inclusionAI/Ring-lite-rl-data huggingface dataset:
The product  $20! \cdot 21! \cdot 22! \cdot \cdot \cdot 28!$  can be expressed in the form  $m$   $\cdot$   $n^3$ , where m and n are positive integers, and m is not divisible by the cube of any prime. Find m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{825}
Three spheres touch each other externally in pairs, and they also touch a certain plane at the vertices of a right triangle with one leg of length 1 and the angle opposite the leg equal to \(30^\circ\). Find the radii of the spheres.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{\sqrt{3}}{4}}
In Figure 1, \( AEFD \) is a unit square. The ratio of the length of the rectangle \( ABCD \) to its width is equal to the ratio of the length of the rectangle \( BCF E \) to its width. If the length of \( AB \) is \( W \) units, find the value of \( W \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1 + \sqrt{5}}{2}}
Given that the sum of all positive integers with exactly two proper divisors, each of which is less than $30$, is $2397$, find the sum of all positive integers with exactly three proper divisors, each of which is less than $30$ (a proper divisor of $n$ is a positive integer that divides but is not equal to $n$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$7282$}
Find the mass percentage of C in Acetone
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62.04\%}
Determine the possible values of $n$ for which an automorphism $\sigma$ of $\mathbb{Q}(\omega)$ can map $\omega$ to $\omega^n$, where $\omega = e^{\frac{2\pi i}{10}}$. Provide your answer as a set of integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{1, 3, 7, 9\}}
How many simple graphs with 10 vertices and 15 edges contain a complete subgraph with 5 vertices?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81709614}
将与105互素的所有正数，从小到大排成数列，试求出这个数列的第1000项．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2186}
Given the reaction: CH4(g) + 2Cl2(g) ⇌ CH2Cl2(g) + 2HCl(g) with a known equilibrium constant Kc, predict the amount of HCl formed in moles at equilibrium when 3 moles of methane (CH4) and 6 moles of chlorine (Cl2) are combined in a 2L container at a certain temperature. Assume that the reaction has not yet reached equilibrium and only forward reaction is taking place initially.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $S = \{c_1, \dots, c_n\}$ be a set of vectors in $\mathbb{R}^M$. Determine the maximum cardinality of a subset $S' \subset S$ such that $\dim(\text{span}(S')) < \dim(\text{span}(S))$. What is the largest size of such a subset?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - 1}
Determine the degree of the field extension \(\mathbb{Q} \left( \zeta_{14} + \zeta_{14}^9 + \zeta_{14}^{11} \right)\) over \(\mathbb{Q}\), where \(\zeta_{14}\) is a 14th primitive root of unity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit \( \lim_{x \rightarrow 2k\pi}\frac{2(1-\cos x)+x\sin x}{2\sqrt{1-\cos x}} \) to find \( f'(2k\pi) \) for the function \( f(x)=x \sqrt{1-\cos(x)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the crystallographic restriction for a discrete group of isometries whose translation group is generated by a single non-zero vector \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the highest power of 50 that divides the expression \(49^{{50}^{51}} + 51^{{50}^{49}} + 50^{{49}^{51}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the winding number of the path \( \gamma:[0,2\pi] \rightarrow \mathbb{C}, \ t \mapsto a\cos(2t)+ib\sin(2t) \) around the origin, where \( a, b \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the number of functions $f(x)$ from $\{1,2,3,4,5\}$ to $\{1,2,3,4,5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1,2,3,4,5\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
756
}
Calculate the limit: \[ \lim_{n\to\infty} \frac{\gamma(n,n)}{\Gamma(n)} \] where \( \gamma(s,x) = \int_0^x t^{s-1} e^{-t} \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Each integer in \( \{ 1,2,3,\ldots ,{2020}\} \) is coloured in such a way that,for all positive integers \( a \) and \( b \)
 such that \( a + b \leq  {2020} \) ,the numbers \( a,b \) and \( a + b \) are not coloured with three different colours
 Determine the maximum number of colours that can be used.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
11
}
已知 $x_{1}, x_{2}, x_{3}, x_{4}$ 是两两不等的实数， $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 是 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意一个排列，若记 $X=\max \left\{\min \left\{x_{i 1}, x_{i 2}\right\}, \min \left(x_{i 3}, x_{i 4}\right\}\right\}, Y=$ $\min \left\{\max \left\{x_{i 1}, x_{i 2}\right\}, \max \left\{x_{i 3}, x_{i 4}\right\}\right\}$ ，则 $X>Y$ 的概率是
$\qquad$.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{4}
A 5-dimensional ant starts at one vertex of a 5-dimensional hypercube of side length 1. A move is when the ant travels from one vertex to another vertex at a distance of $\sqrt{2}$ away. How many ways can the ant make 5 moves and end up on the same vertex it started at?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6240}
On the island of Misfortune with a population of 96 people, the government decided to implement five reforms. Each reform is disliked by exactly half of the citizens. A citizen protests if they are dissatisfied with more than half of all the reforms. What is the maximum number of people the government can expect at the protest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Compute the turning number of the closed plane curve defined by \( \gamma(t) = \left(\left(\frac{1}{2}+3\cos(t)\right)\cos(t), \left(\frac{1}{2}+3\cos(t)\right)\sin(t)\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the asymptotic bound for the expression \( \sum_{k=0}^{\frac{n}{2^{\log(n)}}} (k+1)^{3} + \sum_{i=0}^{\frac{n}{2}} (i+1)^{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n^3)}
Determine the Morley rank of the type $x = x$ in the theory of Algebraically Closed Fields of characteristic zero (ACF$_{0}$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the surface flux of the vector field \( \textbf{F} = x^2\textbf{i} + y\textbf{j} - xz \textbf{k} \) across the surface \( S \) defined by \( 4y=x^2 \), bounded by the planes \( x=2 \) and \( z=4 \), where \( \hat{\textbf{n}} \) points in the direction of increasing \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{16}{3}}
Frank alternately flips two coins. The first coin has a probability of \( \frac{2}{3} \) for heads and \( \frac{1}{3} \) for tails. The second coin has a probability of \( \frac{1}{4} \) for heads and \( \frac{3}{4} \) for tails. Starting with the first coin, what is the probability that Frank sees two consecutive heads before two consecutive tails?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{33}}
Evaluate the limit: $$\lim_{n \to \infty } \left( \frac{1}{\sqrt{n^2+n+1}}+\cdots+\frac{1}{\sqrt{n^2+2n}} \right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the range of values for $p$ such that the interval $(3,4)$ is contained between the roots of the equation \(3^x + 3^{4-x} + p = 0.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -82)}
In the sum shown, different shapes represent different digits. What digit does the square represent? Express your answer as a single digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
在圆周上任意写上 49 个 1 及 50 个 0 , 然后进行下列运算: 在两个相同的数之间写上 0 , 在两个不同数之间写上 1 , 并擦掉原有的数字, 接着再进行同样的运算, 如此继续，问能否经过有限步这种运算使得圆周上的数字都变成 0 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{不论运算多少次,都不能使所有的数字都变成 0.}
In a convex quadrilateral, three of its vertices form a triangle such that the circumcenter of this triangle is located outside the triangle. Given this information, determine the value of angle $x$ in the quadrilateral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Evaluate the limit \( \lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n} \), where for every positive integer \( n > 1 \), \( k(n) \) is the largest positive integer \( k \) such that there exists a positive integer \( m \) with \( n = m^k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the value of  $ { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=odd} $   $ - { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=even} $
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Consider an event with probability \( p \). What is the probability that this event never occurs over an infinite number of trials?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the flux of the vector field \( F = \langle z, y, zx \rangle \) through the surface of the tetrahedron enclosed by the coordinate planes and the plane \( \frac{x}{4}+\frac{y}{2}+\frac{z}{5}=1 \). Use the divergence theorem to find the correct flux value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{40}{3}}
Calculate the Vassiliev invariant for the knot K_1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find a linear operator on \( \mathbb{C} \) (considered as a vector space over \( \mathbb{R} \)) such that the range of the operator is a proper subset of its kernel.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of real solutions to the equation \( f(x) = 0 \), where \( f(x) = \sum_{i=1}^{2020} \frac{i^2}{x-i} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}
In the given figure, there are two trapezoids $ABDE$ and $ACDE$. If the area of the pentagon $ABCDE$ is 2621 and the area of the triangle $ADF$ is 300, what is the area of the triangle $EBC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2021}
Problem 6. (30 points) Ten balls of the same radius are arranged in the form of a triangular pyramid such that each ball touches at least three others. Find the radius of the sphere in which the pyramid of balls is inscribed, if the radius of the sphere inscribed in the center of the pyramid of balls, touching six identical balls, is $\sqrt{6}-1$.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5(\sqrt{2}+1)}
In three out of the six circles of a diagram, the numbers 4, 14, and 6 are written. In how many ways can natural numbers be placed in the remaining three circles such that the product of the triplets along each of the three sides of the triangular diagram are the same?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the number of distinct Celtic knots of size (2, 2), considering knots that differ only by rotations and reflections as identical.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the dimension of the subspace of quadric hypersurfaces in \(\mathbb{P}^4\) that vanish at eight points in general position on a rational normal curve of degree 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A particle moves in the Cartesian plane according to the following rules:
From any lattice point $(a,b),$ the particle may only move to $(a+1,b), (a,b+1),$ or $(a+1,b+1).$
There are no right angle turns in the particle's path.
 How many different paths can the particle take from $(0,0)$ to $(5,5)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83}
Given triangle  $ABC$ , let  $D$ ,  $E$ ,  $F$  be the midpoints of  $BC$ ,  $AC$ ,  $AB$  respectively and let  $G$  be the centroid of the triangle. For each value of  $\angle BAC$ , how many non-similar triangles are there in which  $AEGF$  is a cyclic quadrilateral?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral \( I = \int_{C(0,2)^+} \frac{z^3}{z^3+z^2-z-1} \, dz \) using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi i}
Find the length of the latus rectum for the hyperbola given by the equation \[\frac{(3x-4y-1)^2}{100}-\frac{(4x+3y-1)^2}{225}=1.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Dean, Sara, and Nick went for a swim together. The depth of the water without waves is 10 times each person's height. Dean stands at 6 feet tall, Sara at 5 feet tall, and Nick at 5.5 feet tall. They stand at distances of 5, 10, and 15 feet from the shore, respectively. The waves increase the water depth by an additional 25% at its peak. Calculate the average water depth each swimmer experienced during their swim at their respective distances from shore.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61.875}
Find the greatest common divisor (GCD) of all numbers of the form \( p^8 - 9p^4 + 8 \), where \( p \) is a prime number greater than or equal to 13.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
How many positive integers $n$ are there such that $n!(2n+1)$ and $221$ are relatively prime?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10
}
Evaluate the limit: $$\lim_{x\rightarrow\infty}\Large{\Delta^x_{k=0}}\normalsize{\cos \frac{k\pi}{3x}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of terms needed in the series \(\sum_{k=0}^{\infty}{\frac{(3i)^k}{k!}}\) to ensure the error is less than \(10^{-4}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Find the minimum value of the function \( f(x,y) = 2x + xy + 3y \) subject to the constraints:
\[ \begin{cases} x^2 + y \geq 3 \\ x + 0.5 \geq 0 \\ y \geq 0 \end{cases} \]
Using the Kuhn-Tucker conditions, determine the point \((x, y)\) that minimizes \(f(x, y)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\sqrt{3}, 0)}
A guidance system is designed to deliver a bomb to a target site at coordinates \((3,2)\). The bomb's landing point \((X,Y)\) follows the joint probability density function:
\[ f_{X,Y}(x,y|3,2)=\frac{1}{2\pi}\exp\left\{-\frac{(x-3)^2}{2}-\frac{(y-2)^2}{2}\right\} \]
where \(X\) and \(Y\) are measured in feet. The target will be destroyed if a bomb lands within \(r=2\) feet of it. Calculate the probability that the target will be destroyed if 10 bombs are fired at it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - e^{-20}}
Evaluate the integral \( \int_0^4 r(\sqrt{25-r^2}-3) \, dr \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{26}{3}}
During a fireworks display, a body is launched upwards with an initial velocity of $c=90 \mathrm{m/s}$. We hear its explosion $t=5$ seconds later. At what height did it explode if the speed of sound is $a=340 \mathrm{m/s}$? (Air resistance is neglected.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{289}
Determine the largest possible constant $d$ such that the inequality \[ \frac{2^{2d+3k}\pi\sqrt{k}e\sqrt{3}}{3^{3k}} \geq 2^{dk} \] holds for all integers $k \geq 1$. Provide your answer as a single value of $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 - 3 \log_2 3}
A cube with edge length 3 is divided into 27 unit cubes. The numbers 1, 2, ..., 27 are distributed arbitrarily over the unit cubes, with one number in each cube. Determine the maximum number of odd row sums that can be formed from the 27 possible row sums, where there are nine such sums of three integers for each of the three directions parallel to the edges of the cube.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Evaluate the limit \( \lim_{x\to 0^+}\frac{\int_{x}^{1}\frac{\sin t}{t}\,\mathrm{d}t}{x} \) using L'Hopital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let $f(x)=30-2x-x^3$. Determine the number of positive integer values of $x$ for which $f(f(f(x))) > f(f(-x))$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $n$ be a positive integer, $k \in \mathbb{C}$, and $A \in \mathcal{M}_n(\mathbb{C})$ such that $\text{Tr } A \neq 0$ and \[ \text{rank } A + \text{rank } ((\text{Tr } A) \cdot I_n - kA) = n. \] Find $\text{rank } A$. \( \text{Tr } A \) denotes the trace of matrix $A$, and $I_n$ is the $n \times n$ identity matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Two players play alternately on a  $ 5 \times 5$  board. The first player always enters a  $ 1$  into an empty square and the second player always enters a  $ 0$  into an empty square. When the board is full, the sum of the numbers in each of the nine  $ 3 \times 3$  squares is calculated and the first player's score is the largest such sum. What is the largest score the first player can make, regardless of the responses of the second player?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Compute 
$\sum\limits_{i=0}^\infty(-1)^i\sum\limits_{j=i}^\infty(-1)^j\frac{2}{j^2+4j+3}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Nine lines, parallel to the base of a triangle, divide the other sides into 10 equal segments and the area into 10 distinct parts. Find the area of the original triangle, if the area of the largest of these parts is 76.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
400
}
Calculate the limit as \( n \) approaches infinity for the expression:
\[ \lim_{n\rightarrow+\infty} np^n \sum_{k=n}^{+\infty} \frac{1}{k} \binom{k-1}{n-1} (1-p)^{k-n} \]
where \( 0<p<1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p}
Given the parametric equations for real numbers $t,$
\[(x,y) = \left( \frac{1 - t^2}{1 + t^2}, \frac{2t}{1 + t^2} \right),\]identify the type of curve on which all the plotted points lie.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Circle}
Evaluate the sum \( \sum_{k=1}^n k\cdot(-3)^k{n \choose k} \) for \( n > 0 \) and express it as a function of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3n(-2)^{n-1}}
What is the probability that you never lose in the following dice game? You start with a four-sided die numbered 1-4. In each round, you roll all your dice. If a die shows 1, you lose that die. If it shows 2, you keep the die. If it shows 3 or 4, you gain an additional identical die. You lose the game if you have no dice left. Calculate the probability that you never lose the game over infinite rounds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Professor Oak is feeding his 100 Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is 100 kilograms. Professor Oak distributes 100 kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of their bowl). The dissatisfaction level of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $|N-C|$.
Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute the food in a way that the sum of the dissatisfaction levels over all the 100 Pokémon is at most $D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
What is the general expression, or at least tight upper and lower bounds, for $f(n,k)$, which represents the number of ways to have at least $k$ consecutive heads in $n$ coin tosses? For instance, when $n=5$ and $k=2$, how many possibilities exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Let \( k \) be a positive integer and \( w \) be an integer such that two of the roots of the cubic equation \( x^3 - 17x^2 + kx + w = 0 \) are consecutive positive integers. Find the value of \( k \) that maximizes the absolute value of \( k - w \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Find the value of $f(\sqrt{3})$ for a continuous function $f(x)$ defined for all real numbers $x$ such that the equation \(x^2 + x\{f(x)\} - 3 = \sqrt{3} \, f(x)\) holds, where \(\{t\}\) denotes the fractional part of \(t\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the parabola \( y^2 = 4x \), a line passing through its focus \( F \) intersects the parabola at points \( M \) and \( N \). Let \( E(m,0) \) be a point on the x-axis. The extensions of \( ME \) and \( NE \) intersect the parabola again at points \( P \) and \( Q \), respectively. If the slopes \( k_1 \) and \( k_2 \) of the lines \( MN \) and \( PQ \) satisfy \( k_1 = 3k_2 \), find the value of \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the limit of \( \frac{x^x - (\sin x)^x}{x^3} \) as \( x \to 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5, x+z=\cos^2y\}$ be a submanifold of $\mathbb{R}^3$. Consider the point $p=(5,0,-4)$ and the tangent vector $v=(-C,C,C) \in T_{(5,0,-4)}M$. Define the smooth map $F:M \rightarrow S^1$ by $F(x,y,z) = \left(\frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x^2+y^2}}\right)$. Let $\omega = -\frac{1}{x}dy$ be a 1-form on $S^1$ in a neighborhood of $(1,0)$. Compute $(F^{*}\omega)_p(v)$. Express your answer in terms of $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{C}{5}}
Find the limit: \[ \lim_{ x \to a }\left(\frac{f(x)}{g(x)}-\frac{h(x)}{k(x)}\right) \] where \( f(x), g(x), h(x), k(x) \) are polynomials, and it is given that \( \lim_{ x \to 1 }\left(\frac{1}{x-1}-\frac{3x-1}{x^2-1}\right) = -1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given a sequence of independent and identically distributed (i.i.d.) random variables \(X_i\), each with mean \(M\) and variance \(\sigma^2\), let \(Y(z)\) denote the smallest integer such that the sum of the first \(Y(z)\) variables exceeds \(z\). As \(z\) tends to infinity, \(Y(z)\) is approximately \(z/M\). Determine the limit:
\[\lim_{{z \to \infty}} \frac{V(z) M^3}{\sigma^2 z}\]
where \(V(z)\) is the variance of \(Y(z)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}