inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the number of points at which the function \( f(x) = \begin{cases} x^2+1 & x \in \mathbb{Q}^c \\ \tan x & x \in \mathbb{Q} \end{cases} \) is continuous on the interval \([0, 4\pi]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In an examination consisting of 20 questions, each question can be scored as -1, 0, or 4. Determine the number of distinct total scores a student can achieve.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
Simplify the following complexity expression into a polynomial in terms of \( n \):
\[ O\left (\sum_{i=1}^n\sum_{j=1}^{i^2}\sum_{k=1}^{j^2}k\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^{11})}
Evaluate the path integral \(\int_\mathcal{C} (x^2+y^2) e^{-(x^2+y^2)} (1-2y^2) \,\mathrm{d}x + (x^2+y^2) e^{-(x^2+y^2)} (1-2x^2) \,\mathrm{d}y\) where \(\mathcal{C}\) is the lower semi-circle from \(A=(-1,0)\) to \(B=(1,0)\) with \(x,y\leq0\). Use the substitution \(y=\sin(t)\) when solving \(\int\frac{1-2y^2}{\sqrt{1-y^2}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{2}{3e}}
Find the determinant of the $n \times n$ matrix given by:
\[
D_n = \left| 
 \begin{matrix}
  n & -1 & -3 &  0 &  0 & \cdots & 0 & 0 &  0 \\
  n &  1 &  2 & -3 &  0 & \cdots & 0 & 0 &  0 \\
  n &  0 &  1 &  2 & -3 & \cdots & 0 & 0 &  0 \\
  n &  0 &  0 &  1 &  2 & \cdots & 0 & 0 &  0 \\
  n &  0 &  0 &  0 &  1 & \cdots & 0 & 0 &  0 \\
  \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots \\
  n &  0 &  0 &  0 &  0 & \cdots & 1 & 2 & -3 \\
  n &  0 &  0 &  0 &  0 & \cdots & 0 & 1 &  2 \\
  n &  0 &  0 &  0 &  0 & \cdots & 0 & 0 &  1 \\
 \end{matrix}
\right|
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
Let \( N \) be the number of ordered triple(A,B,C)of integers satisfying the conditions (a) \( 0 \leq  A < B < C \leq  {99} \) , (b) there exist integers \( a,b \) , and \( c \) , and prime \( p \) where \( 0 \leq  b < a < c < p \) , (c) \( p \) divides \( A - a \) , \( B - b \) , and \( C - c \) , and (d) each ordered triple(A,B,C)and each ordered triple(b,a,c)form arithmetic sequences. Find \( N \) ,
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
272
}
A \( 3 \times  3 \) square is partitioned into9unit squares. Each unit square is painted either
 white or black with each color being equally likely, chosen independently and at rando
 The square is the rotated \( {90}^{ \circ  } \) clockwise about its center, and every white square in a
 position formerly occupied by a black square is painted black. The colors of all other
 squares are left unchanged. What is the probability that the grid is now entirely black?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \dfrac{49}{512} \]

}
Given the second-order linear homogeneous differential equation \(2x^2y''-(x+x^2)y'+(x^2-2)y=0\) with two linearly independent solutions \(\phi_1(x)\) and \(\phi_2(x)\), where \(\phi_1(0)=0\), find the smallest positive integer \(n\) such that \(\displaystyle \lim_{x \to 0}x^n\frac{\phi_2(x)}{\phi_1(x)}=0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the rate of convergence with respect to \(\epsilon\) for the norm \(\|f_\epsilon-f\|_{L^1}\), where \(f_\epsilon = \phi_\epsilon * f\) and \(\phi_\epsilon = (1/\epsilon)\phi(x/\epsilon)\) is a mollifier.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(\epsilon)}
Determine the dimension of the moduli space of lines on a general quadric hypersurface in \(\mathbb{P}^n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 5}
Determine the points of discontinuity for the function \( f(x) \), where \[ f(x)=\lim_{a\to x}\left(\frac{\sin a}{\sin x}\right)^{\frac{x}{\sin a-\sin x}}. \] If \( \sin x \neq 0 \), the function evaluates to \( f(x)=e^{\frac{x}{\sin x}} \). Analyze the behavior of \( f(x) \) at points where \( \sin x = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x = n\pi}
Find the minimum value of the function $$f(a,b)= \left(\sqrt{\frac{a^2b}{ab^2+1}}+\sqrt{\frac{ab^2}{a^2b+1}}+\sqrt{\frac{1}{ab(a+b)}}\right)\sqrt{\frac{ab(a+b+15)+1}{ab}}$$ for positive values of $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Identify the topological space that corresponds to the quotient $SO(n+1)/O(n)$, where $SO(n)$ and $O(n)$ are considered as topological groups.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{RP}^n}
Given that the probability of 3 events occurring in a 90-minute interval is given by the Poisson distribution formula:

$$\Pr(X=3) = e^{-3\lambda}\frac{(3\lambda)^{3}}{3!}$$

where \(\lambda\) is the rate of events per 30 minutes, calculate the probability that all 3 events occur in the first 30 minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{27}}
Let \(X_1, X_2, \ldots, X_n \sim \text{Unif}[0, \beta]\). Define the estimators \(\hat{\beta_1} = (n+1)X_{\min}\) and \(\hat{\beta_2} = \frac{n+1}{n} X_{\max}\). Compute the ratio of the mean squared errors of \(\hat{\beta_1}\) and \(\hat{\beta_2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
Evaluate the surface integral \(\iint_{S} \frac{\mathbf{x}}{||\mathbf{x}||} \cdot d\mathbf{S}\), where \(S\) is the surface defined by \(x^3 + y^3 + z^3 = a^3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi a^2}
Let $F$ be a field, $K$ an extension field of $F$, and $a \in K$ such that the minimal polynomial of $a$ over $F$ is $x^n - a$, where $n$ is irreducible. Determine the degree of the minimal polynomial of $b^m$ over $F$, where $b$ is a root of $x^n - a$ and $m$ is a divisor of $n$. What is the degree of this minimal polynomial?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n}{m}}
Calculate the integral of $\operatorname{sgn}(\sin\pi/x)$ over the interval $(0,1)$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - 2\ln 2}
Suppose that $x$ and $y$ are complex numbers such that $x+y=1$ and $x^{20}+y^{20}=20$.  Find the sum of all possible values of $x^2+y^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
-90
}
Calculate the sum of the first six terms of the geometric series $3 + \left(\frac{1}{3}\right) + \left(\frac{1}{3}\right)^2 + \left(\frac{1}{3}\right)^3 + \left(\frac{1}{3}\right)^4 + \left(\frac{1}{3}\right)^5$. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{364}{81}}
Given that \( x, y \in G \) and \( xyx^{-1} = y^{17} \), find the order of \( y \) if \( |x| = 2 \), the order of \( y \) is odd, and \( y \neq e \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Compute \( \iint _S (y^2+x^2)\, dA \) by using a change of variables, where \( S \) is the region in the first quadrant bounded by the curves \( xy=1 \), \( xy=3 \), \( x^2-y^2=1 \), and \( x^2-y^2=4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
What is the inverse Laplace transform of the function $$\mathcal{L}\{f(t)\} = \ln\left(\frac{s^2+a^2}{s^2+b^2}\right)?$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{2\cos(bt) - 2\cos(at)}{t}\)}
Evaluate the limit: \[ \lim_{x\to +\infty} \frac{x^{\sqrt{\ln x}}\cdot(\sqrt{\ln x})^x}{(\sqrt{x})^{\ln x}\cdot(\ln x)^{\sqrt{x}}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{+\infty}
Consider a number $n$ randomly selected from the first 100 positive integers. Let $f(n)$ denote the product of all the positive divisors of $n$, and let $\varphi(n)$ denote the sum of all the positive divisors of $n$. What is the probability that $f(n) - \varphi(n)$ is an even number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{43}{100}}
For a binary number with W bits, initially set to all zeros, determine the number of random bit toggles required to reduce the average value of any bit to approximately 0.499.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.11W}
Calculate \((2023)^{2024} \mod{2025}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1741}
Evaluate the double integral \( S = \int_{-1}^{1} \int_{0}^{2\pi} \sqrt{\frac{1}{16}v^{2}} \,du \,dv \). What is the value of \( S \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Consider an equilateral triangular grid  $G$  with  $20$  points on a side, where each row consists of points spaced  $1$  unit apart. More specifically, there is a single point in the first row, two points in the second row, ..., and  $20$  points in the last row, for a total of  $210$  points. Let  $S$  be a closed non-self-intersecting polygon which has  $210$  vertices, using each point in  $G$  exactly once. Find the sum of all possible values of the area of  $S$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52\sqrt{3}}
Determine the maximal \( \delta > 0 \) such that \( |f(x)| < \frac{1}{2} \) for all \( x \in (1 - \delta, 1 + \delta) \), where \( f(x) = e \cdot \ln(x) - x + 1 \). If a maximal \( \delta \) does not exist, find any such \( \delta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.238}
A retiring employee receives an annual pension proportional to the square root of the number of years of his service. Had he served $a$ years more, his pension would have been $p$ dollars greater, whereas had he served $b$ years more $(b\ne a)$, his pension would have been $q$ dollars greater than the original annual pension. Find his annual pension in terms of $a,b,p$ and $q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{aq^2-bp^2}{2(bp-aq)}$}
Find the cardinality of the set of all tagged partitions of all closed intervals in \( \mathbb{R} \), denoted as \( \bigcup_{a \leq b \in \mathbb{R}} M_{a,b} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
A particle is moving randomly around a plane. It starts at $(0,0)$. Every second, it moves one unit randomly in a direction parallel to the $x$ or $y$ axis. At some time in the first hour, the particle was at the point $(2023,23)$. After $4092$ seconds, the particle is at $(x,y)$. Find the expected value of $x+y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2046}
Find the value of \( \lim _{n \rightarrow \infty} \mathbb{E}\left\{\left|1-X_{n}\right|\right\} \), given that \( X_{1}, X_{2}, \ldots \) are nonnegative random variables such that \( X_{n} \stackrel{\text{a.s.}}{\rightarrow} 0 \) and \( \lim_{n \rightarrow \infty} \mathbb{E}\{X_{n}\} = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Compute the limit: \[ \lim_{n \to \infty} \frac{(2n)!}{(2^n \cdot n!)^2} \] using bounding techniques or simplification methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In space, there are 3 planes and a sphere. How many distinct ways can a second sphere be placed in space so that it touches the three given planes and the first sphere? (In this problem, sphere touching is considered, i.e., it is not assumed that the spheres can only touch externally.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Compute the mixed partial derivative \( u_{yx} \) of the function \( u(x,y) = \frac{xy(x^2-y^2)}{(x^2+y^2)} \) at the point \((0,0)\), given that \( u(0,0) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let $f(x)$ be a cubic polynomial with rational coefficients. Determine the possible values of the degree of the field extension $[\mathbb{Q}(a_1,a_2,a_3):\mathbb{Q}]$, where $a_1, a_2, a_3$ are the roots of $f(x)$. Provide your answer as a set of possible values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{1, 2, 3, 6\}}
Determine the maximum value of the expression \(2m + 7n\) given that there exist distinct positive integers \(x_i\) \((1 \leq i \leq m)\) and \(y_j\) \((1 \leq j \leq n)\) such that the \(x_i\)'s are even, the \(y_j\)'s are odd, and \(\sum_{i = 1}^m x_i + \sum_{j=1}^n y_j = 1986\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{322}
We project one face of a regular octahedron perpendicularly onto the plane of the opposite face. What fraction of the area of the opposite face is covered by the projection?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
Find the indicial equation for the differential equation \(xy'' - 4y' + 5xy = 0\) at the regular singular point \(x=0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r(r - 5) = 0}
Evaluate the limit: $$\lim_{j,M \to \infty}\frac{1}{M}\prod_{k=1}^{M}\left[\prod_{n=j}^{2j}\left(1+\frac{1}{kn}\right)\right]^{\frac{1}{\ln 2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\gamma}}
Exercise 6. A set of $\mathrm{n}$ cells in an $\mathrm{n} \times \mathrm{n}$ grid is said to be distributed if it never includes two cells in the same row or column. In how many ways can one color some (possibly none) cells of an $\mathrm{n} \times \mathrm{n}$ grid so that all distributed sets contain the same number of colored cells?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n+1}-2}
Find the order of the element \(g^{-2}h^{-2}fh^{2}g^{2}\) in the symmetric group \(S_9\), where \(f=(9\ 6\ 3\ 5\ 1\ 4)(2\ 3\ 4\ 5\ 7\ 8)\), \(g=(4\ 6\ 7\ 3\ 1)(1\ 4\ 2\ 6\ 5\ 7\ 8)\), and \(h=(1\ 3\ 5\ 2)(2\ 3\ 6\ 7)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given a point P on the circle C: $x^2+y^2-4ax-2by-5=0$ (where $a>0$, $b>0$), if the symmetric point of P with respect to the line $x+2y-1=0$ is still on the circle C, then the minimum value of $\frac{4}{a} + \frac{1}{b}$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
How many functions $f:\{1,2,3,4,5\} \rightarrow\{1,2,3,4,5\}$ have the property that $f(\{1,2,3\})$ and $f(f(\{1,2,3\}))$ are disjoint?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{94}
Find the radius of convergence for the series \( 1 - x^2 + x^4 - x^6 + \cdots = \sum_{n=0}^{\infty} a_n (x-3)^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{10}}
Find the maximum value of the function \( f(x) = \lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^n \sqrt{k(n-k)} \binom{n}{k} x^k (1-x)^{n-k} \) on the interval \([0, 1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Evaluate the expression \( \frac{\sum_{k=1}^{n} \binom{n}{2k-1}(n-(2k-1))^3}{n^2(n+3)2^n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{16}}
Calculate the line integral \( \int_{\partial F} 2xy\, dx + x^2 \, dy + (1+x-z) \, dz \) for the intersection of the surfaces \( z=x^2+y^2 \) and \( 2x+2y+z=7 \). The curve should be traversed clockwise with respect to the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18\pi}
Let $m$ be a positive integer, and let $T$ denote the set of all subsets of $\{1, 2, \dots, m\}$. Call a subset $S$ of $T$ $\delta$-[I]good[/I] if for all $s_1, s_2\in S$, $s_1\neq s_2$, $|\Delta (s_1, s_2)|\ge \delta m$, where $\Delta$ denotes the symmetric difference (the symmetric difference of two sets is the set of elements that is in exactly one of the two sets). Find the largest possible integer $s$ such that there exists an integer $m$ and $ \frac{1024}{2047}$-good set of size $s$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2048}
A captain has 8 crew members and wants to secure a safe such that it can only be opened by a group of at least 5 crew members. He distributes keys to the crew members and locks the safe with several locks. Determine the minimum number of locks required so that any group of 5 or more crew members can open the safe, but no group of 4 or fewer crew members can open it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
There are three committees of bureaucrats. It is known that for each pair of bureaucrats from different committees, among the members of the remaining committee, there are exactly 10 bureaucrats who know both of them, and exactly 10 bureaucrats who do not know both of them. Find the total number of bureaucrats in the committees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Determine the values of \( \alpha > 0 \) for which the series \[ \sum_{n_{1}=1}^{\infty}\cdots\sum_{n_{k}=1}^{\infty}\frac{1}{\left(n_{1}^{2}+\dots+n_{k}^{2}\right)^{\alpha/k}} \] converges, where \( k > 0 \) is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > \dfrac{k^2}{2}}
Given that the function f(x) is defined on (0, +∞) and is an increasing function, and it satisfies the equation f(x) * f(f(x) + 1/x) = 1, find the exact value of f(1).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1 + \sqrt{5}}{2}}
设$P ( x )$为2n次多项式，$P ( 0 ) = P ( 2 ) = \cdots = P ( 2 n ) = 0 , P ( 1 ) =P ( 3 ) = \cdots = P ( 2 n - 1 ) = 2 , P ( 2 n + 1 ) = - 3 0$，求n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n=2}
Find the second partial derivative of the expression \( A^{ij}y^ix^j + B^{ij}x^iy^j \) with respect to \( \bf x \) and \( \bf y \). Here, \( i,j \) follow the Einstein summation convention, \( A, B \) are constants, \( \bf x = [x^1, x^2, \dots, x^n]^T \), and \( \bf y = [y^1, y^2, \dots, y^n]^T \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A^{ji} + B^{ij}}
5. A school has 3 teachers who can teach English, 2 teachers who can teach Japanese, and 4 teachers who can teach both English and Japanese. Now, 3 English teachers and 3 Japanese teachers are to be selected to participate in off-campus tutoring during the holiday. How many ways are there to select them?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216}
Does the limit $\lim _{x\to 0} \frac 1x \int_0^x \left|\cos \frac 1t \right| dt$ exist? If it does, what is the value?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{2}{\pi}$}
In how many ways can 20 people, labeled A to T, be divided into 4 identical non-empty groups such that A and B are in the same group, and C and D are not in the same group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8460859965}
Determine the number of distinct normal subgroups $H$ in the free group of rank 2, $F_2$, such that the quotient group $F_2/H$ is isomorphic to the Klein four-group $V_4$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the line integral \(\int_C{\omega}\) where \(\omega=\frac{dx \wedge dy}{x^2+y^2}\) and \(C(t_1,t_2)=(t_1+1)(\cos(2\pi t_2),\sin(2\pi t_2)) : I_2 \rightarrow \mathbb{R}^2 - \text{{(0,0)}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi \ln 2}
In a sequence of independent flips of a fair coin, determine the minimum number of flips required to have at least a 90% probability of obtaining three consecutive heads.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Evaluate the limit using a power series approach: $$\lim_{x \to 0} \left[\frac{2+x}{2x*(e^x-1)}-\frac{1}{x^2}\right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{6}}
Determine the accurate rate of convergence of $\bar{X}_n - \mu$ to 0, expressed as $o_p(n^\alpha)$, where $X \sim N(\mu, \sigma^2)$ and $X_1, \cdots, X_n$ are samples from $X$. Express your answer in terms of $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2}}
For which values of \( x \in \mathbb{R} \) is the integral \( \int_0^\infty e^{-t}t^{x-1} \, dt \) convergent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \infty)}
Find the derivative of $X^Ta^TaX$ with respect to $X$, where $X$ is a matrix and $a$ is a vector.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2a^TaX}
Given that the domain of the function $f(x^2-1)$ is $[4,9]$, determine the domain of the function $f(\log_2x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2^{15}, 2^{80}]}
Evaluate the limit:

$$\lim_{t \to 0} \frac{f(t^2, \cosh t, e^t)}{f(t, \cos t, \cosh t)}$$

for a continuously differentiable function $f: \mathbb{R}^3 \to \mathbb{R}$ with the following conditions at the point $(0,1,1)$:

$$f(0,1,1) = 0, \quad \frac{\partial f}{\partial x}(0,1,1) = 1, \quad \frac{\partial f}{\partial y}(0,1,1) = 2, \quad \frac{\partial f}{\partial z}(0,1,1) = 3.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given an arithmetic sequence $\{ a_n \}$, where $S_n$ denotes the sum of the first $n$ terms, $S_4 = \pi$ (where $\pi$ represents the value of pi), and $a_4 = 2a_2$. Now, if we randomly select an element from the first 30 terms of this sequence, what is the probability that the cosine of this element is negative?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{15}}
Find the radical of the ideal $J = (ab, ac-bc)$. Express your answer in terms of generators.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(ab, ac, bc)}
Given that $0 < A < 90^{\circ}$ and \( \sin A + \cos A + \tan A + \sec A + \operatorname{cosec} A + \cot A = 7 \), and that \( \sin A \) and \( \cos A \) are roots of the quadratic equation \( 4x^2 + 3x + k = 0 \), find the value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{31}}
In a convex quadrilateral $ABCD$, determine the number of points in the plane that return to their original positions after being reflected successively across the lines $AB$, $BC$, $CD$, and $DA$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Solve the recurrence relation: \[ T(n) = T(n/3) + T(n/2) + n^3 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n^3)}
Use the divergence theorem to compute \( \int \int \limits_S F \cdot n \,dS \) where \( n \) is the outward normal. The surface \( S \) is defined by \( x^2 + y^2 = z^2 \) with \( z \in [0,1] \), and the vector field is \( F(x,y,z) = (x, xy, z) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{3}}
Let \(a\) be a real number in the interval \([0,314]\) such that \(\displaystyle \int^{3\pi+a}_{\pi+a}|x-a-\pi|\sin \frac{x}{2}dx=-16.\) Determine the number of such real values of \(a.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}