inclusionAI/Ring-lite-rl-data huggingface dataset:
Let \( D \) be the closed unit disc in \( \mathbb{R}^2 \) and \( S^1 = \partial D \). Suppose \( f: S^1 \rightarrow \mathbb{R}^3 \) is a smooth embedding such that \( f(S^1) \) intersects the plane \( z = 0 \) at exactly two points. Define \( \lambda: S^1 \rightarrow S^2 \) by

\[\lambda(x) = \frac{f(x)}{\|f(x)\|}.\]

Compute the degree of \( \lambda \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
一个盒中有 9 个正品和 3 个废品，每次取一个产品，取出后不再放回. 则在取出 2 个正品前已取出 3 个废品的概率为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
On Monday, a school library was attended by 5 students; on Tuesday, by 6; on Wednesday, by 4; on Thursday, by 8; and on Friday, by 7. None of them attended the library two days in a row. What is the least possible number of students who visited the library during the week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let $(X_1,X_2)$ be a bivariate normal random variable with mean vector $\mu = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and covariance matrix $\Sigma = \begin{pmatrix} 3 & -1\\ -1 & 3 \end{pmatrix}$. Calculate the probability that $X_1 + X_2 < 2$ and $X_1 - X_2 > 0$. \(\text{Express your answer as a probability.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1116}
Compute the number of values of $x$ in the interval $[-11\pi,-2\pi]$ that satisfy $\tfrac{5\cos(x)+4}{5\sin(x)+3}=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4
}
Find the range of values of \(n\) for which the equation \(x^n - x - n = 0\) has more than one solution in the interval \((1,2]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
17th Chinese 2002 Problem A3 18 people play in a tournament of 17 rounds. There are 9 games in each round and each person plays in one game in each round. Each person plays every other person just once in the tournament. What is the largest n such that however the tournament is arranged we can find 4 people amongst whom only one game has been played by the end of round n?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
What is the value of n if the sum of the consecutive prime numbers from the smallest prime number until n equals 2500?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{151}
设 $n \equiv 1(\bmod 4)$ 且 $n>1, P=\left\{a_1, a_2, \cdots, a_n\right\}$ 是 $\{1,2,3, \cdots, n\}$ 的任意排列, $k_p$表示使下列不等式成立的最大下标 $k$,

$$
a_1+a_2+\cdots+a_k<a_{k+1}+a_{k+2}+\cdots+a_n .
$$


试对一切可能的不同排列 $P$, 求对应的 $k_p$ 的总和.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{2}(n-1)(n!)$.}
Let \( G \) be a cubic graph with \( q \) edges and girth \( \ge 4 \). Determine the number of paths of length 3 in \( G \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4q}
Let \( f(x) = \sin{x} + x^2 + 1 \). Find the value of \( \frac{d^4}{dy^4}(f^{-1}(y)) \bigg|_{y=1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-140}
Given the line y = 3x - 5 and the square PQRS with R at (2009, 2009), find the number of lattice points on the line that are inside the square. A lattice point is a point with integer coordinates. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{670}
Xiao Wang places some equilateral triangle paper pieces on the table. The first time he places 1 piece; the second time he places three more pieces around the first triangle; the third time he places more pieces around the shape formed in the second placement, and so on. The requirement is: each piece placed in each subsequent placement must share at least one edge with a piece placed in the previous placement, and apart from sharing edges, there should be no other overlaps. Calculate the total number of equilateral triangle pieces used after the 20th placement.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{571}
The number 770 is written on a blackboard. Melody repeatedly performs moves, where a move consists of subtracting either 40 or 41 from the number on the board. She performs moves until the number is not positive, and then she stops. Let \( N \) be the number of sequences of moves that Melody could perform. Suppose \( N = a \cdot 2^b \) where \( a \) is an odd positive integer and \( b \) is a nonnegative integer. Compute \( 100a + b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{318}
Compute \( \mathbb{E} \left( \int_0^1 \left| B_s \right| dB_s \right)^2 \) using Ito's formula, where \(B_s\) is a Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the domain of definition of the function $$f(x)=\int_{0}^{\infty}\frac{t^x}{1+t^4 \sin ^2 t}dt$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1,1)}
Determine the number of group homomorphisms from the dihedral group $D_4 = \langle r_4, f_4 : r_4^4 = f_4^2 = (r_4f_4)^2 = e \rangle$ to the symmetric group $S_3$. Consider the possible images of the generators $r_4$ and $f_4$ under a homomorphism $\phi: D_4 \rightarrow S_3$ that satisfy the relation $\phi((r_4f_4)^2) = \phi(e) = \varepsilon$. What is the total number of such homomorphisms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the minimum number of infinite subsets that must be added to a cofinite topology on a countably infinite set to obtain the discrete topology.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
A candidate mistakenly gave his height as a certain percentage more than his actual height in a certificate. In the interview panel, he clarified that his height was 5 feet 8 inches. The candidate made a 17.012448132780083% correction from his stated height to his actual height. What was the percentage by which he overstated his height?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20.5\%}
Inside a square with side length \(a\), a semicircle is constructed on each side, using the side as the diameter. Find the area of the rosette enclosed by the arcs of the semicircles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^2 (\pi - 2)}{2}}
Let $\kappa$ be an infinite cardinal and let $X$ be a compact Hausdorff space of size $2^\kappa$. Let $U$ be a dense open subset of $X$. Determine a lower bound for the cardinality of $U$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^\kappa}
Given a right triangle \(ABC\) with a right angle at \(C\), a circle is drawn with diameter \(BC\) of length 26. A tangent \(AP\) from point \(A\) to this circle (distinct from \(AC\)) is drawn. The perpendicular \(PH\) dropped onto segment \(BC\) intersects segment \(AB\) at point \(Q\). Find the area of triangle \(BPQ\) given \(BH:CH = 4:9\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Suppose $f(x)$ and $g(x)$ are nonconstant smooth functions satisfying the equations $$\frac{f'(x)}{f(x)} + \frac{g'(x)}{g(x)} = 2$$ $$ \frac{f''(x)}{f(x)} - \frac{f'^2(x)}{f^2(x)} + \frac{g''(x)}{g(x)} - \frac{g'^2(x)}{g^2(x)} = 0$$ (assume $f$ and $g$ are always nonzero). Compute the value of $$ \frac{f''(2023)}{f'(2023)} + \frac{g''(2023)}{g'(2023)}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a regular triangular prism \( S-ABC \) with height \( S O = 3 \) and base side length of 6, a perpendicular is drawn from point \( A \) to the opposite face \( SBC \), with the foot of the perpendicular denoted as \( O' \). A point \( P \) is chosen on \( AO' \) such that \( \frac{AP}{PQ} = 8 \). Find the area of the cross-section passing through point \( P \) and parallel to the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Starting at $12:00:00$ AM on January $1, 2022$, after $13!$ seconds it will be $y$ years (including leap years) and $d$ days later, where $d < 365$. Find $y + d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{317}
Given the cubic equation \( x^3 - 3x + 1 = 0 \), let \( a, b, c \) be the roots such that \( a < b < c \). Find the value of the expression \( \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \) without solving the original equation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6}
a and b can do a work in 4 hours and 12 hours respectively. a starts the work at 6 am and they work alternately for one hour each. When will the work be completed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given that $X$ and $Y$ are independent and identically distributed standard normal random variables, calculate the covariance $Cov(XY|X+Y>0)$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{\pi}}
Given a group homomorphism \( \phi: \mathbb{Z}_{50} \rightarrow \mathbb{Z}_{15} \) with \( \phi(7) = 6 \), determine the expression for \( \phi(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3x \bmod 15}
A grocer starts with 400 pounds of coffee in stock, 20 percent of which is decaffeinated. They then buy another 100 pounds of coffee, of which 60 percent is decaffeinated, and 150 pounds of coffee, of which 45 percent is decaffeinated. Next, the grocer sells 50 pounds of decaffeinated coffee and buys 40 pounds of coffee that is 10 percent decaffeinated, followed by selling 70 pounds of caffeinated coffee. Afterward, the grocer purchases 90 pounds of coffee, of which the decaffeinated percentage fluctuates varying from 25 percent at the beginning to 50 percent at the end. Finally, the grocer sells an additional 30 pounds of caffeinated and 20 pounds of decaffeinated coffee, then buys another 80 pounds of coffee, of which 30 percent is decaffeinated. What percent, by weight, of the grocer's stock of coffee is decaffeinated after all these transactions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28.88\%}
Find the coefficient of the term \((a+d)^{11} b^4 c^2\) in the expansion of \((a - 2b + 3c + d)^{17}\) using the multinomial theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54747463680}
We have four registers, $R_1, R_2, R_3, R_4$, such that $R_i$ initially contains the number $i$ for $1 \leq i \leq 4$. We are allowed two operations:

1. Simultaneously swap the contents of $R_1$ and $R_3$ as well as $R_2$ and $R_4$.

2. Simultaneously transfer the contents of $R_2$ to $R_3$, the contents of $R_3$ to $R_4$, and the contents of $R_4$ to $R_2$. For example, if we do this once, then $(R_1, R_2, R_3, R_4) = (1, 4, 2, 3)$.

Using these two operations as many times as desired and in whatever order, what is the total number of possible outcomes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the leftmost three digits of the number\[1^1+2^2+3^3+...+999^{999}+1000^{1000}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
100}
Evaluate the following limit: $$\lim_{x\to 0} \frac{\sin(\sin(\sin(\sin x))) - x}{x^3}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{2}{3}}
Determine the number of solutions in natural numbers for the equation \((a+1)(b+1)(c+1)=2abc\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Given that $P$ is a point on the circle $x^2+y^2=4$ and is not on the coordinate axes, with $A(2,0)$, $B(0,2)$. The line $PA$ intersects the $y$-axis at point $M$, and the line $PB$ intersects the $x$-axis at point $N$. The minimum value of $|AN|+2|BM|$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the function $f(x)=2\sin (\omega x+\frac{π}{4})$ where $\omega\ \ \gt 0$, and the range of $f(x)$ on the interval $\left[-1,1\right]$ is $\left[m,n\right]$ with $n-m=3$, find the value of $\omega$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5π}{12}}
Let $\phi: \mathbb{Z} \rightarrow \mathbb{Z}_8$ be a homomorphism such that $\phi(1) = 6$. Determine all values of $n$ for which $\phi(n) = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\mathbb{Z}}
Let $X = Y = \mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers. Define functions $f: X \to Y$ and $g: Y \to X$ by $f(x) = 2x$ and $g(y) = 3y$. Given $D = f(X)$ and $A = Y \setminus D$, find an element $x \in X$ that is not in the set $W = \bigcup_{n=0}^\infty g(fg)^n(A)$. Find such an element $x$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine all real numbers $a$ such that the sequence \( \{S_n(a)\}_{n\geq 2} \) converges to a positive real number, where \( S_n(a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}
What is the area of a quadrilateral with vertices at $(0,0)$, $(4,3)$, $(7,0)$, and $(4,4)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.5}
Given that $\sqrt{x+1}$ is the simplest quadratic root and can be combined with $\sqrt{\frac{5}{2}}$, find the value of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Tom went to 4 hockey games this year, but missed some games. He went to 9 games last year. Tom went to 13 hockey games in all. How many hockey games did Tom miss this year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit: \[ \lim_{n\to\infty}\left(\frac{3}{2}\cdot\frac{5}{3}\cdot\dots\frac{2n+1}{n+1}\right)^\frac{1}{n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A 24-ton mixture of crushed rock is needed in a construction job; it will cost $800. The mixture is composed of rock costing $30 per ton and another type of rock costing a certain amount per ton. 8 tons of each should be added. How much does the more expensive rock cost per ton?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$35}
Given the function $f(x)$ and its derivative $f''(x)$ on $\mathbb{R}$, and for any real number $x$, it satisfies $f(x)+f(-x)=2x^{2}$, and for $x < 0$, $f''(x)+1 < 2x$, find the minimum value of the real number $a$ such that $f(a+1) \leqslant f(-a)+2a+1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac {1}{2}}
Evaluate the definite integral: \[ \int_0^{\frac{\pi}{2}}{\sqrt{\sin(2\theta)} \cdot \sin(\theta)\,d\theta} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Evaluate the integral: $$\int_1^3 \sqrt{x^4+1} \, dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.980}
In the arithmetic sequence $\{a_n\}$, $a_3 - a_2 = -2$, and $a_7 = -2$, calculate the value of $a_9$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6}
Evaluate the double integral \( \iint_{D} x \, dx \, dy \) where the region \( D \) is bounded by the lines \( y^{2} - 4y + x^{2} = 0 \), \( y^{2} - 8y + x^{2} = 0 \), \( y = \frac{x}{\sqrt{3}} \), and \( x = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Find an explicit value of $\varepsilon >0$ such that for every $x\in \left [ 0,1 \right ] $, there always is $$\left | \sqrt[]{x} -\sqrt[]{x+\varepsilon} \right | < \frac{1}{200}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/40000}
A player plays 9 games each day with his computer. In each game, he will win, draw, or lose with probabilities \(\frac{1}{2}\), \(\frac{1}{3}\), and \(\frac{1}{6}\) respectively. A day is considered “great” if he wins all 9 games. Approximate the probability that within 128 days, he experiences exactly three “great days.”
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.002}
Let \( d(n) \) denote the number of positive divisors of the positive integer \( n \). For instance, we have \( d(1) = 1 \), \( d(2) = 2 \), and \( d(12) = 6 \). We further define the function \( f(n) = \frac{d(n)}{\sqrt[3]{n}} \).

Given that there exists a unique positive integer \( N \) such that for any positive integer \( n \neq N \), the inequality \( f(N) > f(n) \) holds, find the sum of the digits of \( N \). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
9.1. (12 points) Two circles touch each other externally at point $K$. On their common internal tangent, point $P$ is marked such that $K P=14$. Through point $P$, two secants are drawn to the circles, with one of them intercepting a chord $A B=45$ on the first circle, and the other intercepting a chord $C D=21$ on the second circle, with point $A$ lying between points $B$ and $P$, and point $C$ lying between points $D$ and $P$. Find the ratio $B C: A D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.75}
设 $x_{k} \in[-2,2](k=1,2, \cdots, 2013)$ ，  
且 $x_{1}+x_{2}+\cdots+x_{2013}=0$. 试求  
$$M=x_{1}^{3}+x_{2}^{3}+\cdots+x_{2013}^{3}$$  
的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4026}
Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6, and the distance between their centers is 13. If a plane is made tangent to these two spheres and intersects with the cylinder to form an ellipse, then the sum of the lengths of the major and minor axes of this ellipse is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
7.5. 16 travelers, each of whom is either a liar or a knight (liars always lie, knights always tell the truth), settled into 3 rooms of a hotel. When everyone gathered in their rooms, Basil, who was staying in the first room, said: "There are more liars than knights in this room right now. Although no - there are more knights than liars in this room right now." After that, Basil went to the second room and said the same two statements there. And then he went to the third room and said the same two statements there as well. How many knights could there have been among these 16 travelers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the limit of the sequence: \[ \lim_{n \to +\infty} \frac{1}{\sqrt{n}}\sum_{k=1}^n \frac{k}{\sum_{j=1}^k j^{-1/2}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Approximate the value of \( \int_{0}^{0.5}{\frac{\sin(x)}{x}}dx \) using the Maclaurin series to within an error of 0.0001, where \(x\) is in radians.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.4931}
Let $n$ be a positive integer. There is a collection of cards that meets the following properties:
$\bullet$Each card has a number written in the form $m!$, where $m$ is a positive integer.
$\bullet$For every positive integer $t\le n!$, it is possible to choose one or more cards from the collection in such a way 
$\text{   }$that the sum of the numbers of those cards is $t$.
Determine, based on $n$, the smallest number of cards that this collection can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{n(n-1)}{2} + 1 }
Inside a circle, 16 radii of this circle and 10 circles with the same center as the circle are drawn. Into how many regions do the radii and circles divide the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{176}
记集合 $T=\{0,1,2,3,4,5,6\}, M=\left\{\left.\frac{a_{1}}{7}+\frac{a_{2}}{7^{2}}+\frac{a_{3}}{7^{3}}+\frac{a_{4}}{7^{4}} \right\rvert\, a_{i} \in T, i=1,2,3,4\right\}$ ，将 $M$ 中的元素按从大到小排列，求第 2005 个数。原始答案的形式为 \frac{b_1}{7}+\frac{b_2}{7^{2}}+\frac{b_3}{7^{3}}+\frac{b_4}{7^{4}}，请给出b_1 + b_2 + b_3 + b_4的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the surface area of a Lambertian hemisphere, given that the intensity of radiation is proportional to the absolute value of the cosine of the angle between the surface normal and the direction of radiation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi r^2}
Find the volume of the solid generated by revolving the region bounded by the parabola $y=2x^2$ and the line $y=6x-2$ about the line $y=12$ using the shell method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\pi\sqrt{5}}
Compute \( \mathbb{E}\left(\int_{0}^{T} |B_{t}|^{2} dt\right) \) where \((B_t)_{t\geq0}\) is a standard Brownian motion and \(T\) is the first time the Brownian motion hits \(1\) or \(-1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
We randomly select 5 distinct positive integers from the set \( \{1, 2, \ldots, 90\} \). Calculate \( \lfloor 10 \times E(X_4) \rfloor \), where \( E(X_4) \) is the expected value of the fourth largest number in the selection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{303}
Outside the temple, there is a shop which charges 11 dollars for each object. Please note that one shoe is counted as an object. Same is true for socks, mobiles, and umbrellas. Nathan went to the temple with both parents, his grandmother, and his younger sibling. All of them kept their shoes, socks, mobiles, and umbrellas in the shop. Nathan and his dad have a pair of gloves each which they also kept in the shop. How much do they have to pay?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{374}
The true discount on a bill of Rs. 2660 is Rs. 360. What is the banker's discount?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{416.35}
设 $E 、 F 、 G$ 分别是正四面体 $A B C D$ 的棱 $A B 、 B C 、 C D$ 的中点。求二面角 $C-F G-E$ 的大小，并用表达式 $\pi-\operatorname{arccot} \frac{\sqrt{m}}{n}$ 表示，其中$m$和$n$是整数，且$\frac{\sqrt{m}}{n}$是最简分数。请给出$m + n$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the result of the division $0111/1111$ in the finite field $GF(16)$, where $GF(16)$ is generated by the irreducible polynomial $\Pi(\alpha)=1+\alpha+\alpha^4$. Express $0111$ as $\alpha+\alpha^2+\alpha^3$ and $1111$ as $1+\alpha+\alpha^2+\alpha^3$. Compute the result of this division as an element of $GF(16)$. What is the result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1001}
Given that there are $24$ primes between $3$ and $100$, inclusive, determine the number of ordered pairs $(p, a)$ where $p$ is a prime number such that $3 \le p < 100$, and $1 \le a < p$. The condition is that the sum:
\[
a + a^2 + a^3 + \cdots + a^{(p-2)!}
\]
is not divisible by $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Evaluate the limit: \[ \lim_{n\to\infty} \int_0^\infty \frac{\sin(x/n)}{x+x^2} \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $0 < \theta < 2\pi$ be a real number for which $\cos (\theta) + \cos (2\theta) + \cos (3\theta) + ...+ \cos (2020\theta) = 0$ and $\theta =\frac{\pi}{n}$ for some positive integer $n$. Compute the sum of the possible values of $n \le  2020$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1926
}