inclusionAI/Ring-lite-rl-data huggingface dataset:
A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $12^{{th}}$ pirate receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1925}
Calculate the area of the common part of two rhombuses, where the lengths of the diagonals of the first one are 4 cm and 6 cm, and the second one is obtained by rotating the first one by 90 degrees around its center.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9.6 \ \text{cm}^2}
Each of the 2009 distinct points in the plane is colored either blue or red. On every unit circle centered at a blue point, there are exactly two red points. Find the greatest possible number of blue points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1964}
The tax on a commodity is diminished by 22 % but its consumption is increased by 9 %. What is the decrease percent in the revenue derived from it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.98\%}
Determine the number of non-empty strings of length less than or equal to \( N \) that can be formed using \( k \) different characters, such that no string is a prefix of another. Consider the constraints \( 1 \le N \le 10^9 \) and \( 1 \le k \le 50 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k^N}
Evaluate the integral: \[ \int_{0}^{1}\frac{x^4+1}{x^6+1} \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
Let $f$ be a function on nonnegative integers such that $f(0)=0$ and $$f(3n+2)=f(3n+1)=f(3n)+1=3f(n)+1$$ for all integers $n$ $\ge$ $0.$ Compute the sum of all nonnegative integers $m$ such that $f(m)=13.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
156
}
Determine the closure of the subspace \(D=\{f\in C([0,1]):f(1)=0\}\) in the space \(C([0,1])\) with the norm \[||f||=\left(\int_0^1 |f(x)|^2\right)^{1/2}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{C([0,1])}
Evaluate the double integral \( \iint\limits_D (x^2+y^2) \,dA \) where \( D \) is the region enclosed by the curves \( xy=1 \), \( xy=2 \), \( x^2-y^2 =1 \), and \( x^2-y^2 =2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the contour integral \(\oint_C (1 + z + z^2)(e^\frac{1}{z}+e^\frac{1}{z-1}+e^\frac{1}{z-2}) \, dz\), where \(C\) is a circle of radius 3 centered at the origin. Use the residue theorem to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32\pi i}
Evaluate the integral \( \int_{0}^{3} (x^2+1) \, d[x] \), where \([x]\) is the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Evaluate the integral \( \int_0^1 \left((1-x^{2023})^{\frac{1}{2022}} - (1-x^{2022})^{\frac{1}{2023}}\right)\;dx \) using special functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider the ellipse given by the equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) where \(a > b > 0\). The ellipse intersects the positive direction of the \(y\)-axis at point \(B\). Determine the number of right-angled isosceles triangles that can be inscribed in the ellipse with one vertex at point \(B\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the splitting field of the polynomial \( f(x) = x^3 - x + 1 \) over the field \( \mathbb{Z}_5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{F}_{25}}
Given that \( \int_{0}^{\infty} \left(\frac{\sin x}{x}\right)^3 \, dx = A \), find the value of \( k \) such that \( \int_{0}^{\infty} \frac{x - \sin x}{x^3} \, dx = kA \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Determine the growth rate of the longest worm-like path in a random configuration on the lattice $[1,3n]^2$ in $\mathbb{Z}^2$, where each lattice point with coordinates $\equiv 2 \pmod{3}$ is randomly assigned one of six patterns. Express the growth rate with respect to $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n)}
Determine the minimum growth rate of a sequence $w_n$ such that $P(\overline{X}_n > w_n) \to 1$ as $n \to \infty$, where $X_i$ are independent and identically distributed random variables with $X_1 \geq 0$ and $E[X_1] = \infty$. Provide your answer in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log n}
Let three non-identical complex numbers \( z_1, z_2, z_3 \) satisfy the equation \( 4z_1^2 + 5z_2^2 + 5z_3^2 = 4z_1z_2 + 6z_2z_3 + 4z_3z_1 \). Denote the lengths of the sides of the triangle in the complex plane, with vertices at \( z_1, z_2, z_3 \), from smallest to largest as \( a, b, c \). Find the ratio \( a : b : c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:\sqrt{5}:\sqrt{5}}
How many different lines pass through at least two points in this 4-by-4 grid of lattice points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given N trucks, each assigned a number from 1 to N, assume the prior distribution of N is proportional to \( \frac{1}{x} \) for \( x = 1, \ldots, 500 \). Find the posterior mean of N when observing a truck numbered 50, assuming the likelihood is \( \frac{1}{N} \) for \( N \geq 50 \) and 0 otherwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{127.1}
In triangle $\triangle ABC$, $AB = AC$ and $\angle A > 90^\circ$. Point $D$ lies on segment $AB$ such that $AD = BC$ and $\angle ADC = \frac{3}{4} \cdot \angle ABC$. Determine the measure of $\angle A$. \( \boxed{\text{Answer}} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100^\circ}
Identify the topological space that is homeomorphic to the partition of \(\mathbb{R}^2 - \{0\}\) into lines of the form \(y = mx\) for all real \(m\), including vertical lines.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S^1}
Determine the number of monic primitive quadratic polynomials in \( \mathbb{Z}_7[x] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let  $M$  be a finite subset of the plane such that for any two different points  $A,B\in M$  there is a point  $C\in M$  such that  $ABC$  is equilateral. What is the maximal number of points in  $M?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Carol spends eight hours filling up her pool. During the first hour, the pool fills at a rate of 8 gallons of water per hour. For the next two hours, the pool fills at 10 gallons of water per hour. From the fourth hour to the fifth hour, the pool fills at a rate of 14 gallons of water per hour. During the sixth hour, the pool starts filling at a rate of 12 gallons of water per hour. However, during the seventh hour, the pool develops a leak and loses 8 gallons of water. In the eighth hour, the leak slows down and the pool loses another 5 gallons of water while continuing to fill at 12 gallons of water per hour. At the end of the eight hours, how many gallons of water are still left in the pool?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79}
Calculate the conditional expectation \( E[N(6) \mid N(4) = 4, N(10) = 12] \) for a Poisson process \( \{N(t), t \ge 0\} \) with rate \( \lambda = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{20}{3}}
Find the smallest prime number \( p \) such that 7 is the least quadratic nonresidue modulo \( p \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
Find the joint density $g(u, v)$ for the transformed random variables $U = \exp(X) + 2\exp(Y)$ and $V = 2\exp(X^2) + \exp(Y^2)$, where $X$ and $Y$ are independent Exponential random variables with mean 1. Evaluate $g(3e, 3e)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{6e^4}\)}
Find the smallest \( \lambda  \in  \mathbb{R} \) such that for all \( n \in  {\mathbb{N}}_{ + } \) ,there exists \( {x}_{1},{x}_{2},\ldots ,{x}_{n} \) satisfying \( n = {x}_{1}{x}_{2}\ldots {x}_{2023} \) ,where \( {x}_{i} \) is
 either a prime or a positive integer not exceeding \( {n}^{\lambda } \) for all \( i \in  \{ 1,2,\ldots ,{2023}\} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\(\frac{1}{1012}\).}
A hexagon is inscribed in a circle of radius \( r \). Four of the edges of the hexagon are 10 feet long, and two of the edges are 20 feet long, but the exact arrangement of the edges is unknown. Using either Ptolemy's Theorem or the Law of Cosines, determine the value of \( r \) to three decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13.660}
On a standard die, one of the dots is removed at random, with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots? Express your answer as a fraction in simplest form, \frac{k}{m}, and provide the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
In Philosopher's Chess, the chessboard is an infinite grid of identical regular hexagons. A piece called the Donkey moves as follows: starting from one hexagon, it moves $m$ cells in one of the 6 directions, then turns 60 degrees clockwise and moves $n$ cells in this new direction to reach its final cell. Given $m$ and $n$ are nonnegative integers, determine the maximum number of cells on the chessboard such that one cannot reach any other cell from them using a finite number of Donkey movements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m^2 + mn + n^2}
How many distinct roots does the equation x^{5}+3 x^{4}-4044118 x^{3}-12132362 x^{2}-12132363 x-2011^{2}=0 have, given that the roots of the equation are all integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the integral of \( f(z) = \frac{z^2}{z^3 - z - 1} \) over a circle of radius \( R \) centered at the origin, where \( R \) is greater than the maximum modulus of the roots of \( z^3 - z - 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
A bag of potatoes weighs some amount divided by half of its weight. The bag of potatoes weighs 12 lbs. What is the original weight of the bag of potatoes before dividing by half of its weight?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
In an $n \times n$ idempotent matrix, what is the minimum number of entries that need to be known to determine the rest of the matrix?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 1}
Find the norm of the operator $T: l^q \rightarrow l^q$ defined by $T((b_{n})) = b_{n+1} - 2b_{n} + b_{n-1}$, where $l^q$ is the space of sequences with the $q$-norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the expected value of the integral of a pure birth process from time 0 to the first time it reaches state 4, given that the process starts at state 1 and has birth rates \( \lambda_k = k \) for \( k=1,2, \dots \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The operation $\oslash$, called \u0022reciprocal sum, is useful in many areas of physics. If we say that $x=a\oslash b$, this means that $x$ is the solution to
$$\frac{1}{x}=\frac{1}{a}+\frac{1}{b}$$
Compute$4\oslash2\oslash4\oslash3\oslash4\oslash4\oslash2\oslash3\oslash2\oslash4\oslash4\oslash3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\tfrac{1}{4}}
Determine the maximum number of solutions that the equation \(\max \{a_{1} x+b_{1}, \ldots, a_{10} x+b_{10}\} = 0\) can have, given that \(a_{1}, \ldots, a_{10}, b_{1}, \ldots, b_{10}\) are real numbers and all \(a_{i} \neq 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the residue field for the local ring $\mathbb{Z}_{\langle 5\rangle}$ with maximal ideal $5\mathbb{Z}_{\langle 5\rangle}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{F}_5}
A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given the expression $\frac{\cos \alpha \cos \frac{\beta}{2}}{\cos \left(a-\frac{\beta}{2}\right)} + \frac{\cos \beta \cos \frac{\alpha}{2}}{\cos \left(\beta-\frac{\alpha}{2}\right)} - 1$, find the value of $\cos \alpha + \cos \beta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Borya and Vasha play a game with a positive integer \( n \geq 2 \). Borya selects \( 2n \) non-negative numbers \( x_1, x_2, \ldots, x_{2n} \) such that their sum is 1. Vasha arranges these numbers in a circle and calculates the product of each pair of adjacent numbers, writing the largest product on the blackboard. What is the number written on the blackboard, assuming both players use optimal strategies?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4n^2}}
设数列$\left\{ {{a}_{n}} \right\}$的前$\textit{n}$项之积为${{An}}$，且${{A}_{n}}{{An+1}}+n{{A}_{n}}=\left( n+1 \right){{An+1}}$，${{a}_{1}}=\frac{1}{2}$，${{bn}}={{a}_{n}}+\frac{1}{{{an}}}$，数列$\left\{ {{b}_{n}} \right\}$的前$\textit{n}$项和为${{Sn}}$，则${{S}_{n}}+\frac{1}{{{An}}}=$______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$3n+2-\frac{1}{n+1}$}
Evaluate the limit: $$\lim_{y \to 0} \frac{3y}{5-\left(\displaystyle\sum_{n=0}^{\infty}\frac{(2y)^n+4y^n}{n!}\right)}$$ without using L'Hospital's rule or Taylor series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Determine the approximate value of $T(n)$ for large values of $n$ given the recursive equation $T_n = \sqrt{n}T_{\sqrt{n}} + 1$, excluding $n=1$. Provide your answer in terms of asymptotic notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n)}
Let $p=2017$ be a prime. Find the remainder when \[ \left\lfloor \frac{1^p}{p} \right\rfloor + \left\lfloor \frac{2^p}{p} \right\rfloor + \left\lfloor \frac{3^p}{p} \right\rfloor + \cdots + \left\lfloor \frac{2015^p}{p} \right\rfloor \] is divided by $p$. Here $\lfloor \cdot \rfloor$ denotes the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1010}
Evaluate the limit: \[ \lim_{x\rightarrow0}\frac{1-\left(\cos x\right)^{\ln\left(x+1\right)}}{x^{4}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
A tailor trims 4 feet from opposite edges of a square piece of cloth, and 3 feet from the other two edges. If 120 square feet of cloth remain, what was the length of a side of the original piece of cloth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given the function f(x) = e^x cos x and x in (0, π), find the interval where f(x) is increasing. Express your answer as an interval in standard notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, π/4)}
A quadrilateral connecting the midpoints of the sides of trapezoid $\mathrm{ABCD}$ is a rhombus. Find its area if the height of the trapezoid $\mathrm{BH}=5 \mathrm{c}$, the smaller base $\mathrm{BC}=6 \mathrm{~cm}$, and the angle $\mathrm{ABC}$ is $120^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15 \mathrm{~cm}^2}
The polynomial \( f(x) = x^{2007} + 17 x^{2006} + 1 \) has distinct zeroes \( r_1, \ldots, r_{2007} \). A polynomial \( P \) of degree 2007 has the property that \( P\left( r_j + \frac{1}{r_j} \right) = 0 \) for \( j = 1, \ldots, 2007 \). Determine the value of \( P(1) / P(-1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{289}{259}}
47. In how many ways can $n$ traffic lights be lit, of which $k$ can be in one of three states, and the remaining $n-k$ in one of two?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["3^{k}\\cdot2^{n-k}"]}
Three players are playing table tennis. The player who loses a game gives up their spot to the player who did not participate in that game. In the end, it turns out that the first player played 10 games, and the second player played 21 games. How many games did the third player play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A triangle is composed of circular cells arranged in $5784$ rows: the first row has one cell, the second has two cells, and so on. The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called **diagonal** if the two cells in it aren't in the same row. What is the minimum possible amount of diagonal pairs in the division?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2892}
Let $A$ and $B$ be two sets of non-negative integers, define $A+B$ as the set of the values obtained when we sum any (one) element of the set $A$ with any (one) element of the set $B$. For instance, if $A=\{2,3\}$ and $B=\{0,1,2,5\}$ so $A+B=\{2,3,4,5,7,8\}$.
Determine the least integer $k$ such that there is a pair of sets $A$ and $B$ of non-negative integers with $k$ and $2k$ elements, respectively, and
$A+B=\{0,1,2,\dots, 2019,2020\}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{\sin \left\{ \frac{2}{n} \right\}}{\left[ 2n \tan \frac{1}{n} \right] \left( \tan \frac{1}{n} \right)} + \frac{1}{n^2 + \cos n} \right)^{n^2}, \] where \([.]\) denotes the greatest integer function and \(\{.\}\) denotes the fractional part function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Define a *small* prime to be a prime under $1$ billion. Find the sum of all *small* primes of the form $20^n + 1$, given that the answer is greater than $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160402}
Find the number of 1's in the binary representation of the 1717th term in the sequence of positive integers whose binary representation is a palindrome.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In a regular octagon, diagonals connect the second and third neighboring vertices to form two convex octagons. Determine the ratio of the areas of these two octagons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + \sqrt{2}}
An ice ballerina rotates at a constant angular velocity at one particular point. That is, she does not translationally move. Her arms are fully extended as she rotates. Her moment of inertia is  $I$ . Now, she pulls her arms in and her moment of inertia is now  $\frac{7}{10}I$ . What is the ratio of the new kinetic energy (arms in) to the initial kinetic energy (arms out)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\dfrac{10}{7}$}
Find the exhaustive range of $k$ such that the function $f(x)=\frac{x-1}{k-x^2}$ never takes values within the interval $[-1, -\frac{1}{3}]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, \frac{3}{4})}
King Middle School has1200students. Each student takes5classes a day. Each teacher teaches4classes. Each class has3(
 students and 1 teacher. How many teachers are there at King Middle School?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
50
}
Evaluate the infinite series: \[ S = \sum_{n=1}^{\infty} \left( \frac{2}{n} - \frac{4}{2n+1} \right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 - 4\ln 2}
In the positive geometric sequence {a_n}, if a_1 + a_2 + a_3 = 18 and 1/a_1 + 1/a_2 + 1/a_3 = 2, find the value of a_2. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the set \( \bigcup_{s \in \mathbb{R}} \bigcap_{t \geq s} A_t \), where \( A_t = (\cos t - 2, \cos t + 2) \) for \( t \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
Consider a vector $\mathbf{X} \in \mathbb{R}^n$ where each component is subGaussian with parameter $s$. Determine if the vector $\mathbf{X}$ is subGaussian, and if so, find its subGaussian parameter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s\sqrt{n}}
A list of five positive integers has all of the following properties:

$\bullet$ The only integer in the list that occurs more than once is $7,$ and it occurs three times.
$\bullet$ Its median is $10,$
$\bullet$ Its average (mean) is $11.$

What is the largest possible integer that could appear in the list?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Determine the number of ways to rearrange the letters in the word INDIVISIBILITY such that no two 'I's are adjacent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3386880}
Find the minimal extension field of \(\mathbb{F}_2\) that contains an element of order 21.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{F}_{2^6}}
Given that \( \lim_{x\rightarrow \infty} f(x)=0 \), where \( f'(x)=\dfrac{-f(x)}{\sqrt{f^2(x)+g^2(x)}} \) and \( g'(x)=1-\dfrac{g(x)}{\sqrt{f^2(x)+g^2(x)}} \) for all \( x \in \mathbb{R} \), with initial conditions \( g(0)=0 \) and \( f(0)=10 \), find the value of \( \lim_{x\rightarrow \infty} g(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let \( C \) be the smooth projective curve in \( \Bbb{P}^2 \) defined by the equation \( y^2z = x^3 - xz^2 \). Consider the rational map \( \phi: \Bbb{P}^2 \dashrightarrow \Bbb{P}^1 \) given by \( (x:y:z) \mapsto (x:y) \), which is undefined at the point \( P = (0:0:1) \). The restriction \( \psi = \phi|_C \) extends to a morphism \( \psi: C \to \Bbb{P}^1 \). Determine the value of \( \psi(P) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0:1)}
Let \( g: \mathbb{R}^m \rightarrow \mathbb{R} \) be a differentiable function. Define \( v: \mathbb{R}^n \rightarrow \mathbb{R} \) by \( v(x) = g(s) \), where \( s = \sqrt{\sum_{i=1}^m x_i^2} \), for all \( x \in \mathbb{R}^n \). Calculate \( \|\nabla v\|^2 \) in terms of \( g \) and \( s \) only.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(g'(s))^2}
Evaluate the limit: \[ \lim_{n \to \infty} \int_{1}^{\infty} n e^{-x^{\delta}}(1 - e^{-x^{\delta}})^n \, dx \] where \(0< \delta <1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}