inclusionAI/Ring-lite-rl-data huggingface dataset:
Jim and his cousin are dining at an upscale restaurant. They each order a surf n' turf that costs $38 and a premium wine that is $14 each. They also indulge in a shared dessert platter that costs $18. In celebration of the restaurant's anniversary, they receive a 20% discount on the food items but not on the drinks. They choose to leave a generous tip by rounding the total up to the next ten after applying the discount. They spend 85% of their combined money. If Jim brought $100, given the local tax rate of 10%, how much did his cousin bring?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$41.18}
Given a regular tetrahedron P-ABC, where points P, A, B, and C are all on the surface of a sphere with radius $\sqrt{3}$. If PA, PB, and PC are mutually perpendicular, then the distance from the center of the sphere to the plane ABC is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
In a computer science laboratory, there are 15 workstations labeled $W_1$ through $W_{15}$ and 10 servers labeled $S_1$ through $S_{10}$. Each server can have only one direct connection with a workstation at a time. Determine the minimum number of direct connections required to ensure that any subset of up to 10 workstations can simultaneously access different servers via direct connections.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
40 persons like apple. 7 like orange and mango but dislike apple. 10 like mango and apple and dislike orange. Some people like all. There are 47 people who like apple. How many people like apple, orange, and mango?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $T: V \rightarrow V$ be a linear transformation on a vector space $V$ with $\dim \operatorname{im} T = 2$. If $T^3 = bT$ for some scalar $b$, determine the possible values of $b$ for which $T + I$ is invertible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b \neq 1}
Determine the number of boolean functions of $n$ variables that can be expressed as a product of literals, excluding functions that always yield $0$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^n}
Evaluate the limit \( \lim_{x\to 0} \frac{\sqrt[3]{1+\frac{x}{3}} - \sqrt[4]{1+\frac{x}{4}}}{1-\sqrt{1-\frac{x}{2}}} \) without using L'Hospital's rule, power expansions, or derivatives.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{36}}
For $n=1,2, \ldots$ let
$$
S_{n}=-\log \left(\sqrt[n^{2}]{1^{1} \cdot 2^{2} \cdot \ldots \cdot n^{n}}\right)+\log (\sqrt{n})
$$
where $\log$ denotes the natural logarithm. Find $\lim _{n \rightarrow \infty} S_{n}$.
 If the value can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $L, M, N \in \mathbb{P}^5$ be three general subspaces of codimension 3. Consider three general points $l_i$ on $L$, $m_i$ on $M$, and $n_i$ on $N$ for $i=1,2,3$. Determine the dimension of the space $I_{L \cup M \cup N}(3)$, which consists of all cubic polynomials through these subspaces with all first partial derivatives vanishing at these 9 points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the largest positive integer $N$ such that \( \sqrt{64 + 32^{403} + 4^{N+3}} \) is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2008}
$\log_8 2 = 0.2525$ in base $8$ (to $4$ places of decimals). Find $\log_8 4$ in base $8$ (to $4$ places of decimals).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$0.5252$ 

}
Find all polynomials \( P \) with real coefficients such that \( P\left(X^{2}\right)=P(X) P(X-1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( X^2 + X + 1 \right)^n}
What is the expected value of the number of integers that will be printed from a stream of 0 to 9 (each with equal probability, and repetitions allowed) until we get the first instance of '4' followed by '5' consecutively?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Find the smallest \( n \) such that whenever the elements of the set \(\{1, 2, \ldots, n\}\) are colored red or blue, there always exist \( x, y, z, w \) (not necessarily distinct) of the same color such that \( x + y + z = w \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let $\triangle ABC$ be an equilateral triangle with side length $55$. Points $D$, $E$, and $F$ lie on sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively, with $BD=7$, $CE=30$, and $AF=40$. A unique point $P$ inside $\triangle ABC$ has the property that\[\measuredangle AEP=\measuredangle BFP=\measuredangle CDP.\]Find $\tan^{2}\left(\measuredangle AEP\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
已知数列$\left\{ {{a}_{n}} \right\}$满足${{a1}}=1,{{a}_{2}}=3$，若$\left| {{an+1}}-{{a}_{n}} \right|={{2}^{n}}\left( n\in {{N*}} \right)$，且$\left\{ {{a2n-1}} \right\}$是递增数列、$\left\{ {{a}_{2n}} \right\}$是递减数列，则$\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a2n-1}}}{{{a}_{2n}}}=$_______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$-\frac{1}{2}$}
Given that tetrahedron P-ABC is a 'Bie'zhi' with PA⊥ plane ABC, PA=AB=2, and AC=4, and all four vertices of the tetrahedron P-ABC lie on the surface of sphere O, calculate the surface area of the sphere O.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\pi}
Solve the equation \(\ln(x) = e^{-x}\) for \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.3098}
Given the curve \( z = x^3 \), rotate this curve towards the y-axis about the z-axis to express it as \( z = f(x, y) \). Write \( z = f(x, y) = f(x^3, \theta) \). Determine the expression for \( z \) when \( \theta = 90^\circ \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{z = y^3}
Determine whether the functional differential equation \( D'(t) = 2 D(2t) - D(t) \) with the initial condition \( D(0) = 0 \) can be solved in a closed form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{D(t) = 0}
On an exam with $80$ problems, Roger solved $68$ of them. What percentage of the problems did he solve?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
85%}
Evaluate the integral \( \int_{C} (1 + z + z^2)(e^{\frac{1}{z}}+e^{\frac{1}{z-1}}+e^{\frac{1}{z-2}}) \, dz \), where \( C \) is the circle \(|z|=3\) in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32\pi i}
Given the matrices \(A=\begin{pmatrix} 4 & 1 & 1 \\ -1 & 1 & 0 \\ -2 & -1 & 0 \\ \end{pmatrix}\) and \(B=\begin{pmatrix} 0 & 1 & 0 \\ -3 & 4 & 0 \\ a & 2 & 1 \\ \end{pmatrix}\), determine the necessary condition on the scalar \(a\) for the existence of an invertible matrix \(P\) such that \(B=P^{-1}AP\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \neq -2}
A line $l$ with a slope of $\frac{1}{3}$ passes through the left vertex $A$ of an ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a>b>0)$ and intersects the ellipse at another point $B$. If there exists a point $C$ on the $y$-axis such that $\triangle ABC$ is an isosceles right triangle with $C$ as the right-angled vertex, find the eccentricity of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6}}{3}}
250. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $a$. On the line $A A_{1}$, a point $M$ is taken, and on the line $B C$ - a point $N$ such that the line $M N$ intersects the edge $C_{1} D_{1}$. Find the minimum value of the magnitude | $M N \mid$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3a}
Evaluate the integral \( \int_0^{\pi / 2} \frac{\cos 2\theta}{1 + \sin \theta} \, d\theta \) using the substitution \( u = 1 + \sin \theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi - 3}
Find the limit \( \lim_{x \to \infty} f(x) \) for the function \( f(x) \) defined by \( f(x) = -\ln(f(x)) - x + 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A rope of which a calf is tied is increased from 10 m to some length. The calf can graze an additional 1348.2857142857142 square meters of grassy ground. What is the new length of the rope?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Let \( F_{1} \) and \( F_{2} \) be the two foci of an ellipse. A circle with center \( F_{2} \) is drawn, which passes through the center of the ellipse and intersects the ellipse at point \( M \). If the line \( ME_{1} \) is tangent to circle \( F_{2} \) at point \( M \), find the eccentricity \( e \) of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\sqrt{3}-1"]}
Determine the sum of the infinite series $$\sum_{n=1}^{\infty}\frac{(n+2)}{n(n+1)}x^n$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\left( \frac{1}{x} - 2 \right) \ln(1-x) + 1\)}
Determine the limiting distribution of $Y_n = X_n/n$, where $X_n$ is a binomial random variable with parameters $n$ and $p$, as $n \to \infty$. Provide your answer in terms of a well-known distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta_p}
Given a set $X$ and a finite subset $F$, determine the maximum number of ultrafilters on $X$ that can contain the subset $F$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|F|}
Find the area of quadrilateral \(ABCD\) if \(AB = BC = 3\sqrt{3}\), \(AD = DC = \sqrt{13}\), and vertex \(D\) lies on a circle of radius 2 inscribed in the angle \(ABC\), where \(\angle ABC = 60^\circ\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{3}}
Evaluate the integral \( \int_C \vec{F} \cdot d\vec{r} \) using Stokes' theorem, where \( \vec{F} = [-z, 5x, -y] \) and \( C \) is the ellipse defined by \( x^2 + y^2 = 4, z = x + 2 \). The direction of integration along \( C \) is clockwise as seen by a person standing at the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-24\pi}
Let $P$ and $A$ be $3 \times 3$ real matrices such that $PAP^t = -A^t$, where $P^t$ denotes the transpose of $P$. Find $\det(P)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
In a scalene triangle $ABC$, one of the angles is equal to the difference of the other two angles, and one of the angles is twice as large as another angle. The angle bisectors of angles $A$, $B$, and $C$ intersect the circumcircle of the triangle at points $L$, $O$, and $M$ respectively. Find the area of triangle $LOM$ if the area of triangle $ABC$ is 8. If the answer is not an integer, round it to the nearest whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Calculate the volume of the solid bounded below by the triangle with vertices $(0,0)$, $(1,3)$, and $(3,3)$ in the $xy$-plane, and above by the surface $z = x^2y$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{78}{5}}
Two male students and three female students stand in a row. If male student A does not stand at either end, and exactly two of the three female students are adjacent, then the number of different arrangements is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Using Fourier series, we have the expression for \( x \) in the interval \( (-\pi, \pi) \):
\[ x = \sum_{n=1}^{\infty}\frac{\sin(nx)}{n}(-2\cos(n\pi)). \]
The task is to determine the limit of the following series by relating it to the given Fourier series:
\[ \lim_{N \to \infty}\sum_{n=1}^{N} \frac{\sin(n)}{n}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi - 1}{2}\)}
Four distinct points are arranged on a plane such that they have segments connecting them with lengths $a$, $a$, $a$, $b$, $b$, and $2a$. Determine the ratio $\frac{b}{a}$ assuming the formation of a non-degenerate triangle with one of the side lengths being $2a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
What is the total number of digits used when the first 2500 positive even integers are written?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9448}
Rectangle $EFGH$ has area $4032.$ An ellipse with area $4032\pi$ passes through points $E$ and $G$ and has its foci at points $F$ and $H$. Determine the perimeter of the rectangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\sqrt{2016}}
Given non-zero vectors $\overrightarrow{m}, \overrightarrow{n}$ with an angle of $\frac{\pi}{3}$ between them, and satisfying $|\overrightarrow{n}| = \lambda|\overrightarrow{m}|$ ($\lambda > 0$), consider two sets of vectors $\overrightarrow{x_{1}}, \overrightarrow{x_{2}}, \overrightarrow{x_{3}}$ composed of one $\overrightarrow{m}$ and two $\overrightarrow{n}$ vectors and $\overrightarrow{y_{1}}, \overrightarrow{y_{2}}, \overrightarrow{y_{3}}$ composed of two $\overrightarrow{m}$ and one $\overrightarrow{n}$ vectors. If the minimum possible value of $\overrightarrow{x_{1}} \cdot \overrightarrow{y_{1}} + \overrightarrow{x_{2}} \cdot \overrightarrow{y_{2}} + \overrightarrow{x_{3}} \cdot \overrightarrow{y_{3}}$ is $4|\overrightarrow{m}|^2$, find the value of $\lambda$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{3}}
Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ be the eight vertices of a $30 \times 30 \times 30$ cube. The two figures $ACFH$ and $BDEG$ are congruent regular tetrahedra. Find the volume of the intersection of these two tetrahedra.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4500}
Evaluate the improper integral: $$\int_{0}^{\infty}\frac{x^2}{e^x-1}\,dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\zeta(3)}
Given two random variables $x$ and $y$, and a positive function $h(x)$, suppose the inequalities $-h(x) < f(x,y) < h(x)$ hold with probabilities $P[-h(x) < f(x,y)] = \beta$ and $P[f(x,y) < h(x)] = \beta$. Find the probability that both inequalities hold simultaneously.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\beta - 1}
Two trains travel in opposite directions at 36 kmph and 45 kmph, and a man sitting in the slower train passes the faster train in some time. The length of the faster train is 135.0108 meters. How many seconds does it take for the man in the slower train to pass the faster train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.00048}
Find the limit of the series \( \sum_{n=1}^\infty \frac{n}{n^2+k^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
If \(\Theta\) is a random variable uniformly distributed over \([-\pi, \pi]\), determine the probability density function \(f_\Theta(\theta)\) when \(\Theta\) is restricted to the interval \([-\pi/4, \pi/4]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{\pi}}
Flip $98$ fair coins and $1 \ HH$ coin and $1 \ TT$ coin. Given that you see an $H$, what is the probability that it was the $HH$ coin?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/50}
In the inscribed quadrilateral $ABCD$, the diagonals intersect at point $P$, and triangle $APD$ is acute-angled. Let $E$ and $F$ be the midpoints of sides $AB$ and $CD$, respectively. A perpendicular is drawn from $E$ to $AC$, and another from $F$ to $BD$, intersecting at point $Q$. Determine the angle between lines $PQ$ and $BC$. Express your answer in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Find a holomorphic bijection from the upper half-plane \( H = \{z \in \mathbb{C} \mid \text{Im}(z) > 0\} \) to the slit plane \( M = \mathbb{C} \setminus (-\infty,0] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-z^2}
In $\triangle ABC$, $a=1$, $b=\sqrt{7}$, and the area of $\triangle ABC$ is $\frac{\sqrt{3}}{2}$. Find the value of $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}}
A moving circle C is located inside the parabola $x^2=4y$ (where $x^2 \leq 4y$) and passes through the vertex of the parabola. Calculate the maximum value of the circumference of circle C. Express your answer as a multiple of pi (π).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
A group of people participate in various curriculums, with the following participation numbers: 50 practice Yoga (Y), 30 study Cooking (C), 20 study Weaving (W), 15 are involved in Pottery (P), and 10 participate in Dancing (D). The number of people participating in each pair of activities is as follows: 

- 20 study both Yoga and Cooking (Y∩C)
- 13 study both Yoga and Weaving (Y∩W)
- 9 practice Yoga and Pottery (Y∩P)
- 7 are involved in Yoga and Dancing (Y∩D)
- 10 study both Cooking and Weaving (C∩W)
- 4 study Cooking and Pottery (C∩P)
- 5 practice Cooking and Dancing (C∩D)
- 3 participate in both Weaving and Pottery (W∩P)
- 2 study Weaving and Dancing (W∩D)
- 6 participate in Pottery and Dancing (P∩D)

Additionally, the following number of people are involved in each trio of activities:

- 9 study Yoga, Cooking, and Weaving (Y∩C∩W)
- 3 are involved in Yoga, Cooking, and Pottery (Y∩C∩P)
- 2 practice Yoga, Cooking, and Dancing (Y∩C∩D)
- 4 study Yoga, Weaving, and Pottery (Y∩W∩P)
- 1 practices Yoga, Weaving, and Dancing (Y∩W∩D)
- 2 participate in Cooking, Weaving, and Pottery (C∩W∩P)
- 1 studies Cooking, Weaving, and Dancing (C∩W∩D)
- 3 are involved in Cooking, Pottery, and Dancing (C∩P∩D)

Finally, 5 people participate in all five activities (Y∩C∩W∩P∩D).

How many people study both Cooking and Pottery, but not any of the other activities?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
求最小的正整数n(≥3)，使得平面内任意无三点共线的n个点中，必有3点是非等腰三角形的顶点．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Tina wrote a positive number on each of five pieces of paper. She did not say which numbers she wrote, but revealed their pairwise sums instead: $17,20,28,14,42,36,28,39,25,31$. Which numbers did she write?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3, 11, 14, 17, 25
}
Brandon sold some geckos, parrots, and tarantulas to a local pet shop. He sold 6 geckos for $100 each, 3 parrots for $200 each, and 10 tarantulas for $50 each. The pet store offers a bulk discount of 10% on the purchase of 5 or more animals of the same type. The pet store sells the geckos for $5 more than 3 times their purchase price, the parrots for $10 more than 2 times their purchase price, and the tarantulas for $15 more than 4 times their purchase price. What is the total profit percentage of the pet store from selling all of these animals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{227.67\%}
Find the derivative of the matrix expression $b^tAc$ with respect to the matrix $A$. What is the result in matrix notation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b c^T}
Solve the congruence equation: \(2357x \equiv 3^{527} \pmod{206}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
已知点 \(A\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) 关于 \(y=k x\) 的对称点 \(A^{\prime}\) 落在圆 \((x-2)^{2}+y^{2}=1\) 上，求 \(k\) 的值。答案的形式为\(\frac{\sqrt{a}}{b}\)，请给出a+b的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the minimum number of terms needed in the Taylor series expansion of \( f(x) = \ln(x) \) around \( x = e \) to ensure that the error is less than \( 0.05 \) when \( |x-e| < 1 \). Use the error term \( R_{n+1}(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-e)^{n+1} \) and assume \( x = e-1 \) for the worst-case scenario.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
There are \( n \) planets, and Luke starts his journey on planet \( A \). Each planet, except for planets \( B \) and \( C \), has two unidirectional roads leading to other planets. From at most one of these roads, Luke can find a sequence of roads that brings him back to the planet he just left. Luke's journey ends when he reaches planet \( B \) or \( C \). When leaving a planet, he chooses one of the two roads with equal probability. If the probability that he arrives at planet \( B \) is \( \frac{1}{2016} \), what is the minimum value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
Given a winning probability of 50% with a return of 200% and a losing probability of 50% with a return of 50%, calculate the probability of doubling the initial money at some point using Kelly's criterion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of distinct isomorphism classes of graphs with 3 labeled vertices and allowing loops.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
How many homomorphisms are there from $S_4$ to the symmetric group of $3$ symbols, $S_3$, up to conjugation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the asymptotic distribution of the statistic \(8nB = - 8n\operatorname{ln}(\sqrt{0.2\hat{p}} + \sqrt{0.8(1-\hat{p})} )\), where \(\hat{p} = (1/n)\sum_i X_i = \bar{X}_n\) is the estimator of \(p\) from a random sample \(X_1, ..., X_n\) taken from a Bernoulli\((p)\) distribution. Assume \(H_0: p=0.2\) and \(H_1: p\neq 0.2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\chi^2_1}
Determine the radius of convergence, denoted as $R$, for the power series $$\sum_{n=1}^\infty \frac{\sin(n!)}{n!} x^n.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the limit \( \lim_{x \to 0}\frac{x}{\sqrt{1-e^{-x^2}}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the fractional equation $\frac{x}{x-2}-2=\frac{m}{x-2}$, find the value of $m$ when the equation has a root. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A certain DNA fragment consists of 500 base pairs, with A+T making up 34% of the total number of bases. If this DNA fragment is replicated twice, calculate the total number of free cytosine deoxyribonucleotide molecules required. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1320}
Let $\{W_t\}_{t\ge 0}$ be a standard 2D Brownian motion starting at $(1,1)$ in $\mathbb{R}^2$. Determine the probability that $W_t$ hits the positive x-axis before it hits the negative x-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}