MikaStars39/nano-eval huggingface dataset:
Let \( A \) be an \( n \)-order square matrix, \( r(A) = n-3 \), and \( \alpha_{1}, \alpha_{2}, \alpha_{3} \) are three linearly independent solution vectors of the linear equation system \( Ax = 0 \). Then the fundamental solution set of this equation system is ( ).

Options:
A. \alpha_{1} + \alpha_{2} + \alpha_{3}, \alpha_{2} + \alpha_{3} + \alpha_{1}, \alpha_{3} + \alpha_{1} + \alpha_{2}
B. \( \alpha_{1} + \alpha_{2}, \alpha_{2} + \alpha_{3}, \alpha_{3} + \alpha_{1} \)
C. \alpha_{1} + 2\alpha_{2}, \alpha_{2} + \alpha_{3}, \alpha_{3} + \alpha_{1}
D. \alpha_{1} + 2\alpha_{2}, \alpha_{2} + 2\alpha_{3}, \alpha_{3} + 2\alpha_{1}
E. \( 2lpha_{2} - lpha_{1}, rac{1}{2}lpha_{3} - lpha_{2}, lpha_{1} - lpha_{3} \)
F. \alpha_{1} + \alpha_{2} + \alpha_{3}, \alpha_{2} + \alpha_{1} + \alpha_{3}, \alpha_{3} + \alpha_{2} + \alpha_{1}
G. \alpha_{1} - \alpha_{2}, \alpha_{2} - \alpha_{3}, \alpha_{3} - \alpha_{1}
H. \( \alpha_{1} + 2\alpha_{2}, \alpha_{2} + 2\alpha_{3}, \alpha_{3} + 2\alpha_{1} \)
I. \( lpha_{1} + lpha_{2} + lpha_{3}, lpha_{3} - lpha_{2}, -lpha_{1} - 2lpha_{3} \)
J. \( lpha_{2} - lpha_{1}, lpha_{3} - lpha_{2}, lpha_{1} - lpha_{3} \)
Answer: B
10010110.1001B = ( )BCD

Options:
A. 1001 0110 0000.1001
B. 0001 0101 0000.0101 0110 0010 1011
C. 1001 0110 0000. 0101 0110 0010 0101
D. 0001 0101 0000. 1001
E. 0001 0101 0000.0101 0110 0010 0101
F. 0001 0101 0000.0101 0110 0010 1010
G. 0001 0101 0000.0101 0110 0010 0111
H. 0001 0101 0000.0101 0110 0010 0110
I. 0001 0101 0000.0101 0110 0010 1001
J. 0001 0101 0000.0101 0110 0010 1000
Answer: E
Determine the number of five digit integers $37abc$ (in base $10$ ), such that $37abc, 37bca, 37cab$ is divisible by $37$ .

Options:
A. 34
B. 27
C. 35
D. 30
E. 31
F. 36
G. 33
H. 29
I. 28
J. 32
Answer: I
For all positive integers $n$ , let $1=c_1<c_2<...<c_k=n$ be the positive integer factors of $n$ . Define the function $f$ as follows: $f(1)=1$ , and for all integers $n>1$ the equation $2020f(n) = \displaystyle\sum_{i=1}^{k} f(c_i)$ holds true. The value of $f(3\cdot 2^{2020})$ can be expressed as $p\cdot q^r\cdot s^{-t}$ for positive integers $p$ , $q$ , $r$ , $s$ , and $t$ such that $p+q+s$ is as small as possible and $\gcd{(q,s)}=1$ . Compute the remainder when $p+q+r+s+t$ is divided by $1000$ .

Options:
A. 87
B. 84
C. 89
D. 81
E. 86
F. 88
G. 82
H. 80
I. 83
J. 85
Answer: G
Given that
$$$\sum^{491}_{a=1}\sum^{491}_{b=1}\frac{1}{(a + bi)^4 - 1}$$$
can be expressed in the form $\frac{p+qi}{r}$ , where gcd $(p, q, r) = 1$ and $r > 0$ , find the unique integer $k$ between $0$ and $982$ inclusive such that $983$ divides $(p+q)-kr$ . [Note: $983$ is a prime.]

Options:
A. 858
B. 855
C. 854
D. 853
E. 857
F. 860
G. 856
H. 851
I. 852
J. 859
Answer: F
Akshaj has a special coin. If $n$ is the largest non-negative integer such that his coin's last $n$ flips were all heads, then his coin will flip heads with a probability of $\dfrac{1}{2^{n+1}}$ (and tails with probability $1 - \frac{1}{2^{n+1}}$ ). For example, on the first flip, the coin will flip heads with a probability of $\dfrac{1}{2}$ because $n=0$ . Let $x$ be the expected number of flips it will take Akshaj to flip four heads in a row. Find $\dfrac{1}{2} x$ .

Options:
A. 841
B. 843
C. 842
D. 840
E. 835
F. 837
G. 844
H. 838
I. 836
J. 839
Answer: D
Compute the smallest positive integer $k$ such that the fraction
$$$\frac{7k+100}{5k-3}$$$
is reducible.

Options:
A. 216
B. 213
C. 214
D. 212
E. 217
F. 215
G. 210
H. 211
I. 218
J. 209
Answer: J
We say that a function $f:\mathbb{N}\to\mathbb{R}$ is a MODEL FUNCTION if
$$f(mn) = f(m)f(n)+f(m)+f(n)$$
for all relatively prime positive integers $m$ and $n$ and if
$$\sum_{d\mid n} f(d) = n-\tau(n)$$
for all $n\ge 100$ (here $\tau(n)$ is the number of divisors of $n$ ). Let $M$ be the sum of the possible values of $f(10^{100})$ . Find the sum of the digits of $M$ .

Options:
A. 895
B. 893
C. 892
D. 894
E. 897
F. 899
G. 891
H. 896
I. 898
J. 890
Answer: D
Suppose a positive integer $n$ is the perfect $k^{th}$ power of a prime, where $k$ is a positive integer. If the number of factors of $2020n^7$ is a multiple of the number of factors of $n$ , find the sum of all possible values of $k$ .

Options:
A. 256
B. 251
C. 253
D. 257
E. 250
F. 254
G. 259
H. 258
I. 252
J. 255
Answer: F
Kelvin the Frog generates an infinite sequence $a_n$ , which satisfies the relation
$$$a_{m-n} + a_{m+n} = 2a_ma_n$$$
for any nonnegative integers $m\ge n$ . If $a_0 = 1$ , let $x$ be the largest possible value of $a_1$ such that $\prod^{\lfloor log_2{2014} \rfloor}_{i=0} a_{2^i} = \frac{1}{2048}$ . When $a_1 = x$ , compute the remainder when $|2049a_{2049}|$ is divided by $1000$ .

Options:
A. 52
B. 55
C. 54
D. 49
E. 50
F. 57
G. 53
H. 51
I. 56
J. 58
Answer: D
Twenty red points are equally spaced on the circumference of a circle. How many right triangles are there whose vertices are all red points?

Options:
A. 188
B. 190
C. 168
D. 176
E. 170
F. 172
G. 182
H. 180
I. 184
J. 160
Answer: H
Let $P$ be a monic polynomial of degree $n \ge 1$ such that
$$$[P(x)]^3 = [P(x)]^2 - P(x) + 6$$$
for $x = 1, 2, . . . , n$ . Let $n_0$ be the smallest $n$ such that $P(0) > [P(1) + P(2) + ... + P(n)]^3$ . Find the remainder when $P(0)$ is divided by $1000$ given $n = n_0$ .

Options:
A. 323
B. 321
C. 325
D. 326
E. 324
F. 327
G. 322
H. 328
I. 320
J. 329
Answer: G
Compute the sum of the prime divisors of $1^2 + 2^2 + 3^2 +...+ 2005^2$ .

Options:
A. 679
B. 682
C. 687
D. 684
E. 688
F. 686
G. 680
H. 681
I. 685
J. 683
Answer: G
Let $z$ be a complex number such that $z^7=1$ . Find the remainder when the sum of all possible distinct values of $(z^4+z^2+z+2)^{27}$ is divided by 1000.

Options:
A. 559
B. 556
C. 557
D. 553
E. 552
F. 550
G. 555
H. 551
I. 558
J. 554
Answer: B
Find the sum of the squares of the roots, real or complex, of the system of simultaneous equations
$$$x+y+z=3,~x^2+y^2+z^2=3,~x^3+y^3+z^3 =3.$$$
Clarification:
If the solutions are: $(x_{1},y_{1},z_{1}),$ and $(x_{2},y_{2},z_{2},)$ find $x_{1}^{2}+y_{1}^{2}+z_{1}^{2}+x_{2}^{2}+y_{2}^{2}+z_{2}^{2}.$

Options:
A. 8
B. 4
C. 12
D. 11
E. 3
F. 10
G. 6
H. 5
I. 9
J. 7
Answer: E
Let $\triangle ABC$ have $BC=\sqrt{7}$ , $CA=1$ , and $AB=3$ . If $\angle A=\frac{\pi}{n}$ where $n$ is an integer, find the remainder when $n^{2007}$ is divided by $1000$ .

Options:
A. 188
B. 186
C. 182
D. 197
E. 185
F. 183
G. 181
H. 184
I. 189
J. 187
Answer: J
Let $n$ represent the smallest integer that satisfies the following conditions:
$\frac n2$ is a perfect square.
$\frac n3$ is a perfect cube.
$\frac n5$ is a perfect fifth.
How many divisors does $n$ have that are not multiples of $10$ ?

Options:
A. 242
B. 248
C. 244
D. 241
E. 243
F. 245
G. 249
H. 240
I. 246
J. 247
Answer: A
Draw in the diagonals of a regular octagon. What is the sum of all distinct angle measures, in degrees, formed by the intersections of the diagonals in the interior of the octagon?

Options:
A. 405
B. 435
C. 480
D. 420
E. 455
F. 445
G. 465
H. 450
I. 495
J. 460
Answer: H
A non-rectangle parallelogram is called integrized if it has integer sides and integer altitudes. How many non-congruent integrized parallelograms have a perimeter of $1000$ ?

Options:
A. 209
B. 208
C. 200
D. 207
E. 201
F. 203
G. 204
H. 205
I. 206
J. 202
Answer: J
Justin has a complete set of alphabet refrigerator magnets. That is, he has 26 distinct magnets, one of each letter. Given that there are five vowels $(A, E, I, O, U)$ , and he arranges all of the letters into a line in a random order, the probability that no two vowels are adjacent can be written as $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p+q$ is divided by 1000.

Options:
A. 185
B. 193
C. 195
D. 197
E. 183
F. 191
G. 199
H. 189
I. 181
J. 187
Answer: J
How many $4$ -digit numbers are there such that the $3$ rd digit (from the left) is the sum of the first two and the $4$ th is greater than the sum of the $1$ st and the $3$ rd?

Options:
A. 49
B. 50
C. 46
D. 47
E. 48
F. 42
G. 44
H. 40
I. 45
J. 41
Answer: B
Let $S$ be a nonempty closed bounded convex set in the plane. Let $K$ be a line and $t$ a positive number. Let $L_1$ and $L_2$ be support lines for $S$ parallel to $K_1$, and let $\overline{L}$ be the line parallel to $K$ and midway between $L_1$ and $L_2$. Let $B_S(K, t)$ be the band of points whose distance from $\overline{L}$ is at most $(t/2)w$, where $w$ is the distance between $L_1$ and $L_2$. What is the smallest $t$ such that \[ S \cap \bigcap_K B_S(K, t) \neq \emptyset \] for all $S$? ($K$ runs over all lines in the plane.)

Options:
A. \frac{1}{2}
B. \frac{1}{8}
C. \frac{1}{7}
D. \frac{1}{5}
E. \frac{1}{6}
F. \frac{1}{3}
G. \frac{2}{5}
H. \frac{1}{10}
I. \frac{1}{4}
J. \frac{1}{9}
Answer: F
Determine the count of positive integers $N$ that fulfill all of these criteria: \begin{{enumerate}} \item[(i)] $N$ is a multiple of 252500. \item[(ii)] The number of decimal digits in $N$ does not exceed 252500. \item[(iii)] When written in decimal notation, $N$ consists of a sequence of ones immediately followed by a sequence of zeros (either sequence may be empty). \end{{enumerate}}

Options:
A. 7969341870
B. 7969341875
C. 7969341874
D. 7969341871
E. 7969341879
F. 7969341878
G. 7969341873
H. 7969341877
I. 7969341876
J. 7969341872
Answer: I
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point $(2021, 2021)$?

Options:
A. 581
B. 577
C. 573
D. 575
E. 580
F. 574
G. 576
H. 579
I. 572
J. 578
Answer: J
Two real numbers $h$ and $r$ are chosen at random in the interval (0,1) with respect to the uniform distribution. What is the probability that the closest integer to $h/r$ is even? Express the answer in the form $r+s\pi$, where $r$ and $s$ are rational numbers.

Options:
A. \frac{5 - 3\pi}{4}
B. \frac{5 - 2\pi}{4}
C. \frac{5 - \pi}{4}
D. \frac{5 - 4\pi}{4}
E. \frac{5 - \pi}{8}
F. \frac{5 - \pi}{3}
G. \frac{5 - \pi}{6}
H. \frac{5 - \pi}{5}
I. \frac{5 - \pi}{2}
J. \frac{5 - \pi}{7}
Answer: C
Evaluate \[ \lim_{x \to 1^-} \prod_{n=0}^\infty \left(\frac{1 + x^{n+1}}{1 + x^n}\right)^{x^n}. \]

Options:
A. \frac{1}{2\sqrt{e}}
B. \frac{1}{e}
C. \frac{1}{2e}
D. \frac{1}{\sqrt{e}}
E. \frac{1}{2}
F. \frac{1}{e^2}
G. \frac{1}{2e^2}
H. \frac{2}{e}
I. \frac{1}{2} \cdot \frac{2}{e}
J. \frac{e}{2}
Answer: H
A rectangle, $HOMF$, has sides $HO=11$ and $OM=5$. A triangle $ABC$ has $H$ as the intersection of the altitudes, $O$ the center of the circumscribed circle, $M$ the midpoint of $BC$, and $F$ the foot of the altitude from $A$. What is the length of $BC$?

Options:
A. 27
B. 32
C. 25
D. 34
E. 26
F. 28
G. 22
H. 24
I. 20
J. 30
Answer: F
Find the positive number $m$ such that the area in the first quadrant bounded by the line $y = \frac{1}{35} x$, the $x$-axis, and the ellipse $\frac{1}{9801} x^2 + y^2 = 1$ is equal to the area of the region in the first quadrant bounded by the line $y = mx$, the $y$-axis, and the ellipse $\frac{1}{9801} x^2 + y^2 = 1$.

Options:
A. \frac{35}{9806}
B. \frac{35}{9803}
C. \frac{35}{9807}
D. \frac{35}{9801}
E. \frac{35}{9805}
F. \frac{35}{9808}
G. \frac{35}{9800}
H. \frac{35}{9809}
I. \frac{35}{9802}
J. \frac{35}{9804}
Answer: D
Solve the integral:
$$
\int \frac{ 1 }{ \sin(x)^7 \cdot \cos(x) } \, dx
$$

Options:
A. C + ln(|tan(x)|) - 3/(2*(tan(x))^2) - 3/(4*(tan(x))^4) - 1/(5*(tan(x))^6)
B. C + ln(|tan(x)|) - 3/(2*(tan(x))^2) - 3/(4*(tan(x))^4) - 1/(12*(tan(x))^6)
C. C + ln(|tan(x)|) - 3/(2*(tan(x))^2) - 3/(4*(tan(x))^4) - 1/(9*(tan(x))^6)
D. C + ln(|tan(x)|) - 3/(2*(tan(x))^2) - 3/(4*(tan(x))^4) - 1/(6*(tan(x))^6)
E. C + ln(|tan(x)|) - 3/(2*(tan(x))^2) - 3/(4*(tan(x))^4) - 1/(7*(tan(x))^6)
F. C + ln(|tan(x)|) - 3/(2*(tan(x))^2) - 3/(4*(tan(x))^4) - 1/(14*(tan(x))^6)
G. C + ln(|tan(x)|) - 3/(2*(tan(x))^2) - 3/(4*(tan(x))^4) - 1/(8*(tan(x))^6)
H. C + ln(|tan(x)|) - 3/(2*(tan(x))^2) - 3/(4*(tan(x))^4) - 1/(11*(tan(x))^6)
I. C + ln(|tan(x)|) - 3/(2*(tan(x))^2) - 3/(4*(tan(x))^4) - 1/(10*(tan(x))^6)
J. C + ln(|tan(x)|) - 3/(2*(tan(x))^2) - 3/(4*(tan(x))^4) - 1/(13*(tan(x))^6)
Answer: D
Compute the integral:
$$
\int_{0}^1 \frac{ \sqrt{x}+1 }{ \sqrt[3]{x}+1 } \, dx
$$

Options:
A. 3*ln(2)+3*pi/2-409/77
B. 3*ln(2)+3*pi/2-409/78
C. 3*ln(2)+3*pi/2-409/71
D. 3*ln(2)+3*pi/2-409/70
E. 3*ln(2)+3*pi/2-409/69
F. 3*ln(2)+3*pi/2-409/74
G. 3*ln(2)+3*pi/2-409/76
H. 3*ln(2)+3*pi/2-409/75
I. 3*ln(2)+3*pi/2-409/73
J. 3*ln(2)+3*pi/2-409/72
Answer: D
Find a "reasonable" upper-bound on the error in approximating $f(x) = (x-1) \cdot \ln(x-1)$ by its 3rd order Taylor polynomial $P_{3}(x)$ about $a=2$ valid for all values of $x$ such that $|x-2| \le 0.3$.

Options:
A. 0.003662040962236937
B. 0.0036620409622369345
C. 0.0036620409622369353
D. 0.0036620409622369352
E. 0.003662040962236934
F. 0.0036620409622369358
G. 0.003662040962236936
H. 0.003662040962236938
I. 0.0036620409622369365
J. 0.003662040962236935
Answer: C
Find the Taylor series of $f'(x)$ about $a=0$ if $f(x) = \frac{ \sin(x) - x }{ x^2 }$. Use sigma notation in the final answer.

Options:
A. ∑_{k=1}^∞((-1)^k·(2·k-1)·x^(2·k-2)/(2·k+1)!)
B. ∑_{k=1}^∞((-1)^k·(2·k-1)·x^(2·k-2)/(2·k-1)!)
C. ∑_{k=1}^∞((-1)^k·(2·k-1)·x^(2·k-2)/(2·k+3)!)
D. ∑_{k=1}^∞((-1)^k·(2·k-1)·x^(2·k-2)/(2·k+4)!)
E. ∑_{k=1}^∞((-1)^k·(2·k-1)·x^(2·k-3)/(2·k+1)!)
F. ∑_{k=1}^∞((-1)^k·(2·k-1)·x^(2·k-2)/(2·k+5)!)
G. ∑_{k=1}^∞((-1)^k·(2·k-1)·x^(2·k-1)/(2·k+1)!)
H. ∑_{k=1}^∞((-1)^k·(2·k-1)·x^(2·k-2)/(2·k)!)
I. ∑_{k=1}^∞((-1)^k·(2·k-1)·x^(2·k-2)/(2·k+6)!)
J. ∑_{k=1}^∞((-1)^k·(2·k-1)·x^(2·k-2)/(2·k+2)!)
Answer: A
Using the series expansion for the function $(1+x)^m$, calculate approximately $\sqrt[3]{7}$ with an accuracy of 0.0001.

Options:
A. 1.9127
B. 1.9126
C. 1.9128
D. 1.9129
E. 1.9134
F. 1.9130
G. 1.9135
H. 1.9131
I. 1.9133
J. 1.9132
Answer: D
Find the Maclaurin series for the function: $f(x) = \cos(x) - x \cdot \sin(x)$.

Options:
A. sum_{n=0}^infty left(frac{(-1)^n cdot (2 cdot n - 1) cdot x^{2 cdot n}}{(2 cdot n)!}right)
B. sum_{n=0}^infty left(frac{(-1)^n cdot (2 cdot n + 1) cdot x^{2 cdot n + 1}}{(2 cdot n + 1)!}right)
C. sum_{n=0}^infty left(frac{(-1)^n cdot (2 cdot n + 3) cdot x^{2 cdot n}}{(2 cdot n)!}right)
D. sum_{n=0}^infty left(frac{(-1)^n cdot (2 cdot n + 1) cdot x^{2 cdot n + 2}}{(2 cdot n)!}right)
E. sum_{n=0}^infty left(frac{(-1)^n cdot (2 cdot n + 1) cdot x^{2 cdot n + 4}}{(2 cdot n)!}right)
F. sum_{n=0}^infty left(frac{(-1)^n cdot (2 cdot n + 1) cdot x^{2 cdot n}}{(2 cdot n)!}right)
G. sum_{n=0}^infty left(frac{(-1)^n cdot (2 cdot n) cdot x^{2 cdot n}}{(2 cdot n)!}right)
H. sum_{n=0}^infty left(frac{(-1)^n cdot (2 cdot n + 1) cdot x^{2 cdot n + 3}}{(2 cdot n)!}right)
I. sum_{n=0}^infty left(frac{(-1)^n cdot (2 cdot n + 1) cdot x^{2 cdot n + 1}}{(2 cdot n)!}right)
J. sum_{n=0}^infty left(frac{(-1)^n cdot (2 cdot n + 2) cdot x^{2 cdot n}}{(2 cdot n)!}right)
Answer: F
Compute the volume of the solid formed by rotating about the x-axis the area bounded by the axes and the parabola $x^{\frac{ 1 }{ 2 }}+y^{\frac{ 1 }{ 2 }}=6^{\frac{ 1 }{ 2 }}$.

Options:
A. π⋅80/5
B. π⋅120/5
C. π⋅96/5
D. π⋅70/5
E. π⋅144/5
F. π⋅72/5
G. π⋅108/5
H. π⋅60/5
I. π⋅64/5
J. π⋅76/5
Answer: F
Use two circular permutations of the variables $x$, $y$, and $z$ to write new integral whose value equal the value of the original integral. A circular permutation of $x$, $y$, and $z$ is the arrangement of the numbers in one of the following orders: $y$, $z$, and $x$ or $z$, $x$, and $y$:
$$
\int_{0}^1 \int_{1}^3 \int_{2}^4 \left(x^2 \cdot z^2 + 1\right) \, dx \, dy \, dz
$$

Options:
A. ∫_{0}^1 ∫_{1}^3 ∫_{2}^4 (y^2 * z^2 + 2) dy dz dx
B. ∫_{0}^1 ∫_{1}^3 ∫_{2}^4 (x^2 * z^2 + 1) dy dz dx
C. ∫_{0}^1 ∫_{1}^3 ∫_{2}^4 (x^2 * y^2 + 2) dx dy dz
D. ∫_{0}^1 ∫_{1}^3 ∫_{2}^4 (z^2 * x^2 + 1) dz dy dx
E. ∫_{0}^1 ∫_{1}^3 ∫_{2}^4 (x^2 * z^2 + 1) dz dy dx
F. ∫_{0}^1 ∫_{1}^3 ∫_{2}^4 (x^2 * y^2 + 1) dx dy dz
G. ∫_{0}^1 ∫_{1}^3 ∫_{2}^4 (y^2 * x^2 + 1) dy dz dx, ∫_{0}^1 ∫_{1}^3 ∫_{2}^4 (z^2 * y^2 + 1) dz dx dy
H. ∫_{0}^1 ∫_{1}^3 ∫_{2}^4 (x^2 * z^2 + 2) dx dy dz
I. ∫_{0}^1 ∫_{1}^3 ∫_{2}^4 (y^2 * z^2 + 1) dz dx dy
J. ∫_{0}^1 ∫_{1}^3 ∫_{2}^4 (y^2 * z^2 + 1) dy dz dx
Answer: G
Find the radius of convergence and the interval of convergence for the series:
$$
\sum_{n=0}^\infty \left(\frac{ x^n }{ n^n }\right)
$$

Options:
A. R = 1, I = (-∞,1)
B. R = 1, I = [-1,1]
C. R = 1, I = (-∞,∞)
D. R = 1, I = (-1,1)
E. R = 1, I = (-∞,1]
F. R = 1, I = [1,∞)
G. R = 1, I = (0,∞)
H. R = ∞, I = (-∞,∞)
I. R = 1, I = (-1,1]
J. R = 1, I = (0,1)
Answer: H
Compute the integral:
$$
\int \frac{ -1 }{ x^2 \cdot \left(3+x^3\right)^{\frac{ 5 }{ 3 }} } \, dx
$$

Options:
A. C + \frac{1}{9} \cdot \sqrt[3]{1 + \frac{3}{x^3}} - \frac{1}{18 \cdot \left(1 + \frac{3}{x^3}\right)^{\frac{2}{3}}}
B. C + \frac{1}{9} \cdot \sqrt[3]{1 + \frac{3}{x^3}} - \frac{1}{9 \cdot \left(1 + \frac{3}{x^3}\right)^{\frac{2}{3}}}
C. C + \frac{1}{9} \cdot \sqrt[3]{1 + \frac{3}{x^3}} + \frac{1}{21 \cdot \left(1 + \frac{3}{x^3}\right)^{\frac{2}{3}}}
D. C + \frac{1}{9} \cdot \sqrt[3]{1 + \frac{3}{x^3}} + \frac{1}{36 \cdot \left(1 + \frac{3}{x^3}\right)^{\frac{2}{3}}}
E. C + \frac{1}{9} \cdot \sqrt[3]{1 + \frac{3}{x^3}} + \frac{1}{24 \cdot \left(1 + \frac{3}{x^3}\right)^{\frac{2}{3}}}
F. C + \frac{1}{9} \cdot \sqrt[3]{1 + \frac{3}{x^3}} + \frac{1}{18 \cdot \left(1 + \frac{3}{x^3}\right)^{\frac{2}{3}}}
G. C + \frac{1}{9} \cdot \sqrt[3]{1 + \frac{3}{x^3}} - \frac{1}{36 \cdot \left(1 + \frac{3}{x^3}\right)^{\frac{2}{3}}}
H. C + \frac{1}{9} \cdot \sqrt[3]{1 + \frac{3}{x^3}} + \frac{1}{15 \cdot \left(1 + \frac{3}{x^3}\right)^{\frac{2}{3}}}
I. C + \frac{1}{9} \cdot \sqrt[3]{1 + \frac{3}{x^3}} + \frac{1}{27 \cdot \left(1 + \frac{3}{x^3}\right)^{\frac{2}{3}}}
J. C + \frac{1}{9} \cdot \sqrt[3]{1 + \frac{3}{x^3}} + \frac{1}{20 \cdot \left(1 + \frac{3}{x^3}\right)^{\frac{2}{3}}}
Answer: F
Compute the area of the figure bounded by curves $y = 8 \cdot x^2$, $y = 4 + 4 \cdot x^2$, lines $x = 3$, $x = -2$, and the $x$-axis.

Options:
A. 183/3
B. 181/3
C. 182/3
D. 188/3
E. 189/3
F. 184/3
G. 185/3
H. 180/3
I. 187/3
J. 186/3
Answer: F
Compute $\sqrt[4]{90}$ with accuracy $0.0001$.

Options:
A. 3.0803
B. 3.0801
C. 3.0806
D. 3.0807
E. 3.0802
F. 3.0804
G. 3.0808
H. 3.0805
I. 3.0800
J. 3.0809
Answer: B
Find the area of a triangle bounded by the x-axis, the line $f(x) = 12 - \frac{ 1 }{ 3 } \cdot x$, and the line perpendicular to $f(x)$ that passes through the origin.

Options:
A. 972/9
B. 972/10
C. 972/12
D. 972/11
E. 972/4
F. 972/5
G. 972/7
H. 972/8
I. 972/13
J. 972/6
Answer: F
Compute $\int_{0}^{\frac{ 1 }{ 5 }} e^{-2 \cdot x^2} \, dx$ with accuracy $0.00001$.

Options:
A. 0.1943
B. 0.1949
C. 0.1947
D. 0.1945
E. 0.1941
F. 0.1944
G. 0.1948
H. 0.1946
I. 0.1942
J. 0.1950
Answer: G
When $1 + 7 + 7^2 +...+ 7^{2004}$ is divided by $1000$ , a remainder of $N$ is obtained. Determine the value of $N$ .

Options:
A. 802
B. 700
C. 701
D. 601
E. 803
F. 702
G. 501
H. 901
I. 703
J. 801
Answer: J
Let $P_{n}(x)$ be the n-th order Taylor polynomial of $f(x) = \cos(x)$ about $a = 0$. Find the minimum $n$ such that the approximation $f(x) \approx P_{n}(x)$ is accurate to within $0.00001$ for every value of $x$ satisfying $|x| \le 0.1$.

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

Options:
A. C + \frac{\sqrt{9 \cdot x + 4}}{3 \cdot x} - \frac{3}{4} \cdot \ln\left(\frac{\left|\sqrt{9 \cdot x + 4} + 2\right|}{2 - \sqrt{9 \cdot x + 4}}\right)
B. C + \frac{\sqrt{9 \cdot x + 4}}{3 \cdot x} - \frac{3}{4} \cdot \ln\left(\frac{\left|\sqrt{9 \cdot x + 4} - 2\right|}{2 - \sqrt{9 \cdot x + 4}}\right)
C. C + \frac{\sqrt{9 \cdot x + 4}}{3 \cdot x} - \frac{3}{4} \cdot \ln\left(\frac{\left|\sqrt{9 \cdot x + 4} - 2\right|}{2 + \sqrt{9 \cdot x + 4}}\right)
D. C + \frac{\sqrt{9 \cdot x + 4}}{3 \cdot x} - \frac{3}{4} \cdot \ln\left(\frac{\left|\sqrt{9 \cdot x + 4} + 2\right|}{2 + \sqrt{9 \cdot x + 4}}\right)
Answer: C
Franklyn chooses a random positive divisor of 2016, and calls it $x$ . Then, Franklyn randomly chooses a positive divisor of $x$ , and calls it $y$ . The probability that $y = 42$ can be expressed as $\dfrac{m}{n}$ , where $\gcd(m, n) = 1$ . Find $m$ .

Options:
A. 37
B. 29
C. 31
D. 33
E. 35
Answer: B
Solve the integral:
$$
\int \frac{ -9 \cdot \sqrt[3]{x} }{ 9 \cdot \sqrt[3]{x^2} + 3 \cdot \sqrt{x} } \, dx
$$

Options:
A. -\left(C+\frac{1}{3}\cdot\sqrt[6]{x}^2+\frac{2}{27}\cdot\ln\left(\frac{1}{3}\cdot\left|1+3\cdot\sqrt[6]{x}\right|\right)+\frac{3}{2}\cdot\sqrt[6]{x}^4-\frac{2}{3}\cdot\sqrt[6]{x}^3+\frac{2}{9}\cdot\sqrt[6]{x}\right)
B. -\left(C+\frac{1}{3}\cdot\sqrt[6]{x}^2+\frac{2}{27}\cdot\ln\left(\frac{1}{3}\cdot\left|1+3\cdot\sqrt[6]{x}\right|\right)+\frac{3}{2}\cdot\sqrt[6]{x}^4-\frac{2}{3}\cdot\sqrt[6]{x}^3+\frac{1}{27}\cdot\sqrt[6]{x}\right)
C. -\left(C+\frac{1}{3}\cdot\sqrt[6]{x}^2+\frac{2}{27}\cdot\ln\left(\frac{1}{3}\cdot\left|1+3\cdot\sqrt[6]{x}\right|\right)+\frac{3}{2}\cdot\sqrt[6]{x}^4-\frac{2}{3}\cdot\sqrt[6]{x}^3+\frac{1}{9}\cdot\sqrt[6]{x}\right)
D. -\left(C+\frac{1}{3}\cdot\sqrt[6]{x}^2+\frac{2}{27}\cdot\ln\left(\frac{1}{3}\cdot\left|1+3\cdot\sqrt[6]{x}\right|\right)+\frac{3}{2}\cdot\sqrt[6]{x}^4-\frac{2}{3}\cdot\sqrt[6]{x}^3-\frac{2}{9}\cdot\sqrt[6]{x}\right)
E. -\left(C+\frac{1}{3}\cdot\sqrt[6]{x}^2+\frac{2}{27}\cdot\ln\left(\frac{1}{3}\cdot\left|1+3\cdot\sqrt[6]{x}\right|\right)+\frac{3}{2}\cdot\sqrt[6]{x}^4-\frac{2}{3}\cdot\sqrt[6]{x}^3+\frac{1}{3}\cdot\sqrt[6]{x}\right)
Answer: D
The number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_{10} \le 2007$ such that $a_i-i$ is even for $1\le i \le 10$ can be expressed as ${m \choose n}$ for some positive integers $m > n$ . Compute the remainder when $m$ is divided by $1000$ .

Options:
A. 18
B. 8
C. 14
D. 10
E. 12
F. 16
Answer: B
Find $\frac{ d y }{d x}$ if $y = \frac{ 5 \cdot x^2 - 3 \cdot x }{ \left( 3 \cdot x^7 + 2 \cdot x^6 \right)^4 }$.

Options:
A. \frac{-390\cdot x^2+23\cdot x+138}{x^{24}\cdot(3\cdot x+2)^3}
B. \frac{-390\cdot x^2+23\cdot x+138}{x^{24}\cdot(3\cdot x+2)^7}
C. \frac{-390\cdot x^2+23\cdot x+138}{x^{24}\cdot(3\cdot x+2)^5}
D. \frac{-390\cdot x^2+23\cdot x+138}{x^{24}\cdot(3\cdot x+2)^8}
E. \frac{-390\cdot x^2+23\cdot x+138}{x^{24}\cdot(3\cdot x+2)^6}
F. \frac{-390\cdot x^2+23\cdot x+138}{x^{24}\cdot(3\cdot x+2)^4}
Answer: C
The intersection of the perimeter of a regular octagon and the perimeter of a triangle consists of three disjoint line segments of positive lengths. The ratio of the area of the regular octagon to the area of the triangle can be expressed in the form $a\sqrt{b} - c$ for positive integers $a$ , $b$ , and $c$ , where $b$ is not divisible by the square of any prime. Find $a + b + c$ .

Options:
A. 98
B. 96
C. 97
D. 93
E. 99
F. 95
G. 94
Answer: A
Given \[ 11z^{10}+10iz^9+10iz-11=0, \] find the value of $|z|$. (Here $z$ is a complex number and $i^2=-1$.)

Options:
A. 1.5
B. 2
C. 1.1
D. 0.5
E. 0.75
F. 1.25
G. 1
Answer: G
Let $S$ denote the value of the sum
$$$\sum^{668}_{n=0} (-1)^n {2004 \choose 3n}$$$
Determine the remainder obtained when $S$ is divided by $1000$ .

Options:
A. 98
B. 104
C. 94
D. 96
E. 102
F. 106
G. 6
H. 108
Answer: G
Simplify $E = \left(\sin(2 \cdot a)\right)^3 \cdot \left(\cos(6 \cdot a)\right) + \left(\cos(2 \cdot a)\right)^3 \cdot \left(\sin(6 \cdot a)\right)$.

Options:
A. E=\frac{3}{4}\cdot\sin(10\cdot a)
B. E=\frac{3}{4}\cdot\sin(14\cdot a)
C. E=\frac{3}{4}\cdot\sin(12\cdot a)
D. E=\frac{3}{4}\cdot\sin(4\cdot a)
E. E=\frac{3}{4}\cdot\sin(16\cdot a)
F. E=\frac{3}{4}\cdot\sin(8\cdot a)
G. E=\frac{3}{4}\cdot\sin(18\cdot a)
H. E=\frac{3}{4}\cdot\sin(6\cdot a)
Answer: F
For the curve $x = a \cdot \left(t - \sin(t)\right)$, $y = a \cdot \left(1 - \cos(t)\right)$, determine the curvature. Use $a = 4$.

Options:
A. 1/(4*|sin(t/2)|)
B. 1/(16*|sin(t/2)|)
C. 1/(32*|sin(t/2)|)
D. 1/(8*|sin(t/2)|)
E. 1/(28*|sin(t/2)|)
F. 1/(12*|sin(t/2)|)
G. 1/(24*|sin(t/2)|)
H. 1/(20*|sin(t/2)|)
Answer: B
The sequence $a_n$ is defined by
$$$a_n=\frac{a_{n-1} \cdot a_{n-2}}{x}.$$$
Given that $a_1=350$ and $a_2=1190$ , find the greatest value of $x$ such that $a_n$ is always an integer.

Options:
A. 350
B. 70
C. 210
D. 35
E. 175
F. 105
G. 14
H. 140
I. 100
Answer: B
For all $n \geq 1$, let \[ a_n = \sum_{k=1}^{n-1} \frac{\sin \left( \frac{(2k-1)\pi}{2n} \right)}{\cos^2 \left( \frac{(k-1)\pi}{2n} \right) \cos^2 \left( \frac{k\pi}{2n} \right)}. \] Determine \[ \lim_{n \to \infty} \frac{a_n}{n^3}. \]

Options:
A. \frac{6}{\pi^3}
B. \frac{8}{\pi^3}
C. \frac{16}{\pi^3}
D. \frac{10}{\pi^3}
E. \frac{4}{\pi^3}
F. \frac{12}{\pi^3}
G. \frac{14}{\pi^3}
H. \frac{18}{\pi^3}
I. \frac{8}{\pi^2}
Answer: B
Evaluate $L=\lim_{(x,y) \to (-1,1)}\left(\frac{ x \cdot y \cdot (x+y)-5 \cdot y \cdot (y+5)+5 \cdot x \cdot (x-5) }{ x^2+y^2+2 \cdot x \cdot y+5 \cdot x+5 \cdot y }\right)$.

Options:
A. -37/5
B. -35/5
C. -39/5
D. -36/5
E. -35/6
F. -35/4
G. -38/5
H. -33/5
I. -34/5
Answer: D
How many positive odd integers between $0$ and $999$ inclusive can be written as sum of $3$ odd composite positive integers?

Options:
A. 483
B. 485
C. 482
D. 477
E. 478
F. 479
G. 484
H. 486
I. 480
J. 481
Answer: J
A triple elimination tournament takes place with 100 teams. A team is eliminated once they have lost 3 games and there are no ties. What is the maximum number of games that can take place during such a tournament?

Options:
A. 300
B. 303
C. 299
D. 304
E. 297
F. 301
G. 298
H. 302
I. 295
J. 296
Answer: C
Find the number of polynomials $P(x)$ with positive integer coefficients and with at most $2$ terms such that $P(2)=384$ .

Options:
A. 726
B. 732
C. 734
D. 727
E. 730
F. 725
G. 728
H. 731
I. 729
J. 733
Answer: I
Iggy, Piggy, and Twiggy are all jogging beside a railroad track at constant rates in the same direction such that Iggy is ahead of Piggy and Piggy is ahead of Twiggy. A steam locomotive is traveling on the railroad track behind Twiggy in the same direction as the three joggers and takes $6\tfrac{1}{15}$ minutes to reach Twiggy, then another $5$ seconds to completely pass Twiggy, then another $11\tfrac{1}{12}$ minutes to reach Piggy, then another $4$ seconds to completely pass Piggy, then another $16\tfrac{1}{10}$ minutes to reach Iggy, then another $3$ seconds to completely pass Iggy. How many seconds after Twiggy catches up to Piggy will it take for Twiggy to catch up to Iggy? Assume that Iggy, Piggy, and Twiggy have negligible size.

Options:
A. 735
B. 770
C. 750
D. 760
E. 720
F. 755
G. 745
H. 740
I. 730
J. 765
Answer: C
If $x$ and $y$ are real numbers such that $2xy+2x^{2}=6+x^{2}+y^{2}$ find the minimum value of $(x^{2}+y^{2})^{2}$ .

Options:
A. 13
B. 21
C. 18
D. 19
E. 17
F. 20
G. 22
H. 16
I. 14
J. 15
Answer: C
Let the sum of all odd $n$ such that $\frac{1}{n}$ expressed in base $8$ is a repeating decimal with period $4$ be $S$ . Find the remainder when $S$ is divided by $1000$ .

Options:
A. 639
B. 631
C. 636
D. 630
E. 632
F. 634
G. 638
H. 637
I. 635
J. 633
Answer: E
If $abc-ab-bc-ca+a+b+c=2008$ for positive integers $a$ , $b$ , and $c$ , then determine the minimum value of $a+b+c$ .

Options:
A. 233
B. 235
C. 239
D. 231
E. 238
F. 236
G. 237
H. 234
I. 232
J. 230
Answer: I
In $\triangle RST$ , X is on $\overline {RT}$ , dividing $RX:XT = 1:2$ . Y is on $\overline {ST}$ , dividing $SY:YT = 2:1$ . V is on $\overline {XY}$ , dividing $XV:VY = 1:2$ . It is found that ray $VT$ intersects $\overline {RS}$ at Z. Find
$$128 \left(\frac {TV}{VZ} + \frac {RZ}{ZS}\right)$$

Options:
A. 184
B. 196
C. 178
D. 192
E. 188
F. 176
G. 168
H. 172
I. 160
J. 180
Answer: D
If $a_{n}+i\cdot b_{n}=(1+17i)^{n}$ where $a_{n}$ and $b_{n}$ are real sequences and $c_{n}=\gcd (a_{n}, b_{n})$ , determine the last 3 digits of $c_{2007}$ .

Options:
A. 8
B. 120
C. 108
D. 112
E. 104
F. 102
G. 100
H. 116
I. 128
J. 124
Answer: A
$100$ blocks are selected from a crate containing $33$ blocks of each of the following dimensions:
$$$13 \times 17 \times 21, 13 \times 17 \times 37, 13 \times 21 \times 37 , \,\,\, \text{and} \,\,\, 17 \times 21 \times 37: $$$
The chosen blocks are stacked on top of each other (one per cross section) forming a tower of height $h$ . Compute the number of possible values of $h$ :

Options:
A. 595
B. 592
C. 594
D. 597
E. 598
F. 591
G. 599
H. 593
I. 596
J. 590
Answer: A
Let $N$ denote the amount of positive solutions to the equation $\sqrt[2018]x=2^{\sin x\pi}$ . Find the last three digits of $N$ .

Options:
A. 141
B. 146
C. 145
D. 147
E. 143
F. 148
G. 140
H. 144
I. 149
J. 142
Answer: E
Determine the number of permutations of $1,2,3, 4,...,32$ such that if $m$ divides $n$ , the $m$ -th number divides the $n$ -th number.

Options:
A. 240
B. 235
C. 238
D. 243
E. 239
F. 242
G. 241
H. 236
I. 234
J. 237
Answer: A
Alex and Mitchell decide to play a game. In this game, there are $2010$ pieces of candy on a table, and starting with Alex, the two take turns eating some positive integer number of pieces of candy. Since it is bad manners to eat the last candy, whoever eats the last candy loses. The two decide that the amount of candy a person can pick will be a set equal to the positive divisors of a number less than $2010$ that each person picks (individually) from the beginning. For example, if Alex picks $19$ and Mitchell picks $20$ , then on each turn, Alex must eat either $1$ or $19$ pieces, and Mitchell must eat $1$ , $2$ , $4$ , $5$ , $10$ , or $20$ pieces. Mitchell knows Alex well enough to determine with certainty that Alex will either be immature and pick $69$ , or be cliched and pick $42$ . How many integers can Mitchell pick to guarantee that he will not lose the game?

Options:
A. 163
B. 164
C. 166
D. 162
E. 167
F. 168
G. 161
H. 165
I. 160
J. 169
Answer: E
Define the cost of a lattice point $(i,j)$ to be $i\times j$ . A particle starts at $(0,0)$ and makes a series of $10$ hops. Each hop increases the particle's x-coordinate by $1$ or y-coordinate by $1$ . Let the price of the particle's path be the sum of the costs of the lattice points it meets, including $(0,0)$ and its final position. Find the maximum price of any path the particle can take.

Options:
A. 98
B. 94
C. 95
D. 97
E. 96
F. 90
G. 92
H. 93
I. 91
J. 85
Answer: C
Find the area of the region bounded by $|x - |y|| + |y - |x|| + |y| = 9$ in the Cartesian plane.

Options:
A. 57
B. 60
C. 72
D. 56
E. 63
F. 54
G. 58
H. 65
I. 69
J. 66
Answer: E
Find the number of possible values the expression
$$\left\lfloor\frac{x}{4}\right\rfloor+\left\lfloor\frac{x}{5}\right\rfloor+\left\lfloor\frac{x}{7}\right\rfloor+\left\lfloor\frac{x}{9}\right\rfloor$$
can take over all real numbers $x$ between $0$ and $1260$ , inclusive. ( $\lfloor k \rfloor$ denotes the greatest integer less than or equal to $k$ .)

Options:
A. 685
B. 686
C. 681
D. 684
E. 689
F. 680
G. 682
H. 683
I. 688
J. 687
Answer: A
Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$ . If $2 \cos B = \cos A+\cos C$ , then the area of triangle $ABC$ can be expressed as $\frac{a\sqrt{b}}{ c}$ , where $a$ , $b$ , and $c$ are positive integers such that a and c are relatively prime and $b$ is not divisible by the square of any prime. Compute $a + b + c$ .

Options:
A. 133
B. 136
C. 142
D. 138
E. 135
F. 141
G. 139
H. 143
I. 145
J. 137
Answer: F
Let $A$ , $B$ , $C$ , $D$ , and $E$ be points in 3d space. Each pair of points is connected by a line segment (and the segments only intersect at their endpoints). A move consists of traveling to another point along a line segment. How many possible paths are there that: start at point $A$ , move $5$ times, and end up at point $A$ again.

Options:
A. 194
B. 198
C. 206
D. 204
E. 202
F. 192
G. 196
H. 190
I. 199
J. 200
Answer: D
Let $a_1,a_2,\ldots,a_6$ be a sequence of integers such that for all $1 \le i \le 5$ ,
$$$a_{i+1}=\frac{a_i}{3} \quad \text{or} \quad a_{i+1}={-}2a_i.$$$
Find the number of possible positive values of $a_1+a_2+\cdots+a_6$ less than $1000$ .

Options:
A. 141
B. 145
C. 142
D. 147
E. 149
F. 148
G. 140
H. 146
I. 143
J. 144
Answer: C
$13$ nondistinguishable residents are moving into $7$ distinct houses in Conformistville, with at least one resident per house. In how many ways can the residents be assigned to these houses such that there is at least one house with at least $4$ residents?

Options:
A. 504
B. 570
C. 561
D. 558
E. 546
F. 534
G. 540
H. 552
I. 567
J. 576
Answer: I
We have a unit cube $ABCDEFGH$ where $ABCD$ is the top side and $EFGH$ is the bottom side with $E$ below $A$ , $F$ below $B$ , and so on. Equilateral triangle $BDG$ cuts out a circle from the cube's inscribed sphere. The area of the circle can be written as $\frac{\pi}{n}$ , where $n$ is an integer. Find $n$ .

Options:
A. 4
B. 6
C. 8
D. 9
E. 7
F. 10
G. 14
H. 5
I. 12
J. 11
Answer: B
For all real numbers $x$ satisfying $x^4+99225\leq 2074x^2$, what is the maximum value of $f(x)=x^3-3x$?

Options:
A. 90990
B. 90980
C. 90984
D. 90988
E. 90987
F. 90982
G. 90975
H. 90989
I. 90970
J. 90985
Answer: A
Evaluate \[ \int_0^1 \left( C(-w-1) \sum_{k=1}^11956 \frac{1}{w+k} \right)\,dw, \] where $C(\alpha)$ is the coefficient of $x^11956$ in the expansion of the power series of $(1 + b)^\alpha$ centered at $b=0$.

Options:
A. 11957
B. 11956
C. 11953
D. 11958
E. 11959
F. 11950
G. 11951
H. 11952
I. 11955
J. 11954
Answer: B
Evaluate \[ \int_0^1 \left( C(-p-1) \sum_{k=1}^7468 \frac{1}{p+k} \right)\,dp, \] where $C(\alpha)$ is the coefficient of $x^7468$ in the expansion of the power series of $(1 + n)^\alpha$ centered at $n=0$.

Options:
A. 7468.25
B. 7468.75
C. 7466
D. 7468.375
E. 7465
F. 7468
G. 7468.5
H. 7468.125
I. 7469
J. 7467
Answer: F
Evaluate \[ \int_0^1 \left( C(-j-1) \sum_{k=1}^2616 \frac{1}{j+k} \right)\,dj, \] where $C(\alpha)$ is the coefficient of $x^2616$ in the expansion of the power series of $(1 + d)^\alpha$ centered at $d=0$.

Options:
A. 2611
B. 2610
C. 2615
D. 2612
E. 2619
F. 2618
G. 2617
H. 2616
I. 2613
J. 2614
Answer: H
Consider a sequence of independent random variables $P_1, P_2, \dots$ defined for $0 \leq p \leq 1/2$, where each $P_i$ follows the distribution: \[ P_i = \begin{cases} 1 & \text{with probability $p$,} \\ -1 & \text{with probability $p$,} \\ 0 & \text{with probability $1-2p$.} \end{cases} \] Let $c$ be a positive integer and $z, n_1, \dots, n_c$ be integers. Define $P(z, n_1, \dots, n_c)$ as the probability of the event $n_1 P_1 + \cdots + n_c P_c = z$. What is the largest value of $p$ that ensures \[ P(0, n_1, \dots, n_c) \geq P(z, n_1, \dots, n_c) \] holds true for any choice of positive integer $c$ and integers $z, n_1, \dots, n_c$?

Options:
A. \frac{1}{10}
B. \frac{1}{11}
C. \frac{1}{4}
D. \frac{1}{12}
E. \frac{1}{7}
F. \frac{1}{9}
G. \frac{1}{3}
H. \frac{1}{8}
I. \frac{1}{6}
J. \frac{1}{5}
Answer: C
Consider the function $x(e)$ defined for all positive integers $e$. This function counts the number of 1's that appear when you write $2023 \cdot e$ in binary notation. What is the smallest possible output of $x(e)$?

Options:
A. 10
B. 8
C. 2
D. 4
E. 1
F. 6
G. 7
H. 5
I. 9
J. 3
Answer: J
Consider the set $V$ containing all integers $e$ between 1 and 2021 (inclusive) that are coprime to 2021. Define a function $E(z)$ for non-negative integers $z$ as follows: \[ E(z) = \sum_{e \in V} e^z. \] What is the largest value of $z$ below 2021 for which $E(z)$ is divisible by 2021?

Options:
A. 2017
B. 2020
C. 2011
D. 2019
E. 2014
F. 2013
G. 2012
H. 2018
I. 2016
J. 2015
Answer: B
Given a positive integer $x$, define the function $d_x(a) = \prod_{{k=1}}^x \cos(ka)$.Determine the smallest $x$ such that the absolute value of the second derivative at zero satisfies $|d_x''(0)| > 13357$.

Options:
A. 36
B. 30
C. 32
D. 38
E. 37
F. 33
G. 34
H. 39
I. 31
J. 35
Answer: G
Evaluate the area of the figure bounded by $\rho = a \cdot \cos(2 \cdot \varphi)$.

Options:
A. frac{pi}{6} * a^2
B. frac{pi}{4} * a^2
C. frac{pi}{10} * a^2
D. frac{pi}{16} * a^2
E. frac{pi}{8} * a^2
F. frac{pi}{3} * a^2
G. frac{pi}{12} * a^2
H. frac{pi}{5} * a^2
I. frac{pi}{2} * a^2
J. frac{pi}{1} * a^2
Answer: I
Solve the integral:
$$
\int \frac{ 20 \cdot \cos(-10 \cdot x)^3 }{ 21 \cdot \sin(-10 \cdot x)^7 } \, dx
$$

Options:
A. C + \frac{1}{21} \cdot \left(\frac{1}{2} \cdot \left(\cot(10 \cdot x)\right)^4 + \frac{1}{5} \cdot \left(\cot(10 \cdot x)\right)^6\right)
B. C + \frac{1}{21} \cdot \left(\frac{1}{2} \cdot \left(\cot(10 \cdot x)\right)^4 + \frac{1}{3} \cdot \left(\cot(10 \cdot x)\right)^6\right)
C. C + \frac{1}{21} \cdot \left(\frac{1}{2} \cdot \left(\cot(10 \cdot x)\right)^4 + \frac{1}{10} \cdot \left(\cot(10 \cdot x)\right)^6\right)
D. C + \frac{1}{21} \cdot \left(\frac{1}{2} \cdot \left(\cot(10 \cdot x)\right)^4 + \frac{1}{4} \cdot \left(\cot(10 \cdot x)\right)^6\right)
E. C + \frac{1}{21} \cdot \left(\frac{1}{2} \cdot \left(\cot(10 \cdot x)\right)^4 + \frac{1}{9} \cdot \left(\cot(10 \cdot x)\right)^6\right)
F. C + \frac{1}{21} \cdot \left(\frac{1}{2} \cdot \left(\cot(10 \cdot x)\right)^4 + \frac{1}{6} \cdot \left(\cot(10 \cdot x)\right)^6\right)
G. C + \frac{1}{21} \cdot \left(\frac{1}{2} \cdot \left(\cot(10 \cdot x)\right)^4 + \frac{1}{12} \cdot \left(\cot(10 \cdot x)\right)^6\right)
H. C + \frac{1}{21} \cdot \left(\frac{1}{2} \cdot \left(\cot(10 \cdot x)\right)^4 + \frac{1}{2} \cdot \left(\cot(10 \cdot x)\right)^6\right)
I. C + \frac{1}{21} \cdot \left(\frac{1}{2} \cdot \left(\cot(10 \cdot x)\right)^4 + \frac{1}{7} \cdot \left(\cot(10 \cdot x)\right)^6\right)
J. C + \frac{1}{21} \cdot \left(\frac{1}{2} \cdot \left(\cot(10 \cdot x)\right)^4 + \frac{1}{8} \cdot \left(\cot(10 \cdot x)\right)^6\right)
Answer: B
Compute $\lim_{x \to 0}\left(\frac{ \cos(x)+2 }{ 3 \cdot x^3 \cdot \sin(x) }-\frac{ 3 }{ 3 \cdot x^4 }\right)$. Use the expansion of the function in the Taylor series.

Options:
A. 1/130
B. 1/180
C. 1/200
D. 1/210
E. 1/160
F. 1/190
G. 1/170
H. 1/240
I. 1/150
J. 1/120
Answer: B
Compute the integral using the Substitution Rule:
$$
\int \frac{ x^2+3 }{ \sqrt{(2 \cdot x-5)^3} } \, dx
$$

Options:
A. C + \frac{60 \cdot x + (2 \cdot x - 5)^2 - 271}{12 \cdot \sqrt{2 \cdot x - 5}}
B. C + \frac{60 \cdot x + (2 \cdot x - 5)^2 - 251}{12 \cdot \sqrt{2 \cdot x - 5}}
C. C + \frac{60 \cdot x + (2 \cdot x - 5)^2 - 266}{12 \cdot \sqrt{2 \cdot x - 5}}
D. C + \frac{60 \cdot x + (2 \cdot x - 5)^2 - 265}{12 \cdot \sqrt{2 \cdot x - 5}}
E. C + \frac{60 \cdot x + (2 \cdot x - 5)^2 - 261}{12 \cdot \sqrt{2 \cdot x - 5}}
F. C + \frac{60 \cdot x + (2 \cdot x - 5)^2 - 281}{12 \cdot \sqrt{2 \cdot x - 5}}
G. C + \frac{60 \cdot x + (2 \cdot x - 5)^2 - 256}{12 \cdot \sqrt{2 \cdot x - 5}}
H. C + \frac{60 \cdot x + (2 \cdot x - 5)^2 - 276}{12 \cdot \sqrt{2 \cdot x - 5}}
I. C + \frac{60 \cdot x + (2 \cdot x - 5)^2 - 241}{12 \cdot \sqrt{2 \cdot x - 5}}
J. C + \frac{60 \cdot x + (2 \cdot x - 5)^2 - 255}{12 \cdot \sqrt{2 \cdot x - 5}}
Answer: E
Find zeros of $f(x) = \sqrt{2 \cdot x - 1} - \sqrt{x - 1} - 5$

Options:
A. 144.5
B. 144.25
C. 144.75
D. 145
E. 148
F. 142
G. 147
H. 144
I. 146
J. 143
Answer: D
Where is the parabola $y = x^2$ closest to the point $(2,0)$?

Options:
A. P(0.8351,0.6874)
B. P(0.8351,0.7274)
C. P(0.8351,0.6674)
D. P(0.8351,0.7074)
E. P(0.8351,0.7374)
F. P(0.8351,0.7474)
G. P(0.8351,0.7574)
H. P(0.8351,0.6974)
I. P(0.8351,0.7174)
J. P(0.8351,0.6774)
Answer: H
Compute the integral:
$$
\int \frac{ 10 }{ \sin(4 \cdot x)^6 } \, dx
$$

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

Options:
A. \frac{\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)^3}{2}+\frac{3}{4}\cdot\left(\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)+\frac{x}{6}\right)+C
B. \frac{\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)^3}{2}+\frac{3}{4}\cdot\left(\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)+\frac{x}{3}\right)+C
C. \frac{\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)^3}{2}+\frac{3}{4}\cdot\left(\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)+\frac{x}{4}\right)+C
D. \frac{\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)^3}{2}+\frac{3}{4}\cdot\left(\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)+\frac{x}{7}\right)+C
E. \frac{\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)^3}{2}+\frac{3}{4}\cdot\left(\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)+\frac{x}{5}\right)+C
F. \frac{\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)^3}{2}+\frac{3}{4}\cdot\left(\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)+\frac{x}{12}\right)+C
G. \frac{\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)^3}{2}+\frac{3}{4}\cdot\left(\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)+\frac{x}{10}\right)+C
H. \frac{\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)^3}{2}+\frac{3}{4}\cdot\left(\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)+\frac{x}{8}\right)+C
I. \frac{\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)^3}{2}+\frac{3}{4}\cdot\left(\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)-\frac{x}{2}\right)+C
J. \frac{\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)^3}{2}+\frac{3}{4}\cdot\left(\sin\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{2}\right)+\frac{x}{2}\right)+C
Answer: J
Compute the integral:
$$
\int \frac{ x+2 }{ \sqrt{6+10 \cdot x+25 \cdot x^2} } \, dx
$$

Options:
A. frac{1}{25}cdotsqrt{6+10cdot x+25cdot x^2}+frac{9}{25}cdotlnleft(1+5cdot x+sqrt{1+(5cdot x+3)^2}right)+C
B. frac{1}{25}cdotsqrt{6+10cdot x+25cdot x^2}+frac{9}{25}cdotlnleft(1+5cdot x+sqrt{1+(5cdot x-1)^2}right)+C
C. frac{1}{25}cdotsqrt{6+10cdot x+25cdot x^2}+frac{9}{25}cdotlnleft(1+5cdot x+sqrt{1+(5cdot x+1)^2}right)+C
D. frac{1}{25}cdotsqrt{6+10cdot x+25cdot x^2}+frac{9}{25}cdotlnleft(1+5cdot x+sqrt{1+(5cdot x-2)^2}right)+C
E. frac{1}{25}cdotsqrt{6+10cdot x+25cdot x^2}+frac{9}{25}cdotlnleft(1+5cdot x+sqrt{1+(5cdot x+2)^2}right)+C
F. frac{1}{25}cdotsqrt{6+10cdot x+25cdot x^2}+frac{9}{25}cdotlnleft(1+5cdot x+sqrt{1+(5cdot x+1.5)^2}right)+C
G. frac{1}{25}cdotsqrt{6+10cdot x+25cdot x^2}+frac{9}{25}cdotlnleft(1+5cdot x+sqrt{1+(5cdot x)^2}right)+C
H. frac{1}{25}cdotsqrt{6+10cdot x+25cdot x^2}+frac{9}{25}cdotlnleft(1+5cdot x+sqrt{1+(5cdot x-3)^2}right)+C
I. frac{1}{25}cdotsqrt{6+10cdot x+25cdot x^2}+frac{9}{25}cdotlnleft(1+5cdot x+sqrt{1+(5cdot x+0.5)^2}right)+C
J. frac{1}{25}cdotsqrt{6+10cdot x+25cdot x^2}+frac{9}{25}cdotlnleft(1+5cdot x+sqrt{1+(5cdot x+4)^2}right)+C
Answer: C
Simplify the expression $\sin(x) \cdot \left(\csc(x)-\sin(x)\right)$ by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

Options:
A. 1 - sin(x)^2 + sin(x)
B. 1 - sin(x)
C. 1 - cos(x)^2
D. 1 - sin(x)cos(x)
E. 1 - sin(x)^2
F. cos(x)^2
G. 1 - sin(x)^2 - sin(x)
H. 1 - cos(x)
I. sin(x)^2
J. 1 - 2sin(x)^2
Answer: F
Determine the Taylor series for $y = \left(\cos(2 \cdot x)\right)^2$, centered at $x_{0} = 0$. Write out the sum of the first three non-zero terms, followed by dots.

Options:
A. 1 - 4x^2 + \frac{28}{24}x^4 + \cdots
B. 1 - 4x^2 + \frac{36}{24}x^4 + \cdots
C. 1 - 4x^2 + \frac{20}{24}x^4 + \cdots
D. 1 - 4x^2 + \frac{38}{24}x^4 + \cdots
E. 1 - 4x^2 + \frac{32}{24}x^4 + \cdots
F. 1 - 4x^2 + \frac{24}{24}x^4 + \cdots
G. 1 - 4x^2 + \frac{40}{24}x^4 + \cdots
H. 1 - 4x^2 + \frac{30}{24}x^4 + \cdots
I. 1 - 4x^2 + \frac{34}{24}x^4 + \cdots
J. 1 - 4x^2 + \frac{16}{24}x^4 + \cdots
Answer: E
Find the derivative of the 25th order $y^{(25)}$ for a function $y = 2 \cdot x^2 \cdot \sin(x)$.

Options:
A. \(2\cdot x^2-1200\)\cdot\cos(x)+100\cdot x\cdot\cos(x) + 75\cdot\sin(x)
B. \(2\cdot x^2-1200\)\cdot\cos(x)+100\cdot x\cdot\cos(x) - 25\cdot\sin(x)
C. \(2\cdot x^2-1200\)\cdot\cos(x)+100\cdot x\cdot\cos(x) + 25\cdot\sin(x)
D. \(2\cdot x^2-1200\)\cdot\cos(x)+100\cdot x\cdot\cos(x) + 50\cdot\sin(x)
E. \(2\cdot x^2-1200\)\cdot\cos(x)+100\cdot x\cdot\cos(x)
F. \(2\cdot x^2-1200\)\cdot\cos(x)+100\cdot x\cdot\sin(x)
G. \(2\cdot x^2-1200\)\cdot\cos(x)-100\cdot x\cdot\cos(x)
H. \(2\cdot x^2-1200\)\cdot\cos(x)-100\cdot x\cdot\sin(x) + 25\cdot\sin(x)
I. \(2\cdot x^2-1200\)\cdot\cos(x)-100\cdot x\cdot\sin(x)
J. \(2\cdot x^2-1200\)\cdot\cos(x)+100\cdot x\cdot\cos(x) - 50\cdot\sin(x)
Answer: F
Determine a definite integral that represents the region common to $r=2$ and $r=4 \cdot \cos\left(\theta\right)$.

Options:
A. 4\cdot\int_0^{\frac{\pi}{3}}1d\theta+16\cdot\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\sin\left(\theta\right)^2d\theta
B. 4\cdot\int_0^{\frac{\pi}{3}}1d\theta+16\cdot\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\cos\left(\theta\right)d\theta
C. 4\cdot\int_0^{\frac{\pi}{3}}1d\theta+16\cdot\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\cos\left(\theta\right)^{1.5}d\theta
D. 4\cdot\int_0^{\frac{\pi}{3}}1d\theta+16\cdot\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\cos\left(\theta\right)^4d\theta
E. 4\cdot\int_0^{\frac{\pi}{3}}1d\theta+16\cdot\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\cos\left(\theta\right)^3d\theta
F. 4\cdot\int_0^{\frac{\pi}{3}}1d\theta+16\cdot\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\cos\left(\theta\right)^{0.5}d\theta
G. 4\cdot\int_0^{\frac{\pi}{3}}1d\theta+16\cdot\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\cos\left(2\theta\right)d\theta
H. 4\cdot\int_0^{\frac{\pi}{3}}1d\theta+16\cdot\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\cos\left(\theta\right)^{2.5}d\theta
I. 4\cdot\int_0^{\frac{\pi}{3}}1d\theta+16\cdot\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\cos\left(\theta\right)^2d\theta
J. 4\cdot\int_0^{\frac{\pi}{3}}1d\theta+16\cdot\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\cos\left(\theta\right)^{1.75}d\theta
Answer: I
Evaluate the triple integral $\int_{0}^1{\int_{1}^2{\int_{z}^{z+1}{(y+1) d x} d y} d z}$ by using the transformation $u = x - z$, $v = 3 \cdot y$, and $w = \frac{ z }{ 2 }$.

Options:
A. 9/4
B. 15/8
C. 7/4
D. 11/4
E. 7/2
F. 5/2
G. 13/4
H. 3/2
I. 19/8
J. 17/8
Answer: F
Find $\left(\left(f(a)\right)^{-1}\right)'$ for the function $f(x) = x + \sqrt{x}$ at $a = 2$.

Options:
A. 1/4
B. 1/3 + 1/4
C. 1/6
D. 1/3 + 1/6
E. 2/3
F. 1/2
G. 1/3 + 1/5
H. 1/3 + 1/12
I. 1/3
J. 1/5
Answer: E
Compute the integral:
$$
\int \frac{ -12 }{ \sin(6 \cdot x)^6 } \, dx
$$

Options:
A. C+2\cdot\cot(6\cdot x)+\frac{2}{5}\cdot\left(\cot(6\cdot x)\right)^5+\frac{4}{3}\cdot\left(\cot(6\cdot x)\right)^4
B. C+2\cdot\cot(6\cdot x)+\frac{2}{5}\cdot\left(\cot(6\cdot x)\right)^5+\frac{4}{3}\cdot\left(\cot(6\cdot x)\right)^6
C. C+2\cdot\cot(6\cdot x)+\frac{2}{5}\cdot\left(\cot(6\cdot x)\right)^5+\frac{4}{3}\cdot\left(\cot(6\cdot x)\right)^2
D. C+2\cdot\cot(6\cdot x)+\frac{2}{5}\cdot\left(\cot(6\cdot x)\right)^5+\frac{4}{3}\cdot\left(\cot(6\cdot x)\right)^3
Answer: D
The set $S$ consists of all integers from $1$ to $2007,$ inclusive. For how many elements $n$ in $S$ is $f(n) = \frac{2n^3+n^2-n-2}{n^2-1}$ an integer?

Options:
A. 1
B. 3
C. 0
D. 4
E. 2
Answer: C
A point $P$ lies inside square $ABCD$ such that $PA = 11, PB = 22,$ and $PC = 33.$ The area of $ABCD$ can be expressed in the form $a + b\sqrt{c}$ , where $a, b,$ and $c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a + b + c$ .

Options:
A. 850
B. 853
C. 854
D. 855
E. 851
F. 849
G. 852
Answer: F
Let $p$ be an odd prime number such that $p \equiv 2 \pmod{3}$. Define a permutation $\pi$ of the residue classes modulo $p$ by $\pi(x) \equiv x^3 \pmod{p}$. Given that $\pi$ is an even permutation, what value must p be congruent to modulo 4?.

Options:
A. 15
B. 5
C. 7
D. 1
E. 3
F. 11
G. 9
Answer: E