrt(x-6) + sqrt(x+6))))
F. C + sqrt((x+6)/(x-6)) * (x-28) - 18 * ln(abs((sqrt(x-6) + sqrt(x+6)) / (sqrt(x-6) - sqrt(x+6))))
G. C + sqrt((x+6)/(x-6)) * (x-24) - 18 * ln(abs((sqrt(x-6) - sqrt(x+6)) / (sqrt(x-6) + sqrt(x+6))))
H. C + sqrt((x+6)/(x-6)) * (x-28) - 18 * ln(abs((sqrt(x-6) - sqrt(x+6)) / (sqrt(x-6) + sqrt(x+6))))
I. C + sqrt((x+6)/(x-6)) * (x-30) - 18 * ln(abs((sqrt(x-6) - sqrt(x+6)) / (sqrt(x-6) + sqrt(x+6))))
J. C + sqrt((x+6)/(x-6)) * (x-22) - 18 * ln(abs((sqrt(x-6) - sqrt(x+6)) / (sqrt(x-6) + sqrt(x+6))))
Answer: I
Find the integral:
$$
\int \frac{ \arcsin(4 \cdot x) }{ \sqrt{4 \cdot x+1} } \, dx
$$

Options:
A. 1/2 * sqrt(4 * x + 1) * arcsin(4 * x) - (C + sqrt(1 + 4 * x))
B. 1/2 * sqrt(4 * x + 1) * arcsin(4 * x) - (C - sqrt(1 + 4 * x))
C. 1/2 * sqrt(4 * x + 1) * arcsin(4 * x) - (C + sqrt(1 - 4 * x))
D. 1/2 * sqrt(4 * x + 1) * arcsin(4 * x) - (C - sqrt(1 - 4 * x))
Answer: D
$C$ is on a semicircle with diameter $AB$ and center $O.$ Circle radius $r_1$ is tangent to $OA,OC,$ and arc $AC,$ and circle radius $r_2$ is tangent to $OB,OC,$ and arc $BC$ . It is known that $\tan AOC=\frac{24}{7}$ . The ratio $\frac{r_2} {r_1}$ can be expressed $\frac{m} {n},$ where $m,n$ are relatively prime positive integers. Find $m+n.$

Options:
A. 63
B. 61
C. 59
D. 67
E. 65
Answer: C