inclusionAI/Ring-lite-rl-data huggingface dataset:
A particle starts moving, and from the second mile onwards, its speed is inversely proportional to the distance traveled in miles. For each subsequent mile, the speed is constant. If it takes 2 hours to travel the second mile, then what is the time (in hours) required to travel the nth mile?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(n-1)}
Express the function $f$ in the form $f(e^t, t^2) = c$, where $c$ is a constant, given that the directional derivative of $f$ in the direction of $(e^t, t^2)$ is zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(e^t, t^2) = c}
Two white socks cost 25 cents more than a single brown sock. If two white socks cost 45 cents, how much would you pay for 15 brown socks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$3}
Determine the values of \( \alpha \) for which the matrix \( A = \begin{pmatrix} 1 & 0 & 0 \\ \alpha-3 & 6-2\alpha & 6-3\alpha \\ 2-\alpha & 2\alpha -4 & 3\alpha-4 \\ \end{pmatrix} \) is diagonalizable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha \neq 1}
The map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a north/south road from Centerville to Drake. The distances from Acorn to Centerville, from Centerville to Midland, and from Centerville to Drake are each 60 kilometers. At noon, Aaron starts at Acorn and bicycles east at 17 kilometers per hour, Michael starts at Midland and bicycles west at 7 kilometers per hour, and David starts at Drake and bicycles at a constant rate in a straight line across an open field. All three bicyclists arrive at exactly the same time at a point along the road from Centerville to Midland. Find the number of kilometers that David bicycles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
In triangle $ABC$, the area of the triangle is $70$. Points $E$ and $D$ are on side $AC$ such that $BE = ED = DA$, and points $F$ and $G$ are on side $AB$ such that $BF = FG = GC$. Quadrilateral $HIJK$ is formed within the triangle. What is the area of quadrilateral $HIJK$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given \( x, y, z \geq 0 \) with at most one of them being zero, find the minimum value of
\[ f(x, y, z) = \sum \sqrt{\frac{x^{2}+256 y z}{y^{2}+z^{2}}} \]
where \(\sum\) represents a cyclic sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the radius of convergence for the series \(\sum_{n=1}^\infty \frac{(2z-3i)^n}{(3n-2i)^2}.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
计算下列实变函数定积分．

$\int _ { 0 } ^ { \infty } \frac { x ^ { 2 } d x } { ( x ^ { 2 } + 9 ) ( x ^ { 2 } + 4 ) ^ { 2 } } ,$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\pi/200$}
Let $\mathbb{N}$ denote the set of positive integers. For how many positive integers $k\le 2018$ do there exist a function $f: \mathbb{N}\to \mathbb{N}$ such that $f(f(n))=2n$ for all $n\in \mathbb{N}$ and $f(k)=2018$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1512}
Let $f:[0,\infty) \rightarrow[0,\infty]$ be a continuous function on $[0,\infty]$ and differentiable on $(0,\infty)$. Given that $f(x)=\int_{0}^{x}\sqrt{f(t)}dt$, determine the range of possible values of $f(6)$. \( \text{Provide your answer as an interval or set of values.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 9]}
Will initially had twenty-eight bottles of water. He drank seven bottles each day, shared three bottles with his friends every two days, and bought an extra five bottles every three days. How many days would the bottles last him?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the number of ways to put a number in every unit square of a $3 \times 3$ square such that any number is divisible by the number directly to the top and the number directly to the left of it, and the top-left number is $1$ and the bottom right number is $2013$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$18^3$}
Evaluate the integral \( I = \int_0^{2\pi} \log\left|re^{it} - a\right| \, dt \) where \( a \) is a complex number and \( 0 < r < |a| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi \log |a|}
Determine the order of the automorphism group of the group $A = \mathbb{Z} \oplus \mathbb{Z}_3$. What is $|\text{Aut}(A)|$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Assuming the twin prime conjecture, calculate the value of \( \mathcal{P}_\infty \), where \( \mathcal{P}_n \) is defined as follows: \[ \mathcal{P}_n = \frac{\#\text{twin primes} \le n:\min\{|b_1-a_1|,|b-a|,|b_2-a_2|\}\neq|b-a|}{\#\text{twin primes} \le n} \] Here, \( p_1 \) and \( p_2 \) are twin primes, \( p_1-1=a_1\times b_1 \) and \( p_2+1=a_2\times b_2 \) are factorizations such that \( |b_1-a_1| \) and \( |b_2-a_2| \) are minimized, and \( p_1+1=p_2-1=a\times b \) is a factorization such that \( |b-a| \) is minimized. Note that by convention the pair \((2,3)\) are not twin primes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that \(\cot \alpha + \cot \beta + \cot \gamma = -\frac{4}{5}\), \(\tan \alpha + \tan \beta + \tan \gamma = \frac{17}{6}\), and \(\cot \alpha \cot \beta + \cot \beta \cot \gamma + \cot \gamma \cot \alpha = -\frac{17}{5}\), find the value of \(\tan (\alpha + \beta + \gamma)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Calculate the expected value of the following series:
\[ \sum_{i=4}^\infty 500 \frac{e^{-4} 4^i}{i!} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{283.26}
Let $n = p^e m$, where $p^e$ is the largest power of the prime $p$ dividing $n$, and $p$ does not divide $m$. Consider a set $S$ with $|S| = n$. Compute the number of subsets of $S$ with exactly $p^e$ elements, denoted as $N$. Find $N \mod p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \bmod p}
Consider a $12 \times 12$ matrix $B$ where each diagonal entry is $2$ and all off-diagonal entries are $-1$. Calculate the sum of all entries of the inverse matrix $B^{-1}$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{4}{3}}
Determine the values of $z$ on the unit circle $|z|=1$ for which the series \( \sum_{n=1}^\infty \frac{z^n}{n} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{z \neq 1}
Determine the order of the projective general linear group $PGL_2(F_p)$, where $F_p$ is a finite field with $p$ elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p(p^2 - 1)}
A function \( f \) is defined on the set of positive integers with the properties that for any positive integer \( n \), \( f(f(n)) = 2n \) and \( f(4n + 1) = 4n + 3 \). Determine the last three digits of \( f(2016) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{904}
Calculate the integral: $$\int_{\mathbb{R}^5}\frac{e^{-x^2-y^2-z^2}}{1+w^2+s^2}\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}w\,\mathrm{d}s$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the maximum value of $y$ for a point $(x,y)$ on the curve defined by the equation $2(x^3+y)^4+(x^3+y)^2=2x^3+y$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{8}}
Let $F = \{E_1, E_2, \ldots, E_n\}$ be a collection of $n$ subsets of a set $X$, where $n$ is a positive integer. Determine the number of distinct sets in the sigma-algebra $\sigma(F)$ generated by $F$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2^n}}
While visiting a small city in the United States, I lost my overcoat in a city bus. When I reported the matter to the bus company, I was asked to provide the bus number. Unfortunately, I couldn't remember the exact number, but I did recall certain peculiarities about it. The number plate showed the bus number as a perfect square, and even if the plate was turned upside down, the number remained a perfect square. Also, this number was a palindrome, reading the same backward and forward. There was no leading 0 in the number. I discovered that the bus company operated 2,000 buses, numbered sequentially from 1 to 2,000. Using this information, I was able to work out the bus number. Now your task is to find out which number was on the bus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{484}
In a real vector space $V$ of dimension $2m$ with a non-singular quadratic form $Q$, if $Q$ is hyperbolic, determine the dimension of the subspace on which $Q$ vanishes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m}
Minimize the function \( xy + 49xz + 7yz \) subject to the constraint \( xyz = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Compute the total area covered by a circle of radius \( r \) whose center moves along the path \( x = t, y = \sin(t) \) for non-negative \( t \). Consider any overlaps in the area covered by the circle during its motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the double integral \(\int_{0}^{9}\int_{\frac{y-3}{2}}^{\sqrt{y}} x \, dx \, dy\) by reversing the order of integration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{81}{8}}
If x + y = 250 and x - y equals a certain number, for integers of x and y, y equals 225. What is the difference between x and y?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
过椭圆的左焦点 \(F\) 作倾斜角为 \(45^{\circ}\) 的直线 \(l\) 与该椭圆交于 \(A 、 B\) 两点。若 \(|B F|=2|A F|\)，则该椭圆的离心率是多少？答案应为\(\frac{\sqrt{n}}{m}\)的形式，请计算n+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the norm of the Volterra operator $A$ acting on the space $L_2(0,1)$, where $Ax(t) = \int_0^t x(s) \, ds$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{\pi}}
Calculate the cross product \( \nabla \times (\mathbf{i} \wedge \mathbf{r}) \), where \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), and \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) are the standard unit vectors in three-dimensional space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\mathbf{i}}
Let \( S = \{1, 2, 3, 4\} \) and let the sequence \( a_{1}, a_{2}, \cdots, a_{n} \) have the following property: For any non-empty subset \( B \) of \( S \) (with the number of elements in set \( B \) denoted as \( |B| \)), there exist \( |B| \) consecutive terms in the sequence that exactly form the set \( B \). Find the smallest possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
How many planes are equidistant from four points that do not lie in the same plane?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A coin is tossed until two consecutive heads appear. The probability of the coin landing heads, denoted by \( p \), follows a beta distribution with parameters \( p \sim \beta(6, 8) \). What is the expected number of coin tosses required to achieve two consecutive heads?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{52}{5}}
Let $V$ be a finite-dimensional vector space over the complex numbers $\mathbb{C}$, and let $T: V \rightarrow V$ be a linear transformation with a minimal polynomial of $x^{13}(x-1)$. Determine the minimal polynomial of $T^4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^4(x - 1)}
Find the number of solutions to the equation $x^d = e$ in a cyclic group of order $n$, where $d = \gcd(m,n)$ and $m$ does not divide $n$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d}
In a classroom with 10 girls and 5 boys, each of the 15 kids plays against every other kid exactly once, with a single winner in each game. The probability of winning for any participant is 0.5. Let X represent the total number of wins by the boys. Determine the minimum possible value of the variance, Var(X), considering potential dependencies between game outcomes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
On a test, the boys in the class averaged 90 points and the girls in the class averaged 96 points. If the overall class average was 94 points, and there were 3 students with missing scores whose total average was 92 points, what is the ratio of boys to girls in the class, considering the students with missing scores?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:5}
Consider a cube PQRSTUVW with a side length s. Let M and N be the midpoints of edges PU and RW, and let K be the midpoint of QT. Find the ratio of the area of triangle MNK to the area of one of the faces of the cube.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Calculate the determinant of the matrix \( J = A \left( A^\intercal W A \right)^{-1} A^\intercal W \), where \( A \) is a non-square matrix, and \( W \), \( J \), and \( A^\intercal W A \) are square matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
129. In a parallelogram, there are two circles of radius 1, each touching the other and three sides of the parallelogram. It is also known that one of the segments of a side of the parallelogram from a vertex to the point of tangency is $\sqrt{3}$. Find the area of the parallelogram.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{3}(2\sqrt{3}+3)}
Estimate the smallest integer $n$ such that $\frac{10^n}{n!} < 0.001$ using Stirling's approximation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
How many positive integers between 10 and 2016 are divisible by 3 and have all of their digits the same?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the number of ordered integer solutions \((A, B, C, D, E, F)\) such that \( A \times B \times C \times D \times E \times F = 7 \times 10^7 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120434688}
At 1 minute to midnight, 10 apples fall into a sack. The same happens at half a minute to midnight, then at a quarter minute to midnight, and so on. At each such event, you remove an apple randomly from the ones still present in the sack. What is the probability that the sack will be empty at midnight?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( A(2,0) \) be a fixed point on the plane, \( P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right) \) be a moving point. When \( t \) changes from \( 15^{\circ} \) to \( 45^{\circ} \), the area swept by the line segment \( AP \) is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{6}}
In a math contest, Riley and Ofelia are on one team. Out of the 35 questions, Riley got some mistakes while Ofelia got 5 more than half the score of Riley. Their team got 17 incorrect answers. How many mistakes did Riley make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A sphere with radius 1 has its center at point \( O \). From a point \( A \), located outside the sphere, four rays are drawn. The first ray intersects the surface of the sphere at points \( B_1 \) and \( C_1 \) sequentially, the second ray at points \( B_2 \) and \( C_2 \), the third ray at points \( B_3 \) and \( C_3 \), and the fourth ray at points \( B_4 \) and \( C_4 \). Lines \( B_1B_2 \) and \( C_1C_2 \) intersect at point \( E \), and lines \( B_3B_4 \) and \( C_3C_4 \) intersect at point \( F \). Given that \( AO = 2 \), \( EO = FO = 3 \), and the angle between faces \( AOE \) and \( AOF \) is \( 30^\circ \), find the volume of pyramid \( OAEF \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{35}{24}}
Determine the Galois group \(\mathrm{Gal}(\mathbb{Q}(\sqrt[5]{3})/\mathbb{Q})\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{e\}}
Given the functional $f(x_1, x_2, x_3) = x_1 - 3x_2 + x_3$ and the dual basis $\mathcal{A}^* = \{g_1, g_2, g_3\}$ where $g_1(x_1, x_2, x_3) = x_1 - x_3$ and $g_2(x_1, x_2, x_3) = x_1 - 2x_2$, find the functional $g_3$ such that the third coordinate of $f$ in the basis $\mathcal{A}^*$ is 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-x_1 + x_2 + x_3}
Apple Street begins at Peach Street and runs directly east for some kilometers until it ends when it meets Cherry Street. Apple Street is intersected every 200 meters by a perpendicular street, and each of those streets other than Peach Street and Cherry Street is given a number. The highest-numbered street that intersects Apple Street is 15. How long is Apple Street in kilometers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.2}
We consider sports tournaments with  $n \ge 4$  participating teams and where every pair of teams plays against one another at most one time. We call such a tournament *balanced* if any four participating teams play exactly three matches between themselves. So, not all teams play against one another.
Determine the largest value of  $n$  for which a balanced tournament with  $n$  teams exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
当s、t取遍所有实数时，则$( s + 5 - 3 \vert \cos t \vert ) ^ { 2 } + ( s - 2 \vert \sin t \vert ) ^ { 2 }$所能达到的最小值是
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Compute the probability that a standard Brownian motion hits 1 before hitting -1, and then hits -1 before hitting 2. Formally, find \( P(T_1 < T_{-1} < T_2) \), where \( T_a = \inf\{t \geq 0 \mid B_t = a\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
对任意区间 $I$, 用 $M_{I}$ 表示函数 $y=\sin x$ 在 $I$ 上的最大值. 若正实数 $a$ 满足 $M_{[0, a]}= 2 M_{[a, 2 a]}$, 请提供所有情况中m+n的值的总和，其中原始答案的形式为\frac{m\pi}{n}。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Let \( P(x) \) be a polynomial of degree 3 such that \( P(-1) = 10 \), \( P(1) = -6 \), and \( P(x) \) has a local maximum at \( x = -1 \). Additionally, \( P'(x) \) has a local minimum at \( x = 1 \). Find the distance between the local maximum and local minimum of the curve \( P(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{65}}
Evaluate the expression: \[ \sum_{n=1}^{64}(-1)^n \left\lfloor \dfrac{64}{n} \right\rfloor \varphi(n) \] where \(\varphi(n)\) denotes Euler's totient function. Use the identity \( \sum_{k=1}^n\left\lfloor\dfrac{n}{k}\right\rfloor \varphi(k) = \dfrac{n(n+1)}{2} \) and the fact that 64 is a power of 2 to simplify the sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-652}
Given a convex \( n \)-gon \( A_1 A_2 \ldots A_n \) with area \( A \) and a point \( P \), we rotate \( P \) through an angle \( x \) about \( A_i \) to get the point \( P_i \). Find the area of the polygon \( P_1 P_2 \ldots P_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \sin^2 \left(\frac{x}{2}\right) \cdot A}
Given a matrix $A \in \mathbb{R}^{m \times n}$, find a matrix $X \in \mathbb{R}^{m \times r}$ with orthonormal columns such that the Frobenius norm $\|X^T A\|$ is maximized. Express the solution in terms of the singular value decomposition of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{U_r}
In an integer triangle, two sides are equal to 10. Find the third side, given that the radius of the inscribed circle is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Consider a collection of 12 sets, each containing 8 elements. Every pair of sets shares 6 elements, any collection of 3 sets shares 4 elements, and no collection of 4 sets shares any elements. Determine the total number of distinct elements in this collection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{580}
Given points $A, B, C,$ and $D$ such that segments $AC$ and $BD$ intersect at point $E$. Segment $AE$ is 1 cm shorter than segment $AB$, $AE = DC$, and $AD = BE$. Additionally, $\angle ADC = \angle DEC$. Find the length of $EC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $X$ be the set of the first $n$ positive integers, where $n \geq 3$. The group $\mathcal{G}$ consists of all bijections of $X$, with composition as the group operation. $\mathcal{G}$ acts on $X$ in the standard way: for $\varphi \in \mathcal{G}$ and $i \in X$, $\varphi \cdot i = \varphi(i)$. Consider the induced action of $\mathcal{G}$ on $X \times X \times X$ defined by $\varphi \cdot (i, j, k) = (\varphi(i), \varphi(j), \varphi(k))$. Determine the number of orbits of this action on $X \times X \times X$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given that $F$ is the left focus of the ellipse $C:\frac{{x}^{2}}{3}+\frac{{y}^{2}}{2}=1$, $M$ is a moving point on the ellipse $C$, and point $N(5,3)$, then the minimum value of $|MN|-|MF|$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 - 2\sqrt{3}}
Determine the Galois group of the splitting field of the polynomial \(x^4 - x - 1\) over \(\mathbb{Q}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_4}
The diagonal of a trapezoid divides its area in the ratio of 3:7. In what ratio will the area of this trapezoid be divided if a line parallel to a lateral side is drawn from the end of the shorter base?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:2}
An insurance company receives claims of two types. Let $N_i(t)$ represent the number of type $i$ claims by time $t$, where $\\{N_1(t), t \ge 0\\}$ and $\\{N_2(t), t \ge 0\\}$ are independent Poisson processes with rates $\lambda_1 = 10$ and $\lambda_2 = 1$, respectively. Type 1 claim amounts are independent exponential random variables with a mean of 1000 dollars, and type 2 claim amounts are independent exponential random variables with a mean of 5000 dollars. Given a claim of 4000 dollars, calculate the probability that it is a type 1 claim.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.671}
Given the function \( f(x) = e^{2x}\cos(3x) \), find the minimum number of correct digits required in an approximation of an irrational number \( x^* \) within the interval \([0, \pi]\) such that the computed value \( f(x^*) \) has at least four correct digits compared to the exact value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}