MikaStars39/nano-eval huggingface dataset:
Calculate the area of the surface formed by rotating the astroid $x^{\frac{ 2 }{ 3 }} + y^{\frac{ 2 }{ 3 }} = 2^{\frac{ 2 }{ 3 }}$ about the x-axis.

Options:
A. 48π/8
B. 48π/5
C. 48π/11
D. 48π/6
E. 48π/10
F. 48π/12
G. 48π/3
H. 48π/4
I. 48π/9
J. 48π/7
Answer: B
Solve the integral:
$$
\int -16 \cdot \sin(-3 \cdot x)^4 \cdot \cos(-3 \cdot x)^2 \, dx
$$

Options:
A. -x + \frac{\sin(12 \cdot x)}{12} + \frac{\sin(6 \cdot x)}{12} - \frac{\sin(18 \cdot x)}{72} + C
B. -x + \frac{\sin(12 \cdot x)}{12} + \frac{\sin(6 \cdot x)}{12} - \frac{\sin(18 \cdot x)}{18} + C
C. -x + \frac{\sin(12 \cdot x)}{12} + \frac{\sin(6 \cdot x)}{12} - \frac{\sin(18 \cdot x)}{36} + C
D. -x + \frac{\sin(12 \cdot x)}{12} + \frac{\sin(6 \cdot x)}{12} - \frac{\sin(18 \cdot x)}{48} + C
E. -x + \frac{\sin(12 \cdot x)}{12} + \frac{\sin(6 \cdot x)}{12} - \frac{\sin(18 \cdot x)}{42} + C
F. -x + \frac{\sin(12 \cdot x)}{12} + \frac{\sin(6 \cdot x)}{12} - \frac{\sin(18 \cdot x)}{60} + C
G. -x + \frac{\sin(12 \cdot x)}{12} + \frac{\sin(6 \cdot x)}{12} - \frac{\sin(18 \cdot x)}{30} + C
H. -x + \frac{\sin(12 \cdot x)}{12} + \frac{\sin(6 \cdot x)}{12} - \frac{\sin(18 \cdot x)}{24} + C
I. -x + \frac{\sin(12 \cdot x)}{12} + \frac{\sin(6 \cdot x)}{12} - \frac{\sin(18 \cdot x)}{54} + C
J. -x + \frac{\sin(12 \cdot x)}{12} + \frac{\sin(6 \cdot x)}{12} - \frac{\sin(18 \cdot x)}{45} + C
Answer: C
Find the volume of the solid that lies under the double cone $z^2 = 4 \cdot x^2 + 4 \cdot y^2$, inside the cylinder $x^2 + y^2 = x$, and above the plane $z = 0$.

Options:
A. 10/18
B. 6/9
C. 18/9
D. 16/9
E. 14/9
F. 12/9
G. 10/9
H. 8/9
I. 4/9
J. 2/9
Answer: H
For what values of $a$, $b$, and $c$ does the curve $y = a \cdot x^4 + b \cdot x^3 + c \cdot x^2 + e \cdot x + f$ have points of inflection?

Options:
A. 3 * b^2 - 8 * a * c < 1
B. 3 * b^2 - 8 * a * c > 1
C. 3 * b^2 - 8 * a * c < 0
D. 3 * b^2 - 8 * a * c > -4
E. 3 * b^2 - 8 * a * c > -2
F. 3 * b^2 - 8 * a * c = 0
G. 3 * b^2 - 8 * a * c > -1
H. 3 * b^2 - 8 * a * c > 0
I. 3 * b^2 - 8 * a * c > 2
J. 3 * b^2 - 8 * a * c > -3
Answer: H
The equation $N(t) = \frac{ 500 }{ 1+49 \cdot e^{-0.7 \cdot t} }$ models the number of people in a town who have heard a rumor after $t$ days. To the nearest whole number, how many people will have heard the rumor after $3$ days?

Options:
A. 68
B. 75
C. 69
D. 72
E. 76
F. 71
G. 73
H. 74
I. 77
J. 70
Answer: F
Solve the equation for $x$:
$$
\frac{ 3 }{ \log_{2}(10) }-\log_{10}(x-9)=\log_{10}(44)
$$

Options:
A. 9.181818181818186
B. 9.181818181818188
C. 9.181818181818189
D. 9.181818181818182
E. 9.181818181818184
F. 9.181818181818180
G. 9.181818181818187
H. 9.181818181818185
I. 9.181818181818181
J. 9.181818181818183
Answer: D
Evaluate $\int\int_{D}{\left(\int_{0}^{4 \cdot x^2+4 \cdot y^2}{y \, dz}\right) \, dA}$, where $D = \left\{(x,y) | x^2+y^2 \le 4, y \ge 1, x \ge 0\right\}$ is the projection of $E$ onto the $x \cdot y$-plane.

Options:
A. 278/15
B. 262/15
C. 274/15
D. 270/15
E. 276/15
F. 271/15
G. 266/15
H. 272/15
I. 280/15
J. 268/15
Answer: C
Find the minimum value of $y = \frac{ \left(\cos(x)\right)^2 - 4 \cdot \cos(x) + 5 }{ 3 - 2 \cdot \cos(x) }$.

Options:
A. 1.585786437626905
B. 1.690983005625053
C. 1.527864045000421
D. 1.309016994374947
E. 1.618033988749895
F. 1.472135954999579
G. 1.381966011250105
H. 1.632455532033676
I. 1.660254037844386
J. 1.5618033988749895
Answer: E
Solve the integral:
$$
\int 22 \cdot \cot(-11 \cdot x)^5 \, dx
$$

Options:
A. C + \frac{1}{2} \cdot \left(\cot(11 \cdot x)\right)^4 + \ln\left(1 + \left(\cot(11 \cdot x)\right)^2\right) - \frac{3}{2} \cdot \left(\cot(11 \cdot x)\right)^2
B. C + \frac{1}{2} \cdot \left(\cot(11 \cdot x)\right)^4 + \ln\left(1 + \left(\cot(11 \cdot x)\right)^2\right) - 2 \cdot \left(\cot(11 \cdot x)\right)^2
C. C + \frac{1}{2} \cdot \left(\cot(11 \cdot x)\right)^4 + \ln\left(1 + \left(\cot(11 \cdot x)\right)^2\right) - \frac{2}{3} \cdot \left(\cot(11 \cdot x)\right)^2
D. C + \frac{1}{2} \cdot \left(\cot(11 \cdot x)\right)^4 + \ln\left(1 + \left(\cot(11 \cdot x)\right)^2\right) - \frac{5}{4} \cdot \left(\cot(11 \cdot x)\right)^2
E. C + \frac{1}{2} \cdot \left(\cot(11 \cdot x)\right)^4 + \ln\left(1 + \left(\cot(11 \cdot x)\right)^2\right) - \frac{1}{3} \cdot \left(\cot(11 \cdot x)\right)^2
F. C + \frac{1}{2} \cdot \left(\cot(11 \cdot x)\right)^4 + \ln\left(1 + \left(\cot(11 \cdot x)\right)^2\right) - \left(\cot(11 \cdot x)\right)^2
G. C + \frac{1}{2} \cdot \left(\cot(11 \cdot x)\right)^4 + \ln\left(1 + \left(\cot(11 \cdot x)\right)^2\right) - \frac{1}{2} \cdot \left(\cot(11 \cdot x)\right)^2
H. C + \frac{1}{2} \cdot \left(\cot(11 \cdot x)\right)^4 + \ln\left(1 + \left(\cot(11 \cdot x)\right)^2\right) - \frac{4}{3} \cdot \left(\cot(11 \cdot x)\right)^2
I. C + \frac{1}{2} \cdot \left(\cot(11 \cdot x)\right)^4 + \ln\left(1 + \left(\cot(11 \cdot x)\right)^2\right) - \frac{1}{4} \cdot \left(\cot(11 \cdot x)\right)^2
J. C + \frac{1}{2} \cdot \left(\cot(11 \cdot x)\right)^4 + \ln\left(1 + \left(\cot(11 \cdot x)\right)^2\right) - 3 \cdot \left(\cot(11 \cdot x)\right)^2
Answer: F
Compute the volume of the solid formed by rotating about the x-axis the area bounded by the axes and the parabola $x^{\frac{ 1 }{ 2 }}+y^{\frac{ 1 }{ 2 }}=5^{\frac{ 1 }{ 2 }}$.

Options:
A. π⋅25/5
B. π⋅25/9
C. π⋅25/10
D. π⋅25/8
E. π⋅25/6
F. π⋅25/7
G. π⋅25/12
H. π⋅25/15
I. π⋅25/3
J. π⋅25/4
Answer: I
Find the volume of the region of points $(x,y,z)$ such that \[ (x^2 + y^2 + z^2 + 8)^2 \leq 36(x^2 + y^2). \]

Options:
A. 12 \pi^2 - 36 \pi
B. 12 \pi^2 - 18 \pi
C. 12 \pi
D. 12 \pi^2 - 12 \pi
E. 12 \pi^2 - 24 \pi
F. 12 \pi^2 - 6 \pi
G. 18 \pi
H. 6 \pi^2
I. 18 \pi^2
J. 12 \pi^2
Answer: H
Consider the function $m(f)$ defined for all positive integers $f$. This function counts the number of 1's that appear when you write $2023 \cdot f$ in binary notation. What is the smallest possible output of $m(f)$?

Options:
A. \boxed{7}
B. \boxed{8}
C. \boxed{10}
D. \boxed{3}
E. \boxed{1}
F. \boxed{6}
G. \boxed{5}
H. \boxed{2}
I. \boxed{9}
J. \boxed{4}
Answer: D
Which expression is equivalent to $\left(\frac{ 1 }{ \sin(x) }+\frac{ 1 }{ \tan(x) }\right)^2$?

Options:
A. T=\frac{1+\cos(x)}{1-\cos(x)}
B. T=\frac{1-\cos(x)}{\sin^2(x)}
C. T=\frac{1-\cos(x)}{1+\cos(x)}
D. T=\frac{1+\cos(x)}{\sin^2(x)}
Answer: A
Consider a horizontal strip of $N+2$ squares in which the first and the last square are black and the remaining $N$ squares are all white. Choose a white square uniformly at random, choose one of its two neighbors with equal probability, and color this neighboring square black if it is not already black. Repeat this process until all the remaining white squares have only black neighbors. Let $w(N)$ be the expected number of white squares remaining. Find \[ \lim_{N \to \infty} \frac{w(N)}{N}. \]

Options:
A. \frac{1}{e}
B. \frac{1}{2}
C. \frac{1}{2e}
D. \frac{1}{3e}
E. \frac{1}{4e}
Answer: A
Find $\lim_{n \to \infty} \left(x_{n}\right)$, where $x_{n} = \left(\frac{ 2 \cdot n^3 }{ 2 \cdot n^2+3 }+\frac{ 1-5 \cdot n^2 }{ 5 \cdot n+1 }\right)$ is the general term of a sequence.

Options:
A. 1/5
B. 1/15
C. 1/10
D. 1/4
E. 1/2
Answer: A
Compute the integral:
$$
-\int \frac{ \sin\left(\frac{ x }{ 3 }\right)^4 }{ \cos\left(\frac{ x }{ 3 }\right)^2 } \, dx
$$

Options:
A. -\frac{3\cdot\sin\left(\frac{x}{3}\right)^3}{\cos\left(\frac{x}{3}\right)}+\frac{3}{2}\cdot x-\frac{9}{4}\cdot\sin\left(\frac{x}{3}\right)+C
B. -\frac{3\cdot\sin\left(\frac{x}{3}\right)^3}{\cos\left(\frac{x}{3}\right)}+\frac{3}{2}\cdot x-\frac{9}{4}\cdot\sin\left(\frac{5\cdot x}{3}\right)+C
C. -\frac{3\cdot\sin\left(\frac{x}{3}\right)^3}{\cos\left(\frac{x}{3}\right)}+\frac{3}{2}\cdot x-\frac{9}{4}\cdot\sin\left(\frac{3\cdot x}{3}\right)+C
D. -\frac{3\cdot\sin\left(\frac{x}{3}\right)^3}{\cos\left(\frac{x}{3}\right)}+\frac{3}{2}\cdot x-\frac{9}{4}\cdot\sin\left(\frac{2\cdot x}{3}\right)+C
E. -\frac{3\cdot\sin\left(\frac{x}{3}\right)^3}{\cos\left(\frac{x}{3}\right)}+\frac{3}{2}\cdot x-\frac{9}{4}\cdot\sin\left(\frac{6\cdot x}{3}\right)+C
F. -\frac{3\cdot\sin\left(\frac{x}{3}\right)^3}{\cos\left(\frac{x}{3}\right)}+\frac{3}{2}\cdot x-\frac{9}{4}\cdot\sin\left(\frac{4\cdot x}{3}\right)+C
Answer: D
Calculate the sum $S=\cos(a)+\cos(2 \cdot a)+\cos(3 \cdot a) + \ldots + \cos(n \cdot a)$, where $a \ne 2 \cdot \pi \cdot k$ for integers $k$.

Options:
A. S=\frac{\sin\left(\frac{(n+1)\cdot a}{2}\right)\cdot\cos\left(\frac{(n-1)\cdot a}{2}\right)}{\sin\left(\frac{a}{2}\right)}
B. S=\frac{\sin\left(\frac{(n+1)\cdot a}{2}\right)\cdot\cos\left(\frac{(n+2)\cdot a}{2}\right)}{\sin\left(\frac{a}{2}\right)}
C. S=\frac{\sin\left(\frac{n\cdot a}{2}\right)\cdot\cos\left(\frac{n\cdot a}{2}\right)}{\sin\left(\frac{a}{2}\right)}
D. S=\frac{\sin\left(\frac{(n-1)\cdot a}{2}\right)\cdot\cos\left(\frac{(n+1)\cdot a}{2}\right)}{\sin\left(\frac{a}{2}\right)}
E. S=\frac{\sin\left(\frac{(n+1)\cdot a}{2}\right)\cdot\cos\left(\frac{n\cdot a}{2}\right)}{\sin\left(\frac{a}{2}\right)}
F. S=\frac{\sin\left(\frac{n\cdot a}{2}\right)\cdot\cos\left(\frac{(n+1)\cdot a}{2}\right)}{\sin\left(\frac{a}{2}\right)}
Answer: F
Determine the smallest positive integer $a > 0$ for which there exists a nonnegative continuous function $f(x)$ defined on $[0,a]$ with the property that the region \[ R = \{ (x,y) ; 0 \le x \le a, 0 \le y \le f(x) \} \] has perimeter $k$ units and area $k$ square units for some real number $k$.

Options:
A. 4
B. 5
C. 8
D. 3
E. 2
F. 6
G. 7
Answer: D
Find $I=\int_{0}^\pi \int_{0}^{4 \cdot \sin\left(\theta\right)} \frac{ 4 }{ \sqrt{16-r^2} } \cdot r \, d r \, d \theta$.

Options:
A. 16 * (pi - 1/2)
B. 16 * (pi - 1)
C. 16 * (pi - 2)
D. 32 * (pi / 2 - 1)
E. 16 * (pi / 2 - 2)
F. 16 * (pi / 2 - 1)
G. 16 * (pi - 3/2)
Answer: D
Suppose that $f$ is a function on the interval $[1,3]$ such that $-1 \leq f(x) \leq 1$ for all $x$ and $\int_1^3 f(x)\,dx = 0$. How large can $\int_1^3 \frac{f(x)}{x}\,dx$ be? \,

Options:
A. \log \frac{5}{4}
B. \log \frac{5}{3}
C. \log \frac{4}{3}
D. \log \frac{3}{2}
E. \log \frac{3}{4}
F. \log \frac{8}{5}
G. \log \frac{6}{5}
H. \log \frac{7}{4}
Answer: C
Find the volume of the prism with vertices $A(0,0,0)$, $B(4,0,0)$, $C(4,6,0)$, $D(0,6,0)$, $E(0,0,1)$, and $F(4,0,1)$.

Options:
A. 24
B. 12
C. 8
D. 16
E. 10
F. 18
G. 20
H. 28
Answer: B
Kelvin the Frog is a triangle, so he likes triangular numbers. Specifically Frog's absolute favorite number involves every single triangular number, the infinite sum $S$ defined as
$$$S = \sum^{\infty}_{i=1}\frac{T_i}{3^{i-1}} = T_1 +\frac{T_2}{3}+\frac{T_3}{3^2} + ...$$$
Kelvin the Frog's absolute favorite number can be expressed in the form $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Compute $m + n$ .

Options:
A. 33
B. 36
C. 37
D. 39
E. 35
F. 32
G. 38
H. 34
I. 31
Answer: E
We have a permutation of the integers from $1$ to $100$ , inclusive. Consider the sum of all triples of consecutive integers in this permutation. At most how many of these sums are odd?

Options:
A. 92
B. 91
C. 95
D. 97
E. 94
F. 99
G. 93
H. 98
I. 96
Answer: D
Let $P$ be the positive integer consisting of 1998 decimal digits, all of which are 1; that is, \[P=1111\cdots 11.\] Determine the $303$th digit after the decimal point in the decimal expansion of $\sqrt{P}$.

Options:
A. 9
B. 3
C. 5
D. 7
E. 2
F. 1
G. 6
H. 4
I. 8
Answer: B
Compute the sum of all positive integers $n$ such that $n^4$ does not divide $n!$ and $n$ is neither prime nor a product of two distinct primes.

Options:
A. 68
B. 56
C. 58
D. 60
E. 64
F. 63
G. 62
H. 66
I. 54
J. 59
Answer: G
Let $a, b$ be positive integers such that
$$$\gcd(2019!, 2020!, a, b) \cdot \text{lcm}(2019!, 2020!, a, b) =\sqrt{ab \cdot 2019! \cdot 2020!}.$$$
Find the minimum possible value of $a+b.$

Options:
A. 129
B. 122
C. 128
D. 121
E. 123
F. 127
G. 126
H. 125
I. 120
J. 124
Answer: D
Given a point $P$ in the coordinate plane, let $T_k(P)$ be the $90^\circ$ rotation of $P$ around the point $(2000-k,k)$ . Let $P_0$ be the point $(2007,0)$ and $P_{n+1}=T_n(P_n)$ for all integers $n\ge 0$ . If $P_m$ has a $y$ -coordinate of $433$ , what is $m$ ?

Options:
A. 443
B. 445
C. 444
D. 440
E. 439
F. 441
G. 448
H. 442
I. 446
J. 447
Answer: E
The sequence of polynomials $P_1(x), P_2(x), \cdots $ is defined such that
$$$P_k(x) = (x-k)^4-16^{\left \lfloor{\log_2 k}\right \rfloor}$$$
where $\left \lfloor{x} \right \rfloor$ denotes the greatest integer less than or equal to $x$ . Let $n$ be the number of distinct complex numbers $z$ such that
$$$P_1(z)P_2(z) \cdots P_{2019}(z) = 0$$$
Find $n \bmod{1000}$ .

Options:
A. 545
B. 543
C. 549
D. 547
E. 541
F. 540
G. 548
H. 542
I. 544
J. 546
Answer: H
Let $N$ be the number of pairs of sequences of integers $a_1$ , $a_2$ , $...$ , $a_5$ and $b_1$ , $b_2$ , $...$ , $b_5$ such that $a_1 + a_2 + ...+ a_5$ and $b_1 + b_2 + ... + b_5$ are both multiples of $5$ , and for each integer $1 \le k \le 5$ ,
$\bullet$ $0 \le a_k, b_k \le 4$ , and
$\bullet$ the integer $a^2_kb^2_k+1$ is a multiple of $5$ .
Find the remainder when $N$ is divided by $1000$ .

Options:
A. 571
B. 573
C. 568
D. 576
E. 572
F. 578
G. 563
H. 569
I. 574
J. 567
Answer: C
Find the second largest prime factor of 1,599,999,999.

Options:
A. 182
B. 171
C. 179
D. 175
E. 178
F. 173
G. 176
H. 183
I. 177
J. 181
Answer: J
Let $\tau(n)$ be the number of positive divisors of $n$ . Let $a$ , $b$ , and $c$ be positive integers in geometric progression such that $a<b<c\le 400$ and $\tau(a)$ , $\tau(b)$ , and $\tau(c)$ form an arithmetic progression in that order. Find the sum of all possible integer values of $\dfrac{b}{a}$ .

Options:
A. 110
B. 113
C. 118
D. 111
E. 114
F. 115
G. 117
H. 116
I. 112
J. 119
Answer: E
In $\triangle ABC,~AB=19,~BC=20,$ and $CA=21.$ let $E$ be the foot of the perpendicular from $B$ onto $AC$ and let $F$ be the foot of the perpendicular from $C$ onto $AB.$ Let $G$ be the foot of the perpendicular from $A$ onto $EF.$ If the length of $AG$ can be written in the form $\frac{m\sqrt{p}}{n},$ where $m$ and $n$ are relatively prime positive integers and $p$ is a positive integer that is not divisible by the square of any prime, find $m+p+n.$

Options:
A. 357
B. 369
C. 362
D. 365
E. 353
F. 355
G. 361
H. 363
I. 367
J. 359
Answer: I
Find $xy$ if $x^2y - 21x^2 + 77y = 2018$ and $x, y$ are positive integers.

Options:
A. 394
B. 395
C. 398
D. 397
E. 390
F. 392
G. 391
H. 396
I. 399
J. 393
Answer: H
A function $f(n)$ returns the number of prime numbers less than or equal to $n$ , for some integer $n$ . Compute
$$$\sum_{n=1}^{99} (f(n)-f(n-1))(f(100-n)-f(99-n)).$$$

Options:
A. 16
B. 13
C. 14
D. 11
E. 12
F. 17
G. 18
H. 10
I. 19
J. 15
Answer: E
A mother purchases $5$ blue plates, $2$ red plates, $2$ green plates, and $1$ orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the $2$ green plates to be adjacent?

Options:
A. 580
B. 588
C. 596
D. 570
E. 592
F. 576
G. 584
H. 572
I. 564
J. 560
Answer: B
The polynomial $P(x, y)$ has integer coefficients, and satisfies the equations:
$$$P(23, 23) - P(23, 16) - P(16, 23) = 83,197,283$$$
$$$P(19, 19) - P(19, 8) - P(8, 19) = 54,730,575$$$
$$$P(17, 17) - P(17, 4) - P(4, 17) = 42,456,749$$$
It is given that $0 \leq P(30, 30) \leq 999$ . Find $P(30, 30)$ .

Options:
A. 784
B. 780
C. 789
D. 782
E. 787
F. 783
G. 786
H. 785
I. 788
J. 781
Answer: F
Suppose $n$ is a positive integer. Let $f(n)$ be the sum of the distinct positive prime divisors of $n$ less than $50$ (e.g. $f(12) = 2+3 = 5$ and $f(101) = 0$ ). Evaluate the remainder when $f(1) + f(2) + · · · + f(99)$ is divided by $1000$ .

Options:
A. 378
B. 352
C. 358
D. 363
E. 362
F. 372
G. 359
H. 388
I. 377
J. 368
Answer: J
Let $n$ be the number of $10$ digit integers divisible by $5$ whose sum of digits is also divisible by $5$ . Find the number of divisors of $n$ .

Options:
A. 243
B. 230
C. 244
D. 245
E. 240
F. 235
G. 246
H. 250
I. 241
J. 242
Answer: E
Let $\textbf{u}=4\textbf{i}+3\textbf{j}$ and $\textbf{v}$ be two perpendicular vectors in the $x-y$ plane. If there are $n$ vectors $\textbf{r}_i$ for $i=1, 2, \ldots, n$ in the same plane having projections of $1$ and $2$ along $\textbf{u}$ and $\textbf{v}$ respectively, then find
$$\sum_{i=1}^{n}\|\textbf{r}_i\|^2.$$
(Note: $\textbf{i}$ and $\textbf{j}$ are unit vectors such that $\textbf{i}=(1,0)$ and $\textbf{j}=(0,1)$ , and the projection of a vector $\textbf{a}$ onto $\textbf{b}$ is the length of the vector that is formed by the origin and the foot of the perpendicular of $\textbf{a}$ onto $\textbf{b}$ .)
tc1729

Options:
A. 24
B. 15
C. 22
D. 19
E. 18
F. 26
G. 20
H. 23
I. 21
J. 25
Answer: G
Joe flips a coin $20$ times, if the probability that he never gets $2$ consecutive heads or $3$ consecutive tails is $\frac{m}{n}$ , $m$ and $n$ are relatively prime positive integers, find $m$ .

Options:
A. 460
B. 469
C. 461
D. 464
E. 466
F. 463
G. 468
H. 462
I. 465
J. 467
Answer: I
Let $f(x,y)$ be a multivariable function defined on base $10$ positive integers that returns the sum of the digits of $x_y$ . For example $f(5,3)=3$ and $f(11,12)=11$ . Let $N$ be the number of ordered pairs of base $10$ positive integers $(a,b)$ that satisfy $f(a,b)=f(a+1,b)$ , $1\leq{a}\leq{2017^2}$ , and $2\leq{b}\leq{2017}$ . Find the remainder when $N$ is divided by $1000$ .
Author: Vismay Sharan

Options:
A. 77
B. 79
C. 78
D. 70
E. 76
F. 73
G. 71
H. 72
I. 75
J. 74
Answer: F
If $f(n)$ denotes the number of divisors of $2024^{2024}$ that are either less than $n$ or share at least one prime factor with $n$ , find the remainder when
$$$\sum^{2024^{2024}}_{n=1} f(n)$$$
is divided by $1000$ .

Options:
A. 224
B. 244
C. 240
D. 248
E. 216
F. 232
G. 256
H. 228
I. 220
J. 236
Answer: A
Find the last three digits of $(\tfrac{5^{2014} - 1}{4})!$ when all its trailing zeroes are truncated.

Options:
A. 448
B. 456
C. 440
D. 438
E. 450
F. 452
G. 446
H. 432
I. 442
J. 444
Answer: A
Let $N$ be the number of ordered triples $(a,b,c)$ of positive integers such that $\gcd(a,b,c)=1$ and all of
$$$\frac{2020 a}b,\,\frac {2021b}c,\,\frac{2022c}{a}$$$
are integers. Find the remainder when $N$ is divided by 1000.

Options:
A. 295
B. 292
C. 260
D. 270
E. 288
F. 275
G. 290
H. 282
I. 285
J. 280
Answer: G
Let $\phi(n)$ denote the amount of positive integers less than or equal to, and relatively prime to $n$ . If
$$$M=\sum_{d|10^{2018}}\phi(d)^{2018}$$$
, compute the remainder when $M$ is divided by $1000$ .

Options:
A. 920
B. 926
C. 929
D. 924
E. 927
F. 922
G. 923
H. 921
I. 925
J. 928
Answer: F
Let S be the sum of all positive integer k such that k is relatively prime to 100 and k is less than 100. If $10^n\leq S < 10^{n+1}$ for a positive integer n, then find $\lfloor \frac{S}{10^{n-2}}\rfloor$

Options:
A. 180
B. 199
C. 195
D. 197
E. 200
F. 198
G. 194
H. 192
I. 196
J. 190
Answer: E
Let $A$ be the area of the region in the first quadrant bounded by the line $y = \frac{1}{2} x$, the $x$-axis, and the ellipse $\frac{1}{9} x^2 + y^2 = 1$. Find the positive number $m$ such that $A$ is equal to the area of the region in the first quadrant bounded by the line $y = mx$, the $y$-axis, and the ellipse $\frac{1}{9} x^2 + y^2 = 1$.

Options:
A. \frac{1}{10}
B. \frac{1}{12}
C. \frac{1}{7}
D. \frac{1}{8}
E. \frac{1}{9}
F. \frac{1}{6}
G. \frac{1}{3}
H. \frac{1}{18}
I. \frac{1}{4}
J. \frac{2}{9}
Answer: J
Consider the sequence defined by $q_0=1$, $q_1=2$, and $q_d=4q_{d-1}-q_{d-2}$ for $d\geq 2$. Determine the smallest odd prime factor of $q_{255}$.

Options:
A. 177
B. 178
C. 175
D. 179
E. 183
F. 187
G. 173
H. 171
I. 181
J. 185
Answer: I
Find the sum of all the $x$, $y$ values in the pairs of real numbers $(x,y)$ satisfying the system of equations \begin{align*} \frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2) \\ \frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4). \end{align*}

Options:
A. 3^{2/5}
B. 3^{1/3}
C. 3^{1/6}
D. 3^{3/5}
E. 3^{2/7}
F. 3^{1/4}
G. 3^{1/8}
H. 3^{1/7}
I. 3^{4/5}
J. 3^{1/5}
Answer: J
Find the least positive integer $Q$ such that a regular tetrahedron with vertices having integer coordinates can be inscribed within the sphere defined by the equation \[ d^2 + q^2 + p^2 = Q. \]

Options:
A. \boxed{7}
B. \boxed{9}
C. \boxed{11}
D. \boxed{6}
E. \boxed{4}
F. \boxed{8}
G. \boxed{10}
H. \boxed{3}
I. \boxed{5}
J. \boxed{2}
Answer: H
For a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary (for example, $d(13) = 1+1+0+1=3)$. Let \[ S = \sum_{k=1}^{2020} (-1)^{d(k)} k^3. \] Determine $S$ modulo 2020.

Options:
A. 1982
B. 1985
C. 1990
D. 1995
E. 1970
F. 1992
G. 1988
H. 1980
I. 1994
J. 1975
Answer: C
Given a positive integer $x$, define the function $p_x(o) = \prod_{{k=1}}^x \cos(ko)$.Determine the smallest $x$ such that the absolute value of the second derivative at zero satisfies $|p_x''(0)| > 89688$.

Options:
A. 60
B. 63
C. 62
D. 66
E. 68
F. 61
G. 64
H. 65
I. 67
J. 69
Answer: H
Identify the smallest natural number (positive integer) $f$ such that for every integer $g$ with $0 < g < 2423$, there exists an integer $d$ for which satisfies the expression \[ \frac{g}{2423} < \frac{d}{f} < \frac{m+1}{2424}. \]

Options:
A. 4841
B. 4842
C. 4843
D. 4846
E. 4848
F. 4849
G. 4844
H. 4847
I. 4840
J. 4845
Answer: H
Let $f_1, \dots, f_{10}$ be bijections of the set of integers such that for each integer $n$, there is some composition $f_{i_1} \circ f_{i_2} \circ \cdots \circ f_{i_m}$ of these functions (allowing repetitions) which maps 0 to $n$. Consider the set of 1024 functions \[ \mathcal{F} = \{f_1^{e_1} \circ f_2^{e_2} \circ \cdots \circ f_{10}^{e_{10}}\}, \] $e_i = 0$ or 1 for $1 \leq i \leq 10$. ($f_i^0$ is the identity function and $f_i^1 = f_i$.) Given that $A$ is any nonempty finite set of integers, find the maximum number the functions in $\mathcal{F}$ that map to $A$ to itself.

Options:
A. 1022
B. 256
C. 384
D. 255
E. 1023
F. 64
G. 1024
H. 128
I. 512
J. 768
Answer: I
Find zeros of $f(x) = \sqrt{x-3} + \sqrt{x+2} - \sqrt{2 \cdot x-5} - 2$

Options:
A. 6.9
B. 6
C. 8
D. 5
E. 6.2
F. 6.5
G. 7
H. 4
I. 6.7
J. 6.8
Answer: G
Evaluate $\lim_{x \to 0^{+}} \left( \left( \frac{ \tan(x) }{ x } \right)^{\frac{ 1 }{ x^2 }} \right)$ using l'Hospital's Rule.

Options:
A. e^(1/6)
B. e^(1/5)
C. e^(1/2)
D. e^(1/10)
E. e^(1/8)
F. e^(1/9)
G. e^(1/3)
H. e^(1/12)
I. e^(1/4)
J. e^(1/7)
Answer: G
Evaluate the triple integral $\int \int \int f(x,y,z) \, dx \, dy \, dz$ over the solid $f(x,y,z) = e^{\sqrt{x^2+y^2}}$, $1 \le x^2+y^2 \le 4$, $y \le 0$, $x \le y \cdot \sqrt{3}$, $2 \le z \le 3$.

Options:
A. π * e^2 / 7
B. π * e^2 / 8
C. π * e^2 / 4
D. π * e^2 / 10
E. π * e^2 / 3
F. π * e^2 / 5
G. π * e^2 / 12
H. π * e^2 / 9
I. π * e^2 / 6
J. π * e^2 / 18
Answer: I
Compute $\sqrt[3]{130}$ with an estimate error of $0.0001$ using series expansion.

Options:
A. 5.06577774
B. 5.06577777
C. 5.06577778
D. 5.06577770
E. 5.06577779
F. 5.06577773
G. 5.06577776
H. 5.06577775
I. 5.06577772
J. 5.06577771
Answer: C
Evaluate $\int\int\int_{E}{(x+2 \cdot y \cdot z) \, dV}$, where $E = \left\{(x,y,z) | 0 \le x \le 1, 0 \le y \le x, 0 \le z \le 5-x-y \right\}$.

Options:
A. 439/127
B. 439/128
C. 439/123
D. 439/125
E. 439/122
F. 439/126
G. 439/121
H. 439/119
I. 439/124
J. 439/120
Answer: J
Solve the integral:
$$
\int \left(\frac{ x+3 }{ x-3 }\right)^{\frac{ 3 }{ 2 }} \, dx
$$

Options:
A. C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) + sqrt(x+3)) / (sqrt(x-3) - 3*sqrt(x+3))))
B. C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) + 2*sqrt(x+3)) / (sqrt(x-3) - 2*sqrt(x+3))))
C. C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) + 3*sqrt(x+3)) / (sqrt(x-3) - sqrt(x+3))))
D. C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) + 2*sqrt(x+3)) / (sqrt(x-3) - sqrt(x+3))))
E. C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) - 2*sqrt(x+3)) / (sqrt(x-3) + sqrt(x+3))))
F. C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) + sqrt(x+3)) / (sqrt(x-3) - 2*sqrt(x+3))))
G. C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) - 2*sqrt(x+3)) / (sqrt(x-3) + 2*sqrt(x+3))))
H. C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) + sqrt(x+3)) / (sqrt(x-3) + 2*sqrt(x+3))))
I. C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) + sqrt(x+3)) / (sqrt(x-3) - sqrt(x+3))))
J. C + sqrt((x+3)/(x-3)) * (x-15) - 9 * ln(abs((sqrt(x-3) - sqrt(x+3)) / (sqrt(x-3) + sqrt(x+3))))
Answer: J
Find the zeros of $f(x) = (2 - x)^4 + (2 \cdot x - 1)^4 - (x + 1)^4$.

Options:
A. x_1=2, x_2=1.5
B. x_1=2, x_2=0.375
C. x_1=2, x_2=0.875
D. x_1=2, x_2=0.75
E. x_1=2, x_2=0.5
F. x_1=2, x_2=0.625
G. x_1=2, x_2=0.6
H. x_1=2, x_2=0.4
I. x_1=2, x_2=0.25
J. x_1=2, x_2=1
Answer: E
Find the Taylor series for $f(x) = \frac{ x }{ (2+x)^3 }$, centered at $x=-1$. Write out the sum of the first four non-zero terms, followed by dots.

Options:
A. x * (1 - 3 * (x + 1) + 6 * (x + 1)^2 - 12 * (x + 1)^3 + ...)
B. x * (1 - 3 * (x + 1) + 6 * (x + 1)^2 - 14 * (x + 1)^3 + ...)
C. x * (1 - 3 * (x + 1) + 6 * (x + 1)^2 - 9 * (x + 1)^3 + ...)
D. x * (1 - 3 * (x + 1) + 6 * (x + 1)^2 - 8 * (x + 1)^3 + ...)
E. x * (1 - 3 * (x + 1) + 6 * (x + 1)^2 - 13 * (x + 1)^3 + ...)
F. x * (1 - 3 * (x + 1) + 6 * (x + 1)^2 - 15 * (x + 1)^3 + ...)
G. x * (1 - 3 * (x + 1) + 6 * (x + 1)^2 - 11 * (x + 1)^3 + ...)
H. x * (1 - 3 * (x + 1) + 6 * (x + 1)^2 - 16 * (x + 1)^3 + ...)
I. x * (1 - 3 * (x + 1) + 6 * (x + 1)^2 - 10 * (x + 1)^3 + ...)
J. x * (1 - 3 * (x + 1) + 6 * (x + 1)^2 - 7 * (x + 1)^3 + ...)
Answer: I
The integral has been converted to polar coordinates. Verify that the identity is true and choose the easiest way to evaluate the integral, in rectangular or polar coordinates:
$$
\int_{0}^1 \int_{x^2}^x \frac{ y }{ \sqrt{x^2+y^2} } \, dy \, dx = \int_{0}^{\frac{ \pi }{ 4 }} \int_{0}^{\sec(\varphi) \cdot \tan(\varphi)} r \cdot \sin(\varphi) \, dr \, d\varphi
$$

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

Options:
A. 19/140
B. 13/140
C. 16/140
D. 9/140
E. 15/140
F. 14/140
G. 17/140
H. 12/140
I. 11/140
J. 10/140
Answer: I
Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{\frac{ -\pi }{ 3 }}^{\frac{ \pi }{ 3 }} \sqrt{\left(\sec(x)\right)^2-1} \, dx
$$

Options:
A. 2\sqrt{2}
B. 2\sqrt{3}
C. 2\sqrt{3} + 2
D. 2\\sqrt{3} - 5
E. 2\sqrt{3} - 3
F. 0
G. -2\sqrt{3}
H. 2\sqrt{3} - 2
I. 2\sqrt{3} - 1
J. 2\sqrt{3} - 4
Answer: F
Compute the integral:
$$
-\int \cos(6 \cdot x)^6 \, dx
$$

Options:
A. C + \frac{1}{288} \cdot \left(\sin(12 \cdot x)\right)^3 - \frac{1}{24} \cdot \sin(12 \cdot x) - \frac{1}{128} \cdot \sin(24 \cdot x) - \frac{11}{16} \cdot x
B. C + \frac{1}{288} \cdot \left(\sin(12 \cdot x)\right)^3 - \frac{1}{24} \cdot \sin(12 \cdot x) - \frac{1}{128} \cdot \sin(24 \cdot x) - \frac{9}{16} \cdot x
C. C + \frac{1}{288} \cdot \left(\sin(12 \cdot x)\right)^3 - \frac{1}{24} \cdot \sin(12 \cdot x) - \frac{1}{128} \cdot \sin(24 \cdot x) - \frac{5}{16} \cdot x
D. C + \frac{1}{288} \cdot \left(\sin(12 \cdot x)\right)^3 - \frac{1}{24} \cdot \sin(12 \cdot x) - \frac{1}{128} \cdot \sin(24 \cdot x) - \frac{19}{16} \cdot x
E. C + \frac{1}{288} \cdot \left(\sin(12 \cdot x)\right)^3 - \frac{1}{24} \cdot \sin(12 \cdot x) - \frac{1}{128} \cdot \sin(24 \cdot x) - \frac{3}{16} \cdot x
F. C + \frac{1}{288} \cdot \left(\sin(12 \cdot x)\right)^3 - \frac{1}{24} \cdot \sin(12 \cdot x) - \frac{1}{128} \cdot \sin(24 \cdot x) - \frac{7}{16} \cdot x
G. C + \frac{1}{288} \cdot \left(\sin(12 \cdot x)\right)^3 - \frac{1}{24} \cdot \sin(12 \cdot x) - \frac{1}{128} \cdot \sin(24 \cdot x) - \frac{1}{16} \cdot x
H. C + \frac{1}{288} \cdot \left(\sin(12 \cdot x)\right)^3 - \frac{1}{24} \cdot \sin(12 \cdot x) - \frac{1}{128} \cdot \sin(24 \cdot x) - \frac{13}{16} \cdot x
I. C + \frac{1}{288} \cdot \left(\sin(12 \cdot x)\right)^3 - \frac{1}{24} \cdot \sin(12 \cdot x) - \frac{1}{128} \cdot \sin(24 \cdot x) - \frac{17}{16} \cdot x
J. C + \frac{1}{288} \cdot \left(\sin(12 \cdot x)\right)^3 - \frac{1}{24} \cdot \sin(12 \cdot x) - \frac{1}{128} \cdot \sin(24 \cdot x) - \frac{15}{16} \cdot x
Answer: C
Find all real solutions of the following equation:
$$
\frac{ x }{ 2 x+6 }-\frac{ 2 }{ 5 x+5 }=\frac{ 5 x^2-3 x-7 }{ 10 x^2+40 x+30 }
$$

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

Options:
A. 104/5
B. 108/5
C. 114/5
D. 116/5
E. 112/5
F. 102/5
G. 111/5
H. 110/5
I. 106/5
J. 109/5
Answer: E
In trapezoid $ABCD,$ $AB||CD,$ and $AB\perp BC.$ There is a point $P$ on side $AD$ such that the circumcircle of triangle $BPC$ is tangent to $AD.$ If $AB=3, AD=78, CD=75,$ $CP-BP$ can be expressed in the form $\frac{a\sqrt{b}} {c},$ where $a,b,c$ are positive integers and $a,c$ are relatively prime. Find $a+b+c.$

Options:
A. 101
B. 99
C. 106
D. 107
E. 105
F. 103
G. 104
H. 108
I. 102
J. 100
Answer: B
Compute the integral:
$$
\int \sin(2 \cdot x)^6 \cdot \cos(2 \cdot x)^2 \, dx
$$

Options:
A. \frac{1}{32}\cdot x-\frac{1}{32}\cdot\frac{1}{8}\cdot\sin(8\cdot x)-\frac{1}{8}\cdot\frac{1}{4}\cdot\frac{1}{3}\cdot\sin(4\cdot x)^3+\frac{1}{128}\cdot x-\frac{1}{128}\cdot\frac{1}{16}\cdot\sin(16\cdot x)^3+C
B. \frac{1}{32}\cdot x-\frac{1}{32}\cdot\frac{1}{8}\cdot\sin(8\cdot x)-\frac{1}{8}\cdot\frac{1}{4}\cdot\frac{1}{3}\cdot\sin(4\cdot x)^3+\frac{1}{128}\cdot x-\frac{1}{128}\cdot\frac{1}{16}\cdot\sin(16\cdot x)^2+C
C. \frac{1}{32}\cdot x-\frac{1}{32}\cdot\frac{1}{8}\cdot\sin(8\cdot x)-\frac{1}{8}\cdot\frac{1}{4}\cdot\frac{1}{3}\cdot\sin(4\cdot x)^3+\frac{1}{128}\cdot x-\frac{1}{128}\cdot\frac{1}{16}\cdot\sin(16\cdot x)+C
D. \frac{1}{32}\cdot x-\frac{1}{32}\cdot\frac{1}{8}\cdot\sin(8\cdot x)-\frac{1}{8}\cdot\frac{1}{4}\cdot\frac{1}{3}\cdot\sin(4\cdot x)^3+\frac{1}{128}\cdot x-\frac{1}{128}\cdot\frac{1}{16}\cdot\sin(16\cdot x)^4+C
Answer: C
Let $F_0(x) = \ln x$. For $n \geq 0$ and $x > 0$, let $F_{n+1}(x) = \int_0^x F_n(t)\,dt$. Evaluate \[ \lim_{n \to \infty} \frac{n! F_n(1)}{\ln n}. \]

Options:
A. -1/3
B. -2
C. -1
D. -1/2
E. -1/4
Answer: C
Let $a$ and $b$ positive integers such that the lines $y=ax$ and $x=by$ form an acute angle of measure $\tan^{-1} 7$ degrees. Find the sum of all possible values of $a$ .

Options:
A. 145
B. 130
C. 135
D. 140
E. 125
F. 120
G. 155
Answer: C
For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \sqrt{2k})$. Evaluate \[ \sum_{k=1}^\infty (-1)^{k-1} \frac{A(k)}{k}. \]

Options:
A. \frac{\pi^2}{8}
B. \frac{\pi^2}{12}
C. \frac{\pi^2}{32}
D. \frac{\pi^2}{20}
E. \frac{\pi^2}{24}
F. \frac{\pi^2}{16}
G. \frac{\pi^2}{28}
Answer: F
Evaluate the integral:
$$
\int_{0}^1 \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \int_{-\sqrt{1-x^2-y^2}}^{\sqrt{1-x^2-y^2}} 1 \, dz \, dy \, dx
$$

Options:
A. 4π/15
B. 2π/3
C. 8π/9
D. 4π/5
E. 4π/3
F. 4π/12
G. 4π/6
H. 4π/9
Answer: B
Determine the remainder obtained when $1000!$ is divided by $2003$ .

Options:
A. 1999
B. 1998
C. 3
D. 1997
E. 1996
F. 1
G. 1995
H. 2
I. 1994
Answer: H
Evaluate the iterated integral:
$$
\int_{e}^{e^2} \int_{\ln(u)}^2 \left(v + \ln(u)\right) \, dv \, du
$$

Options:
A. e^2 - 5e/4
B. e^2 - 7e/4
C. e^2 - 3e/2
D. e^2 - 4e/2
E. e^2 - e/4
F. e^2 - e/2
G. e^2 - 2e/4
H. e^2 - 3e/4
I. e^2 - 2e/2
Answer: F
There are 10 guests at a party: 5 pairs of identical twins. The 10 guests sit around a circular table such that each pair of identical twins have at least 2 other people in between them. Assuming the identical twins in each pair are indistinguishable from each other, and the table is fixed to the ground, what are the last 3 digits of the number of ways the 10 guests can be seated?

Options:
A. 165
B. 140
C. 120
D. 190
E. 110
F. 150
G. 170
H. 130
I. 180
J. 160
Answer: J
Let $ABC$ be a triangle such that $AB = 68$ , $BC = 100$ , and $CA = 112$ . Let $H$ be the orthocenter of $\vartriangle ABC$ (intersection of the altitudes). Let $D$ be the midpoint of $BC$ , $E$ be the midpoint of $CA$ , and $F$ be the midpoint of $AB$ . Points $X$ , $Y$ , and $Z$ are constructed on $HD$ , $HE$ , and $HF$ , respectively, such that $D$ is the midpoint of $XH$ , $E$ is the midpoint of $Y H$ , and $F$ is the midpoint of $ZH$ . Find $AX + BY + CZ$ .

Options:
A. 334
B. 338
C. 340
D. 344
E. 337
F. 342
G. 332
H. 336
I. 339
J. 341
Answer: C
$a_{n}$ is a sequence such that $4\cdot a_{n}=a_{2n}$ and $a_{2n}=2\cdot a_{2n-1}+\frac{1}{4}$ for all $n\in\mathbb{N}$ . Find the sum
$$$S=a_{1}+a_{2}+...+a_{31}$$$

Options:
A. 602
B. 601.375
C. 601.25
D. 601.75
E. 601.9375
F. 601.8125
G. 601.125
H. 601.5
I. 601.625
J. 601.875
Answer: A
Consider a monic cubic polynomial $P(x)$ with roots $a,b,c$ such that $a,b,c\leq 1.$ If $a+b+c=-8$ and there is one root at least $4$ greater than another, find the maximum possible value of the sum of the coefficients of $P(x).$ (The constant term is counted.)

Options:
A. 34
B. 36
C. 38
D. 31
E. 37
F. 32
G. 33
H. 30
I. 35
J. 39
Answer: B
Find the number of positive integers $n \le 1024$ such that $(n + 2)(n + 4)(n + 8)(n + 16)$ is a multiple of $1024$ .

Options:
A. 160
B. 176
C. 170
D. 162
E. 166
F. 174
G. 169
H. 172
I. 164
J. 168
Answer: J
Determine the number of five-digit positive integer multiples of $11$ whose digits sum to $19$ .

Options:
A. 363
B. 396
C. 377
D. 374
E. 376
F. 386
G. 387
H. 384
I. 375
J. 385
Answer: J
Let $P$ be a polyhedron with $37$ faces, all of which are equilateral triangles, squares, or regular pentagons with equal side length. Given there is at least one of each type of face and there are twice as many pentagons as triangles, what is the sum of all the possible number of vertices $P$ can have?

Options:
A. 252
B. 257
C. 259
D. 258
E. 253
F. 251
G. 255
H. 256
I. 250
J. 254
Answer: G
Call a polynomial $P(x)$ 3-star if all of its coefficients are nonnegative integers and has maximum degree $3.$ (For example, $x^3+12x+19$ and $x^2+8x$ are 3-star). Furthermore, consider a 3-star polynomial extra special if all the coefficients are at most $20,$ and $P(-1)=-16.$ Let $K$ be the number of extra special 3-star polynomials. What is the remainder when $K$ is divided by $1000$ ?

Options:
A. 765
B. 773
C. 764
D. 766
E. 771
F. 770
G. 769
H. 772
I. 768
J. 767
Answer: J
If $p(x)=x^{3}-5x^{2}+4x-1$ has roots $a$ , $b$ , and $c$ , then find $\left(a^{2}-\frac{1}{a^{2}}\right)\left(b^{2}-\frac{1}{b^{2}}\right)\left(c^{2}-\frac{1}{c^{2}}\right)$ .

Options:
A. 274
B. 279
C. 272
D. 275
E. 277
F. 278
G. 270
H. 273
I. 276
J. 271
Answer: D
$\triangle ABC $ has circumradius $6$ and inradius $3$ . The incircle is tangent to the sides of the triangle at points $ X,Y,Z $ . Find $ \frac{[ABC]}{[XYZ]} $ where $ [ABC] $ denotes the area of $ \triangle ABC $ .

Options:
A. 4
B. 3.78
C. 3.5
D. 3.25
E. 3.75
F. 3.9
G. 3.72
H. 3.6
I. 3.8
J. 3.7
Answer: A
Determine the value of $10(x+y)$ , given that $[x]+[y]+y=43.8$ , and $x+y-[x]=18.4$ , where $[$ $]$ means greatest integer less than or equal to $x$ or $y$ .

Options:
A. 278
B. 270
C. 276
D. 266
E. 274
F. 288
G. 272
H. 284
I. 264
J. 268
Answer: E
$ABC$ is a triangle with integer side lengths. Extend $\overline{AC}$ beyond $C$ to point $D$ such that $CD=120$ . Similarly, extend $\overline{CB}$ beyond $B$ to point $E$ such that $BE=112$ and $\overline{BA}$ beyond $A$ to point $F$ such that $AF=104$ . If triangles $CBD$ , $BAE$ , and $ACF$ all have the same area, what is the minimum possible area of triangle $ABC$ ?

Options:
A. 120
B. 84
C. 144
D. 96
E. 72
F. 112
G. 168
H. 156
I. 108
J. 160
Answer: B
If $x \sqrt[3]{4} + \frac{ \sqrt[3]{2}}{x}= 3$ and $x$ is real, evaluate $64x^9 +\frac{8}{x^9}$ .

Options:
A. 515
B. 514
C. 513
D. 511
E. 510
F. 512
G. 517
H. 519
I. 516
J. 518
Answer: C
Let $P(x) = (1 + x)(1 + 2x^2)(1 + 3x^4)(1 + 4x^8)(1 + 5x^{16})$ . Find the three rightmost nonzero digits of the product of the coefficients of $P(x)$ .

Options:
A. 413
B. 410
C. 418
D. 416
E. 419
F. 412
G. 415
H. 420
I. 414
J. 417
Answer: D
Regular octahedron $ABCDEF$ (such that points $B$ , $C$ , $D$ , and $E$ are coplanar and form the vertices of a square) is divided along plane $ \mathcal{P} $ , parallel to line $ BC $ , into two polyhedra of equal volume. The cosine of the acute angle plane $ \mathcal{P} $ makes with plane $ BCDE $ is $ \frac{1}{3} $ . Given that $ AB=30 $ , find the area of the cross section made by plane $ \mathcal{P} $ with octahedron $ ABCDEF $ .

Options:
A. 540
B. 600
C. 550
D. 580
E. 570
F. 560
G. 576
H. 572
I. 590
J. 575
Answer: B
The symmetric difference $A \Delta B$ of two sets $A$ and $B$ is the set of elements in exactly one of $A$ and $B$ . Find the remainder when the number of triples of sets $(A, B, C)$ with $A, B, C \subseteq \{1, 2, \dots, 8\}$ and $|A \Delta B| + |B \Delta C| = |A \Delta C|$ is divided by $1000$ .

Options:
A. 615
B. 619
C. 616
D. 613
E. 612
F. 610
G. 618
H. 617
I. 614
J. 620
Answer: C
Let $P(x)=\sum_{i=1}^{20}(-x)^{20-i}(x+i)^i$ . Let $K$ be the product of the roots. How many digits are does $\lfloor K \rfloor$ have where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ ?

Options:
A. 21
B. 26
C. 25
D. 24
E. 29
F. 23
G. 27
H. 22
I. 28
J. 20
Answer: D
How many integers $n$ between $0$ and $2006$ inclusively, satisfy the condition that $3$ divides
$$$n + \left\lfloor \frac{n}{3}\right\rfloor + \left\lfloor \frac{n}{9}\right\rfloor + \left\lfloor \frac{n}{27}\right\rfloor?$$$

Options:
A. 667
B. 668
C. 666
D. 672
E. 669
F. 670
G. 673
H. 671
I. 675
J. 674
Answer: E
Find the positive number $m$ such that the area in the first quadrant bounded by the line $y = \frac{1}{56} x$, the $x$-axis, and the ellipse $\frac{1}{441} x^2 + y^2 = 1$ is equal to the area of the region in the first quadrant bounded by the line $y = mx$, the $y$-axis, and the ellipse $\frac{1}{441} x^2 + y^2 = 1$.

Options:
A. \frac{56}{442}
B. \frac{56}{441}
C. \frac{56}{449}
D. \frac{56}{447}
E. \frac{56}{444}
F. \frac{56}{448}
G. \frac{56}{445}
H. \frac{56}{443}
I. \frac{56}{440}
J. \frac{56}{446}
Answer: B
If $X$ is a finite set, let $X$ denote the number of elements in $X$. Call an ordered pair $(S, T)$ of subsets of $\{1, 2, \dots, n\}$ {\em admissible} if $s > |T|$ for each $s \in S$, and $t > |S|$ for each $t \in T$. How many admissible ordered pairs of subsets of $\{1, 2, \dots, 10\}$ are there?

Options:
A. 17710
B. 17717
C. 17712
D. 17713
E. 17714
F. 17711
G. 17719
H. 17716
I. 17718
J. 17715
Answer: F
Find the sum of all positive integers $n$ that are within 250 of exactly 15 perfect squares.

Options:
A. 10003
B. 9996
C. 10000
D. 9998
E. 9999
F. 9997
G. 9995
H. 10002
I. 10001
J. 10004
Answer: E
Identify the smallest natural number (positive integer) $m$ such that for every integer $d$ with $0 < d < 1743$, there exists an integer $w$ for which satisfies the expression \[ \frac{d}{1743} < \frac{w}{m} < \frac{m+1}{1744}. \]

Options:
A. 3487
B. 3484
C. 3481
D. 3485
E. 3486
F. 3488
G. 3482
H. 3483
I. 3489
J. 3480
Answer: A
Find the upper bound of the real number $c$ such that there is a straight line that intersects the curve \[ x^4 + 9x^3 + cx^2 + 9x + 4 \] in four distinct points?

Options:
A. \frac{243}{12}
B. \frac{243}{18}
C. \frac{243}{24}
D. \frac{243}{32}
E. \frac{243}{8}
F. \frac{243}{14}
G. \frac{243}{10}
H. \frac{243}{16}
I. \frac{243}{6}
J. \frac{243}{4}
Answer: E
Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule. Alice's goal is to maximize the number of uncovered squares when the game ends; Bob's goal is to minimize it. What is the greatest number of uncovered squares that Alice can ensure at the end of the game, no matter how Bob plays?

Options:
A. 284
B. 293
C. 292
D. 289
E. 291
F. 288
G. 285
H. 290
I. 287
J. 286
Answer: H
A \emph{base $10$ over-expansion} of a positive integer $N$ is an expression of the form \[ N = d_k 10^k + d_{k-1} 10^{k-1} + \cdots + d_0 10^0 \] with $d_k \neq 0$ and $d_i \in \{0,1,2,\dots,10\}$ for all $i$. For instance, the integer $N = 10$ has two base 10 over-expansions: $10 = 10 \cdot 10^0$ and the usual base 10 expansion $10 = 1 \cdot 10^1 + 0 \cdot 10^0$. Find the sum of all positive integers less than 150 that have a unique base 10 over-expansion.

Options:
A. 9179
B. 9181
C. 9182
D. 9178
E. 9170
F. 9160
G. 9175
H. 9190
I. 9180
J. 9185
Answer: I
Suppose that $X_1, X_2, \dots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S = \sum_{i=1}^k X_i/2^i$, where $k$ is the least positive integer such that $X_k < X_{k+1}$, or $k = \infty$ if there is no such integer. Find the expected value of $S$.

Options:
A. 2e^{\frac{1}{2}}-8
B. 2e^{\frac{1}{2}}-6
C. 2e^{\frac{1}{2}}-3
D. 2e^{\frac{1}{2}}-9
E. 2e^{\frac{1}{2}}-10
F. 2e^{\frac{1}{2}}-2
G. 2e^{\frac{1}{2}}-5
H. 2e^{\frac{1}{2}}-4
I. 2e^{\frac{1}{2}}-1
J. 2e^{\frac{1}{2}}-7
Answer: C
Find the sum of the series $\sum_{n=1}^\infty \frac{ x^{4 \cdot n-1} }{ 4 \cdot n-1 }$ using the integration of series.

Options:
A. frac{1}{4}cdotlnleft(frac{|x+1|}{|x-1|}right)+frac{1}{4}cdotarctan(x)
B. frac{1}{4}cdotlnleft(frac{|x+1|}{|x-1|}right)-frac{1}{6}cdotarctan(x)
C. frac{1}{4}cdotlnleft(frac{|x+1|}{|x-1|}right)-frac{1}{4}cdotarctan(x)
D. frac{1}{4}cdotlnleft(frac{|x+1|}{|x-1|}right)+frac{1}{2}cdotarctan(x)
E. frac{1}{4}cdotlnleft(frac{|x+1|}{|x-1|}right)-frac{1}{7}cdotarctan(x)
F. frac{1}{4}cdotlnleft(frac{|x+1|}{|x-1|}right)-frac{1}{3}cdotarctan(x)
G. frac{1}{4}cdotlnleft(frac{|x+1|}{|x-1|}right)-frac{1}{5}cdotarctan(x)
H. frac{1}{4}cdotlnleft(frac{|x+1|}{|x-1|}right)-frac{1}{8}cdotarctan(x)
I. frac{1}{4}cdotlnleft(frac{|x+1|}{|x-1|}right)-frac{1}{2}cdotarctan(x)
J. frac{1}{4}cdotlnleft(frac{|x+1|}{|x-1|}right)-frac{1}{9}cdotarctan(x)
Answer: I
Compute the derivative $y^{(5)}$ of the function $y = 2 \cdot e^{3 \cdot x} \cdot \sin(2 \cdot x)$.

Options:
A. 2 * e^(3 * x) * (122 * cos(2 * x) - 599 * sin(2 * x))
B. 2 * e^(3 * x) * (122 * cos(2 * x) - 590 * sin(2 * x))
C. 2 * e^(3 * x) * (122 * cos(2 * x) - 594 * sin(2 * x))
D. 2 * e^(3 * x) * (122 * cos(2 * x) - 596 * sin(2 * x))
E. 2 * e^(3 * x) * (122 * cos(2 * x) - 598 * sin(2 * x))
F. 2 * e^(3 * x) * (122 * cos(2 * x) - 595 * sin(2 * x))
G. 2 * e^(3 * x) * (122 * cos(2 * x) - 593 * sin(2 * x))
H. 2 * e^(3 * x) * (122 * cos(2 * x) - 591 * sin(2 * x))
I. 2 * e^(3 * x) * (122 * cos(2 * x) - 597 * sin(2 * x))
J. 2 * e^(3 * x) * (122 * cos(2 * x) - 592 * sin(2 * x))
Answer: I
Recall the formula for calculating the magnitude of an earthquake, $M=\frac{ 2 }{ 3 } \cdot \log_{10}\left(\frac{ S }{ S_{0} }\right)$. One earthquake has magnitude $3.9$ on the MMS scale. If a second earthquake has $750$ times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth.

Options:
A. 5.52
B. 5.87
C. 5.75
D. 5.67
E. 5.77
F. 5.85
G. 5.72
H. 5.92
I. 5.82
J. 5.62
Answer: I
Find the zeros of the polynomial function with complex roots: $f(x) = 2 \cdot x^3 + 4 \cdot x^2 + 3 \cdot x + 6$.

Options:
A. x = -i·√(3/2), i·√(3/2), -1.5
B. x = -i·√(3/2), i·√(3/2), 0
C. x = -i·√(3/2), i·√(3/2), -2
D. x = -i·√(3/2), i·√(3/2), -1.75
E. x = -i·√(3/2), i·√(3/2), 2
F. x = -i·√(3/2), i·√(3/2), 1
G. x = -i·√(3/2), i·√(3/2), -1.25
H. x = -i·√(3/2), i·√(3/2), -1
I. x = -i·√(3/2), i·√(3/2), -2.5
J. x = -i·√(3/2), i·√(3/2), -3
Answer: C
A tank initially holds 100 gallons of soapy water, containing 5 oz of soap. At $t = 0$, more soap solution is poured in containing 10 oz of soap/gallon at a rate of 1 gal/min. Allowing for the even dispersal of the soap, the new solution leaves the tank at the same rate. Find the amount of soap in the water after 5 minutes.

Options:
A. 52.5267
B. 53.5269
C. 54.5267
D. 53.5264
E. 53.5262
F. 53.5266
G. 53.5263
H. 53.5267
I. 53.5268
J. 53.5265
Answer: H
Use implicit differentiation to find an equation of the tangent line to the curve
$x^4 - x^3 \cdot y^2 - y^3 \cdot x + 5 \cdot y^4 = 17$ at the point
$P(-2,-1)$.

Options:
A. y+1=-\frac{43}{36}\cdot(x+2)
B. y+1=-\frac{43}{34}\cdot(x+2)
C. y+1=-\frac{43}{37}\cdot(x+2)
D. y+1=-\frac{43}{32}\cdot(x+2)
E. y+1=-\frac{43}{29}\cdot(x+2)
F. y+1=-\frac{43}{33}\cdot(x+2)
G. y+1=-\frac{43}{28}\cdot(x+2)
H. y+1=-\frac{43}{31}\cdot(x+2)
I. y+1=-\frac{43}{35}\cdot(x+2)
J. y+1=-\frac{43}{30}\cdot(x+2)
Answer: J
Find the equation of the tangent line to the curve: $r = 3 + \cos(2 \cdot t)$, $t = \frac{ 3 \cdot \pi }{ 4 }$.

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

Options:
A. C - \frac{1}{7} \cdot \left(1 + \frac{3}{x^2}\right) \cdot \sqrt{1 + \frac{3}{x^2}}
B. C - \frac{1}{18} \cdot \left(1 + \frac{3}{x^2}\right) \cdot \sqrt{1 + \frac{3}{x^2}}
C. C - \frac{1}{15} \cdot \left(1 + \frac{3}{x^2}\right) \cdot \sqrt{1 + \frac{3}{x^2}}
D. C - \frac{1}{12} \cdot \left(1 + \frac{3}{x^2}\right) \cdot \sqrt{1 + \frac{3}{x^2}}
E. C - \frac{1}{8} \cdot \left(1 + \frac{3}{x^2}\right) \cdot \sqrt{1 + \frac{3}{x^2}}
F. C - \frac{1}{6} \cdot \left(1 + \frac{3}{x^2}\right) \cdot \sqrt{1 + \frac{3}{x^2}}
G. C - \frac{1}{11} \cdot \left(1 + \frac{3}{x^2}\right) \cdot \sqrt{1 + \frac{3}{x^2}}
H. C - \frac{1}{10} \cdot \left(1 + \frac{3}{x^2}\right) \cdot \sqrt{1 + \frac{3}{x^2}}
I. C - \frac{1}{9} \cdot \left(1 + \frac{3}{x^2}\right) \cdot \sqrt{1 + \frac{3}{x^2}}
J. C - \frac{1}{3} \cdot \left(1 + \frac{3}{x^2}\right) \cdot \sqrt{1 + \frac{3}{x^2}}
Answer: I
Find the arc length of the parametric curve given by $x(t) = \frac{ 1 }{ 6 } \cdot t^3$, $y(t) = \frac{ 1 }{ 9 } \cdot t^3$ on $[1,3]$.

Options:
A. 17.477777777777781
B. 17.47777777777777
C. 17.47777777777775
D. 17.477777777777785
E. 17.47777777777778
F. 17.47777777777774
G. 17.47777777777776
H. 17.47777777777779
I. 17.47777777777780
J. 17.477777777777775
Answer: E
Solve the integral:
$$
\int \left(\frac{ x+1 }{ x-1 }\right)^{\frac{ 3 }{ 2 }} \, dx
$$

Options:
A. C + sqrt((x+1)/(x-1)) * (x-9) - 3 * ln(abs((sqrt(x-1) - sqrt(x+1)) / (sqrt(x-1) + sqrt(x+1))))
B. C + sqrt((x+1)/(x-1)) * (x-5) - 3 * ln(abs((sqrt(x-1) + sqrt(x+1)) / (sqrt(x-1) - sqrt(x+1))))
C. C + sqrt((x+1)/(x-1)) * (x-5) - 3 * ln(abs((sqrt(x-1) - sqrt(x+1)) / (sqrt(x-1) + sqrt(x+1))))
D. C + sqrt((x+1)/(x-1)) * (x-8) - 3 * ln(abs((sqrt(x-1) - sqrt(x+1)) / (sqrt(x-1) + sqrt(x+1))))
E. C + sqrt((x+1)/(x-1)) * (x-4) - 3 * ln(abs((sqrt(x-1) - sqrt(x+1)) / (sqrt(x-1) + sqrt(x+1))))
F. C + sqrt((x+1)/(x-1)) * (x-7) - 3 * ln(abs((sqrt(x-1) - sqrt(x+1)) / (sqrt(x-1) + sqrt(x+1))))
G. C + sqrt((x+1)/(x-1)) * (x-10) - 3 * ln(abs((sqrt(x-1) - sqrt(x+1)) / (sqrt(x-1) + sqrt(x+1))))
H. C + sqrt((x+1)/(x-1)) * (x-6) - 3 * ln(abs((sqrt(x-1) - sqrt(x+1)) / (sqrt(x-1) + sqrt(x+1))))
I. C + sqrt((x+1)/(x-1)) * (x-2) - 3 * ln(abs((sqrt(x-1) - sqrt(x+1)) / (sqrt(x-1) + sqrt(x+1))))
J. C + sqrt((x+1)/(x-1)) * (x-3) - 3 * ln(abs((sqrt(x-1) - sqrt(x+1)) / (sqrt(x-1) + sqrt(x+1))))
Answer: C
Compute $\sqrt[5]{240}$ with an accuracy of $0.0001$.

Options:
A. 2.9920
B. 2.9922
C. 2.9921
D. 2.9926
E. 2.9927
F. 2.9929
G. 2.9925
H. 2.9924
I. 2.9928
J. 2.9923
Answer: D
Compute the first 3 nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of $f(x) = \ln(\cos(x))$.

Options:
A. -\frac{x^2}{2}-\frac{x^4}{12}-\frac{x^6}{40}
B. -\frac{x^2}{2}-\frac{x^4}{12}-\frac{x^6}{55}
C. -\frac{x^2}{2}-\frac{x^4}{12}-\frac{x^6}{60}
D. -\frac{x^2}{2}-\frac{x^4}{12}-\frac{x^6}{48}
E. -\frac{x^2}{2}-\frac{x^4}{12}-\frac{x^6}{50}
F. -\frac{x^2}{2}-\frac{x^4}{12}-\frac{x^6}{47}
G. -\frac{x^2}{2}-\frac{x^4}{12}-\frac{x^6}{54}
H. -\frac{x^2}{2}-\frac{x^4}{12}-\frac{x^6}{30}
I. -\frac{x^2}{2}-\frac{x^4}{12}-\frac{x^6}{42}
J. -\frac{x^2}{2}-\frac{x^4}{12}-\frac{x^6}{45}
Answer: J
Compute $\sqrt[4]{80}$ with accuracy $0.0001$.

Options:
A. 2.9903
B. 2.9909
C. 2.9900
D. 2.9906
E. 2.9907
F. 2.9905
G. 2.9901
H. 2.9904
I. 2.9902
J. 2.9908
Answer: E