inclusionAI/Ring-lite-rl-data huggingface dataset:
Suppose $Y(t)$ is the price of SomeCo stock at time $t$ years from now. Assume that $Y(t)$ follows a geometric Brownian motion with a drift rate of 0.05 per year and a volatility of $\sigma = 0.3$ per year. If the current price of SomeCo stock is 10.00 USD, calculate the probability that the price will be at least 11.00 USD one year from now.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.3817}
From the $8$ vertices of a cube, choose any $4$ vertices, the probability that these $4$ points lie in the same plane is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{6}{35}"]}
Determine a condition on the integer $N$ such that an $N \times N$ matrix, constructed as follows, has full row and column rank. For each entry in the matrix, compute $v$ by multiplying the row number with the column number. Convert $v$ to binary, and for each bit $v_b$, raise the $b$th prime number to the power of $v_b$. For example, the entry at row 5 and column 7 would be $2 \cdot 3 \cdot 13 = 78$, and at row 4 and column 2 would be $7$. What is the condition on $N$ for the matrix to have full rank?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N \leq 6}
Given the ellipse $C$: $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$, with its upper and lower vertices denoted as $A_{1}$ and $A_{2}$, respectively, suppose point $P$ lies on $C$ and the slope of line $PA_{2}$ falls within the range $[-2,-1]$. Determine the range of the slope of line $PA_{1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac{3}{8}, \frac{3}{4}]}
Let $S = \{m + \frac{1}{n} \mid m, n \in \mathbb{N}\}$. Determine the derived set (the set of cluster points) of $S$. \( \mathbb{N} \) denotes the set of natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{N}}
设 $M=\{1,2,3, \cdots, 40\}$, 求最小正整数 $n$, 使可将 $M$ 分成 $n$ 个两两不相交的子集且同一子集内任取 3 个数 $a, b, c$ (不必不相同) 都有 $a \neq b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the conditions under which the integral \( \int\limits_{e_k}^{\infty} \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x \cdots (\ln^{(k)} x)^s } \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s > 1}
A certain fruit is mostly water, and when it dries, it becomes raisins which are 17% water. A quantity of raisins currently weighs 14 kilograms and weighed 116.2 kilograms when they were the fruit. What is the percentage of water in the fruit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90\%}
What condition on \(\beta\) ensures that a function, which is Hölder continuous with order \(\beta\), is differentiable?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\beta > 1}
Two teams, Team A and Team B, each consist of 10 people. For some integer \( n \) where \( 1 \leq n \leq 9 \), Team A chooses \( n \) people to remove from the team. Then, for some integer \( k \) where \( 0 \leq k \leq n \), Team B replaces \( k \) of its members with \( k \) of the removed members from Team A. After these changes, there are \( N \) distinct possible combinations of teams. Find the remainder when \( N \) is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{696}
Determine the limit of the sequence \(x_n\) defined by the equation \(\sum_{k=1}^n kx_k = \prod_{k=1}^n x_k^k\) for all integers \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Estimate the probability that the sum of the faces showing on one hundred fair dice exceeds 370, using a continuity correction in your analysis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.115}
Determine the number of positive integers \( n \) less than 200 for which \( n^n \) is a cube and \( (n+1)^{n+1} \) is a square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Find the equation of the image of the line $x+y=1$ under the Möbius transformation $w=\frac{z+1}{z-1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{u - v = 1}
Let $\triangle{ABC}$ be a triangle with integer side lengths and the property that $\angle{B} = 2\angle{A}$. What is the least possible perimeter of such a triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let $f\left(x\right)$ be the greatest prime number at most $x$. Let $g\left(x\right)$ be the least prime number greater than $x$. Find$$\sum\limits_{i=2}^{100}\frac{1}{f\left(i\right)g\left(i\right)}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \frac{99}{202} \]}
Let $A \in \mathbb{R}^{m \times k}$ be a real matrix, and let $\sigma_{max}$ and $\sigma_{min}$ be its largest and smallest singular values, respectively. If $A$ satisfies the inequality \((1-\delta) \|x\|_2^2 \leq \|Ax\|_2^2 \leq (1+\delta)\|x\|_2^2\) for all $x \in \mathbb{R}^{k}$ and some $\delta \in (0,1)$, determine the rank of $A$. \(\text{Provide your answer as a single number.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k}
Evaluate the integral: $$\int_{\frac{\pi}{12}}^{\frac{\pi}{4}}\dfrac{8\cos2x}{(\tan x+\cot x)^3}\,dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{128}}
Let $\times$ represent the cross product in $\mathbb{R}^3.$ For what positive integers $n$ does there exist a set $S\subset\mathbb{R}^3$ with exactly $n$ elements such that
$$S=\{v\times w:v,w\in S\}?$$ Please provide the sum of all possible values of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Triangle $ABC$ has angle bisectors $\overline{AD},$ $\overline{BE},$ and $\overline{CF}.$  Given that $\angle EDF = 90^\circ,$ enter all possible values of $\angle BAC$ (in degrees), separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
$\text{A sequence of positive integers is defined by}a_0=1\mathrm{and}a_{n+1}=a_n^2+1\mathrm{for~each}n\geq0.\mathrm{Find}\gcd(a_{999},a_{2004}).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
677}
Compute the product
\[\frac{(1998^2 - 1996^2)(1998^2 - 1995^2) \dotsm (1998^2 - 0^2)}{(1997^2 - 1996^2)(1997^2 - 1995^2) \dotsm (1997^2 - 0^2)}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3996}
Given the tetrahedron \(OPQR\), let the points \(P, Q, R\) be reflected over \(O\) to respectively form points \(P', Q', R'\). The edges of tetrahedron \(ABCD\) are bisected by the points \(P, Q, R, P', Q', R'\). How many times larger is the volume of tetrahedron \(ABCD\) compared to the volume of tetrahedron \(OPQR\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given that integers \( x \) and \( y \) satisfy the equation
\[
\left(\frac{1}{x} + \frac{1}{y}\right) \left(\frac{1}{x^2} + \frac{1}{y^2}\right) = -\frac{2}{3} \left(\frac{1}{x^4} - \frac{1}{y^4}\right),
\]
determine how many possible values of \( x + y \) exist.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
What is the correct equivalent expression for the following sum without using sigma notation?

$$\sum\limits_{k = 1}^{10}k\binom{10}{k}\binom{20}{10-k}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10\binom{29}{9}}
Given $a > 0$ and $b > 0$, find the minimum value of the expression $6\sqrt{ab} + \frac{3}{a} + \frac{3}{b}$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In quadrilateral $\mathrm{ABCD}$, $\mathrm{AB} = \mathrm{BC}$, and angles $\mathrm{ABC}$ and $\mathrm{ADC}$ are right angles. Find its area if the distance from vertex $\mathrm{B}$ to line $\mathrm{AD}$ is 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
UBA Capital recently bought brand new vehicles for office use. They only went for Toyota and Honda and bought a certain ratio of Toyota to Honda. In the purchase, 40% of the Toyota bought and 60% of the Honda bought were SUVs. UBA Capital bought 52 SUVs in the aforementioned purchase. What is the ratio of Toyota to Honda vehicles bought?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5:1}
Calculate the following limit, where \( I_{n} = \int_{0}^{1} x^{n} \arctan(x) \, dx \):
\[ \lim_{n \rightarrow \infty} \left( n((n+1)I_{n}-\frac{\pi}{4}) \right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Given that an imaginary number \( z \) satisfies \( w = z + \frac{1}{z} \) where \( w \) is a real number, and \(-1 < w < 2\). Let \( u = \frac{1 - z}{1 + z} \). What is the minimum value of \( \left|w - u^{2}\right| \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In triangle \( \triangle ABC \), \( AB=AC \) and \( \angle A=80^{\circ} \). Points \( D, E \), and \( F \) are on sides \( BC, AC \), and \( AB \) respectively, and \( CE=CD \) and \( BF=BD \). What is the measure of \( \angle EDF \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50^{\circ}}
已知正整数 $y$ 不超过 $2022$ 且满足 $100$ 整除 $2^{y}+y$ ，则这样的 $y$ 的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
If ${{x}^{2018}}={{a}_{0}}+{{a}_{1}}(x-1)+{{a}_{2}}{{(x-1)}^{2}}+\cdots +{{a}_{2018}}{{(x-1)}^{2018}},$ then $\dfrac{{a}_{1}}{3}+\dfrac{{a}_{2}}{{3}^{2}}+\cdots +\dfrac{{a}_{2018}}{{3}^{2018}}=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \dfrac{4}{3}\right)^{2018}-1}
Evaluate the following limit using only simple algebraic tricks and asymptotic equivalences, without using L'Hopital's rule:
$$\lim_{x \rightarrow0} \frac {\sqrt[4]{1+\sin{2x}} - \sqrt[4]{1-\cos{2x}}} {\sqrt{1-\sin{3x}} - \sqrt{1+\cos{3x}}} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that $40\%$ of students initially answered "Yes", $40\%$ answered "No", and $20\%$ were "Undecided", and $60\%$ answered "Yes" after a semester, $30\%$ answered "No", and $10\%$ remained "Undecided", determine the difference between the maximum and minimum possible values of $y\%$ of students who changed their answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40\%}
Compute the limit: $$\lim_{x\to0^+}\frac{2\arctan(x)\sqrt{1-x^2}+\frac{x^3-2x^4}{\ln(2x+2x^2)}-\ln(1+2x+2x^2)}{e^{x-\frac{x^2}2}-\cos(x^2)+\tan(2x^4-x)+x^2e^{-\frac1{x}}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
The average salary per head of all the workers in a workshop is Rs. 1000. If the average salary per head of 10 technicians is Rs. 1200 and the average salary per head of the rest is Rs. 820, what is the total number of workers in the workshop?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Given the equations for a point \((x, y, z)\) in \(\mathbb{R}^3\):

\[ x\cos\alpha - y\sin\alpha + z = 1 + \cos\beta \]
\[ x\sin\alpha + y\cos\alpha + z = 1 - \sin\beta \]
\[ x\cos(\alpha+\beta) - y\sin(\alpha+\beta) + z = 2 \]

where \(\alpha, \beta \in (0, 2\pi)\), find the minimum value of \(x^2 + y^2 + z^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A trapezoid is inscribed in a circle with radius \( R \), where the lower base is twice as long as each of the other sides. Find the area of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 \sqrt{3}}{4} R^2}
Evaluate the line integral \( \int_{\vert x \vert + \vert y \vert = 5 } -\frac{y}{x^2 + y^2} \, dx + \frac{x}{x^2 + y^2} \, dy \) for the vector field \( F(x,y)=\left(-\frac{y}{x^2 + y^2} , \frac{x}{x^2+y^2}\right) \) in \( \mathbb{R}^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Odell and Kershaw run for $30$ minutes on a circular track. Odell runs clockwise at $250 m/min$ and uses the inner lane with a radius of $50$ meters. Kershaw runs counterclockwise at $300 m/min$ and uses the outer lane with a radius of $60$ meters, starting on the same radial line as Odell. How many times after the start do they pass each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$47$}
In triangle $ABC$, $\\angle ABC$ is obtuse.  Point $D$ lies on side $AC$ such that $\\angle ABD$ is right, and point $E$ lies on side $AC$ between $A$ and $D$ such that $BD$ bisects $\\angle EBC$.  Find $CE$ given that $AC=35$, $BC=7$, and $BE=5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Calculate the derivative $DF(0,0,0)$ using the chain rule, where $F(x,y,z)=f(g(x,y),h(z),f(x,h(x),g(x,x)))$. Given that $f:\mathbb{R}^{3}\rightarrow \mathbb{R}$, $g:\mathbb{R}^{2} \rightarrow \mathbb{R}$, and $h:\mathbb{R} \rightarrow \mathbb{R}$ are functions of class $C^{\infty}$ with $f(0,0,0)=g(0,0)=h(0)$, $Df(0,0,0)=(1,2,3)$, $Dg(0,0)=(4,5)$, and $h'(0)=6$. Determine $DF(0,0,0)$. Use the substitutions $u=g(x,y)$, $v=h(z)$, and $w=f(x,h(x),g(x,x))$ if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(124, 5, 12)}
Evaluate the double integral \(\iint_R \frac{xy}{x^2 + y^2} \, dx \, dy\) over the region \( R=\{(x,y) \in \mathbb{R} : y \geq x, 1 \leq x^2 + y^2 \leq 2 \} \) using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the value of $n$ such that the following identity holds for all real $x \not\in \{\pm j: j=1,2,\dots,n\}$:
\[4\sum_{j=1}^n \frac{j}{x^2-j^2}=k_n\sum_{j=1}^n \frac{1}{(x-j)(x+n+1-j)}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the value of \( \lim_{R \to \infty} \int_{C_R} \frac {e^{iz}}{z}dz \), where \( C_R \) is a curve in the complex plane parameterized as \( \gamma(t)= Re^{it}, 0\leq t\leq\pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $m$ and $n$ satisfy $mn = 6$ and $m+n = 7$. Additionally, suppose $m^2 - n^2 = 13$. Find the value of $|m-n|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{13}{7}}
Given the hyperbola $C: mx^{2}+ny^{2}=1$ ($m > 0,n < 0$), one of its asymptotes is tangent to the circle $x^{2}+y^{2}-6x-2y+9=0$. Calculate the eccentricity of the hyperbola $C$. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{4}}
Estimate the value of the integral \( \int_{0}^{0.7} \cos(x^3) \, dx \) using a Taylor series expansion such that the error is less than \( 10^{-4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.6941}
Given that \( \phi_1(x) \) and \( \phi_2(x) \) are two linearly independent solutions of the differential equation:
\[ 2x^2y'' - (x + x^2)y' + (x^2 - 2)y = 0 \]
with the condition \( \phi_1(0) = 0 \), find the smallest positive integer \( n \) such that:
\[ \lim_{x \to 0} x^n \frac{\phi_2(x)}{\phi_1(x)} = 0 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the number of solutions to the equation \( e^{2x} = 10 \cdot \ln(x^4 + 1) \) using the Intermediate Value Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $\overrightarrow{{e}_{1}}$ and $\overrightarrow{{e}_{2}}$ be two non-collinear vectors in space. Given that $\overrightarrow{AB}=\overrightarrow{{e}_{1}}+k\overrightarrow{{e}_{2}}$, $\overrightarrow{BC}=5\overrightarrow{{e}_{1}}+4\overrightarrow{{e}_{2}}$, $\overrightarrow{DC}=-\overrightarrow{{e}_{1}}-2\overrightarrow{{e}_{2}}$, and points $A$, $B$, $D$ are collinear, find the real number $k=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
II. Answer Questions (Total 56 points)
9. (16 points) Given a tetrahedron $P-ABC$ with each edge of the base being 1 and the height being $\sqrt{2}$, the center of its inscribed sphere is $O$, and the radius is $r$. Find the area of the planar region within the base $ABC$ consisting of points that are no more than $2r$ away from point $O$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}+\frac{\pi}{24}}
Define a function $f(x)$ on $\mathbb{R}$ that satisfies $f(x) + f(x+5) = 16$. When $x \in (-1, 4]$, $f(x) = x^2 - 2^x$. Find the number of zeros of the function $f(x)$ on the interval $[0, 2013]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{604}
Evaluate the limit: \[ \lim_{n\to \infty} n^2 \int_0^1 \left(\frac{1}{1+x^2}\right)^n dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Don and his wife each receive an 8 percent annual raise. If Don receives a certain amount as a raise and his wife receives a raise of Rs. 840, the difference between their annual income after their raises is Rs. 540. How much raise did Don receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{880}
Given that the graph of the power function $y=f(x)$ passes through the point $(4, \frac{1}{2})$, and $f(a+1) < f(10-2a)$, find the range of values for the real number $a$. Express your answer in standard interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3,5)}
Determine the number of ways to express $121$ as a sum of strictly increasing positive Fibonacci numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
How many ways are there to distribute 16 identical balloons and 11 different toys to 9 different children, ensuring that each child receives at least one toy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{308255312534400}
Along the road connecting Masha's and Sasha's houses, there are 17 apple trees and 20 poplars. When Masha was walking to visit Sasha, she was photographing all the trees. Right after the tenth apple tree, Masha’s phone memory was full, and she couldn't take photos of the remaining 13 trees. The next day, when Sasha was walking to visit Masha, starting from the eighth apple tree, he picked one leaf from each tree. How many leaves did Sasha pick?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Compute the conditional probability \( A = P(X_6 > X_2 \mid X_1 = \max[X_1, X_2, X_3, X_4, X_5]) \) for iid random variables \( X_i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{12}}
Consider the initial value problem \(y' = 2 - \sin(xy), \quad 1 \leq x \leq 3, \quad y(1) = -\frac{1}{2}\). Find a bound for \(|y''(x)|\) on the interval \([1,3]\].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{29}{2}}
Determine the constant $B$ that minimizes the error between the prime counting function $\pi(x)$ and the approximation $\frac{x}{\ln x - B}$ for large $x$, given that the error is bounded by $O\left(\frac{x}{\ln^3 x}\right)$. What is the value of $B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
James works for 240 minutes. He takes a water break every 20 minutes and a sitting break every certain minutes. He takes 10 more water breaks than sitting breaks. How often does James take a sitting break?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Let \( P \) be any interior point of an equilateral triangle \( ABC \). Let \( D, E, \) and \( F \) be the feet of the perpendiculars dropped from \( P \) to the sides \( BC, CA, \) and \( AB \) respectively. Determine the value of the ratio

$$
\frac{PD + PE + PF}{BD + CE + AF}.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
There were ninety-eight onions on the shelf. Sixty-five were sold, which accounts for 75% of the remaining onions. 20 more onions were added to the shelf from a new shipment. After that, 1/4 of the total onions were given away to a charitable cause. How many onions are on the shelf now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
In rectangle ABCD, point E is on side BC and point F is on side CD, satisfying AE = AF. Given that the area of triangle ADF is 2, the area of triangle ABE is 4, and the area of triangle CEF is 8, calculate the value of 7 cos ∠EAF. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
12.126. Find the angle in the axial section of a cone if a sphere with its center at the vertex of the cone, touching its base, divides the volume of the cone in half.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\arccos\frac{1+\sqrt{17}}{8}}
Given real numbers \( x_{1}, x_{2}, \cdots, x_{2011} \) satisfying \( \left|x_{1}\right|=99 \) and \( \left|x_{n}\right|=\left|x_{n-1}+1\right| \) for \( n = 2, 3, \cdots, 2014 \), find the minimum value of \( x_{1} + x_{2} + \cdots + x_{2014} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5907}
You have one factory that produces cars and one builder factory that can produce either car factories or other builder factories. Producing a builder factory that makes other builder factories takes twice as long as producing one that makes car factories. Determine the optimal number of builder factories to produce before switching to car factories in order to maximize the number of cars produced over a 4-year period.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the triple integral \( I = \int_{R} (x-y) \,dV \) over the region \( R = \{(x,y,z) \in \mathbb{R^3} \mid 1 \leq x^2 + y^2 \leq 16, 0 \leq z \leq y + 4\} \). Express your answer to three decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-200.277}
Evaluate the surface integral \( \iint\limits_S \vec{A} \cdot \hat{n}\,dS \) over the entire surface \( S \) of the region bounded by the cylinder \( x^2 + z^2 = 9 \), \( x = 0 \), \( y = 0 \), \( y = 8 \), \( z = 0 \), where \( \vec{A} = 6z\,\hat{i} + (2x+y)\,\hat{j} - x\,\hat{k} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18\pi}
Determine the number of terms required in the series expansion to estimate the integral \( \int_{0}^{0.4}\sqrt{1+x^4} \, dx \) such that the absolute error is less than or equal to \( \frac{0.4^9}{72} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
11. A ball is propelled from corner $A$ of a square snooker table of side 2 metres. After bouncing off three cushions as shown, the ball goes into a pocket at $B$. The total distance travelled by the ball is $\sqrt{k}$ metres. What is the value of $k$ ?
(Note that when the ball bounces off a cushion, the angle its path makes with the cushion as it approaches the point of impact is equal to the angle its path makes with the cushion as it moves away from the point of impact as shown in the diagram below.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
Representatives from three research institutes are to form a commission of 9 researchers. Determine the number of ways to form this commission such that no institute has an absolute majority.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11130}