inclusionAI/Ring-lite-rl-data huggingface dataset:
已知 a_1,a_2,\cdots,a_{22}\in[1,2]，求\df{\ds{i=1}{22}a_ia_{i+1}}{\left(\ds{i=1}{22}a_i\right)^2}的最大值，其中 a_{23}=a_1.（王正供题）原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Consider a symmetric random walk on a regular hexagon with vertices labeled as $A, B, C, D, E, F$ and a center point $O$. The walk starts at vertex $A$. At each step, if the walk is at a vertex, it moves to one of its two neighboring vertices or the center $O$ with equal probability of $\frac{1}{3}$. Calculate the probability that the walk returns to vertex $A$ without passing through the center $O$ before its first return to $A$. Express your answer as a probability.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{27}}
On a $4\times 4\times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing
$A$, $B$, and $C$ is closest to 

$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.1}
Find the last three digits of the sum of all positive integers less than \( 10^6 \) that can be expressed as \( m! + n! \), where \( m \) and \( n \) are nonnegative integers.
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Answer: \boxed{130}
What is the maximum number of numbers we can choose from the first 1983 positive integers such that the product of any two chosen numbers is not among the chosen numbers?
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Answer: \boxed{1939}
There are relatively prime positive integers $s$ and $t$ such that $$\sum_{n=2}^{100}\left(\frac{n}{n^2-1}- \frac{1}{n}\right)=\frac{s}{t}$$ Find $s + t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25249}
Find the limit $$\lim_{x\to\frac{\pi}{4}} \frac{1-\tan^2(x)}{\sqrt{2}\cos(x)-1}$$ without using L'Hôpital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given positive integers $a_1, a_2, \ldots, a_{2006}$ such that the ratios $\frac{a_1}{a_2}, \frac{a_2}{a_3}, \ldots, \frac{a_{2005}}{a_{2006}}$ are pairwise distinct, determine the minimum number of distinct positive integers in the set $\{a_1, a_2, \ldots, a_{2006}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
Given the equation $2^7\cos^5x \cdot \sin^3x = a\sin8x - b\sin 6x + c\sin 4x + d\sin 2x$, where $x$ is a real number, find the value of $a^4 + b^4 + c^4 + d^4$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1329}
55 boxers participated in a knockout tournament. The matches were held consecutively. It is known that in each match, the number of previous victories of the participants differed by no more than 1. What is the maximum number of matches the tournament winner could have fought in?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
If
 \( \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{\frac{1}{1} + \frac{1}{2} + \cdots  + \frac{1}{n}}{\left( \begin{matrix} n + {100} \\  {100} \end{matrix}\right) } = \frac{p}{q} \)
 for relatively prime positive integers \( p,q \) , find \( p + q \) .
 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
9901}
There is a 100-digit number consisting of ones and twos. It is allowed to swap the first five digits with the next five digits in any sequence of ten consecutive digits. Two such numbers are called similar if one of them can be obtained from the other by several such operations. What is the maximum number of pairwise dissimilar numbers that can be chosen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21^5}
Given points A, B, and C on the surface of a sphere O with a radius of 2, and the angles formed by OA, OB, and OC are equal pairwise. When the volume of the triangular pyramid O-ABC is maximum, the area of the circular cross-section obtained by the plane ABC intersecting sphere O is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8π}{3}}
Evaluate the limit of the improper integral: \[ \lim_{x\to \infty} \int_1^x x\,e^{t^2-x^2} \,dt. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given the conditions:
\[
\operatorname{gcd}(63, n+120) = 21
\]
\[
\operatorname{gcd}(n+63, 120) = 60
\]
find the smallest positive integer n greater than 1000 that satisfies both conditions, and then calculate the sum of its digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Find all integer solutions \((a, b, c)\) such that \(3^a + 5^b = 7^c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Let \( f : \{ 1,2,\ldots ,{2019}\}  \rightarrow  \{  - 1,1\} \) be a function,such that
 for every \( k \in  \{ 1,2,\ldots ,{2019}\} \) , there exists an
 \( \ell  \in  \{ 1,2,\ldots ,{2019}\} \) such that
 \[ \mathop{\sum }\limits_{{i \in  \mathbb{Z} : \left( {\ell  - i}\right) \left( {i - k}\right)  \geqslant  0}}f\left( i\right)  \leqslant  0. \]
 Determine the maximum possible value of
 \[\mathop{\sum }\limits_{{i \in  \mathbb{Z} : 1 \leqslant  i \leqslant  {2019}}}f\left( i\right) . \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{673

}
\[\frac{\tan 96^{\circ} - \tan 12^{\circ} \left( 1 + \frac{1}{\sin 6^{\circ}} \right)}{1 + \tan 96^{\circ} \tan 12^{\circ} \left( 1 + \frac{1}{\sin 6^{\circ}} \right)} =\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
Find the differentiation of the Lagrangian function \( L(X, \lambda) = \mathop{Trace}(XH) + \lambda \mathop{Trace}(X^{\dagger}X) \) with respect to \( X^* \), where all matrices are complex, \( \dagger \) denotes the conjugate transpose, \( * \) denotes the conjugate, and \( \mathop{Trace} \) denotes the trace of a matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda X}
Given that a, b, and c are the roots of the cubic equation dx^3 - ex^2 + fx - g = 0, where d, e, f, and g are real numbers, find the expression for the sum of squares of the roots, a^2 + b^2 + c^2, in terms of d, e, f, and g.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{e^2 - 2df}{d^2}}
Given $m$ orthonormal vectors $u_1, ..., u_m \in \mathbb{R}^n$ where $m < n$, find a linear transformation $W: \mathbb{R}^n \rightarrow \mathbb{R}^m$ such that the vectors $Wu_1, ..., Wu_m \in \mathbb{R}^m$ are orthonormal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{W = U^\top}
Calculate \(1234^{5678} \mod 2345\) using the repeated squaring method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{851}
Determine the dimension of the quotient ring \( \mathbb{Z}[x,y,z]/\langle xy+2, z+4 \rangle \), where \( \mathbb{Z} \) is the ring of integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral $$\int\limits_0^{+\infty}\frac{\ln{x}}{x^2-x+1}\,\mathrm{d}x\,.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
John learned that Lisa scored exactly 85 on the American High School Mathematics Examination (AHSME). Due to this information, John was able to determine exactly how many problems Lisa solved correctly. If Lisa's score had been any lower but still over 85, John would not have been able to determine this. What was Lisa's score? Remember, the AHSME consists of 30 multiple choice questions, and the score, $s$, is given by $s = 30 + 4c - w$, where $c$ is the number of correct answers, and $w$ is the number of wrong answers (no penalty for unanswered questions).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
Q. A light source at the point $(0, 16)$ in the coordinate plane casts light in all directions. A disc (circle along with its interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the X-axis. The length of the shadow can be written in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is square-free. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
In a flow network with a source, a sink, and $n$ internal nodes, each arc from the source to an internal node and from an internal node to the sink has a capacity of 2. Determine the number of different cuts with minimum capacity in this network.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n}
Given a positive integer \( n (n \geqslant 2) \). Find the largest positive real number \( C \) such that for any \( x_{1}, x_{2}, \cdots, x_{n} \), the following inequality holds:
$$
\sum_{i=1}^{n} x_{i}^{2} \geqslant \frac{1}{n+1}\left(\sum_{i=1}^{n} x_{i}\right)^{2}+C\left(\sum_{i=1}^{n} i x_{i}\right)^{2}.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{12}{n(n+1)(n+2)(3n+1)}}
In a javelin throwing competition, each athlete throws independently, and the distances are exponentially distributed with parameter \( \lambda \). The first competitor will lead the competition until the \( N \)-th opponent throws the javelin a greater distance and takes the lead. Calculate the expected value \( E(N) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Solve for \( x \) in the equation:
\[
26280 = 2 \times \sqrt{\frac{149,598,000,000 - x}{1.4}} + \frac{x}{10,217,588,910,387,300,000}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{149356276560}
设M={1,2,3,⋯,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≠b+c.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Joanie takes a $\$6,\!000$ loan to pay for her car.  The annual interest rate on the loan is $12\%$.  She makes no payments for 4 years, but has to pay back all the money she owes at the end of 4 years. How much more money will she owe if the interest compounds quarterly than if the interest compounds annually?  Express your answer as a dollar value to the nearest cent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$187.12}
Evaluate the integral \( \int^{1}_{0}\left(\frac{1}{1-x}+\frac{1}{\ln x}\right)dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\gamma}
Let N be the greatest positive integer that can be expressed using all seven Roman numerals $I, V, X, L, C,D$, and $M$ exactly once each, and let n be the least positive integer that can be expressed using these numerals exactly once each. Find $N - n$. Note that the arrangement $CM$ is never used in a number along with the numeral $D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{222}
James goes to school at 12 pm. On Tuesday, James's classes are in the following order : Maths, History, Geography, Science and Music. Just now James's Science class got over. Can you estimate the time now in pm?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Jake's dad can drive the distance from their house to the water park in 30 minutes. He spends half that journey driving at a certain speed and the other half driving 60 miles per hour on the highway. If Jake can bike 11 miles per hour, it will take him 2 hours to bike to the water park. What is the speed of Jake's dad during the first half of the journey?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Draw 63 lines on a coordinate plane: \( y = b, y = \sqrt{3} x + 2b, y = -\sqrt{3} x + 2b \), where \( b = -10, -9, -8, \cdots, 8, 9, 10 \). Determine the number of equilateral triangles with side length \( \frac{2}{\sqrt{3}} \) formed by these lines.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{660}
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 \( BF=BD \). Calculate the measure of \( \angle EDF \) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50^\circ}
Determine the multiplicities of the zeros of the polynomial \( p(z) = 1 + 2z^4 + \frac{7}{10}z^{10} \) within the unit disc by using the greatest common divisor of \( p(z) \) and its derivative.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $M$ be a set of six distinct positive integers whose sum is $60$. These numbers are written on the faces of a cube, one number to each face. A move consists of choosing three faces of the cube that share a common vertex and adding $1$ to the numbers on those faces. Determine the number of sets $M$ for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Given a sequence \( \{a_n\} \) of real numbers for \( n \ge 1 \) such that:
\[ a_{n+2} = a_{n+1} - a_n \]
\[ \sum_{n=1}^{999} a_n = 1003 \]
\[ \sum_{n=1}^{1003} a_n = -999 \]
Find the value of \( \sum_{n=1}^{2002} a_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2002}
There are eight identical Black Queens in the first row of a chessboard and eight identical White Queens in the last row. The Queens move one at a time, horizontally, vertically or diagonally by any number of squares as long as no other Queens are in the way. Black and White Queens move alternately. What is the minimal number of moves required for interchanging the Black and White Queens?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Given \( f(x) = \sum_{k=1}^{2017} \frac{\cos k x}{\cos^k x} \), find \( f\left(\frac{\pi}{2018}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Evaluate the limit \( \lim_{n\to\infty} \sum\limits_{i=1}^{n^2} \frac{e^{i/n}}{ne^n} \) by recognizing it as a Riemann sum of an integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the infinite series $\sum_{k=1}^{\infty} \frac{\Gamma(k)}{\Gamma(p+k+1)}$, where $\Gamma(n)$ is the Gamma function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{p \, \Gamma(p+1)}\)}
What the smallest number of circles of radius $\sqrt{2}$ that are needed to cover a rectangle of size $5\times 3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In a triangular pyramid, all faces are equilateral triangles. A plane passing through one side of the base divides the volume of the pyramid in the ratio 1:3, counting from the base. Find the angle between this plane and the base plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arctan \frac{\sqrt{2}}{5}}
Find the smallest natural number \( k \) such that among \( k \) different and pairwise relatively prime natural numbers less than 2021, there is at least one prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let $t \in \mathbb{R}$ and consider the points $$A(t) = (t,t^3,t), \, B(t) = (t,t,t), \, C(t) = (0,2t,t)$$ Find the volume of $$\Omega = \bigcup_{t \in [0,1]} T(t)$$ where $T(t)$ is the triangle generated by the three points $A(t), \, B(t), \, C(t)$. If the points are aligned the set $T(t)$ is just assumed to be the empty set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/15}
Consider the hyperbola \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \). A tangent at any point \( P \) on this hyperbola meets another hyperbola at points \( A \) and \( B \). If \( P \) is the midpoint of \( AB \) for every choice of \( P \), find the floor of the sum of all possible values of the eccentricities of this new hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let triangle $ABC$ have side lengths $AB = 13$, $BC = 14$, $AC = 15$. Let $I$ be the incenter of $\triangle ABC$. The circle centered at $A$ with radius $AI$ intersects the circumcircle of $\triangle ABC$ at points $H$ and $J$. Let $L$ be a point that lies on both the incircle of $\triangle ABC$ and line $HJ$. If the minimal possible value of $AL$ is $\sqrt{n}$, where $n \in \mathbb{Z}$, find $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Let \( A_{1} A_{2} \ldots A_{6} \) be a regular hexagon with side length \( 11 \sqrt{3} \), and let \( B_{1} B_{2} \ldots B_{6} \) be another regular hexagon completely inside \( A_{1} A_{2} \ldots A_{6} \) such that for all \( i \in \{1,2, \ldots, 5\} \), \( A_{i} A_{i+1} \) is parallel to \( B_{i} B_{i+1} \). Suppose that the distance between lines \( A_{1} A_{2} \) and \( B_{1} B_{2} \) is 7, the distance between lines \( A_{2} A_{3} \) and \( B_{2} B_{3} \) is 3, and the distance between lines \( A_{3} A_{4} \) and \( B_{3} B_{4} \) is 8. Compute the side length of \( B_{1} B_{2} \ldots B_{6} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \sqrt{3}}
求 1,2,\cdots,2014 的排列数目的最大值，使得对任意不同的 a,b\in\{1,2,\cdots,2014\}，最多在一个排列中，b 恰位于 a 的后面.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
In how many ways can you divide the set of eight numbers $\{2, 3, \ldots, 9\}$ into $4$ pairs such that no pair of numbers has a greatest common divisor (gcd) equal to $2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
In a social party, there are 8 girls and 7 boys sitting around a round table. Calculate the average number of pairs of seats that are adjacent and consist of one girl and one boy.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the function f(x) = 2/(x+1) for a positive number x, calculate the sum of f(100) + f(99) + f(98) + ... + f(2) + f(1) + f(1/2) + ... + f(1/98) + f(1/99) + f(1/100).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{199}
Given a tetrahedron \(ABCD\), all of whose faces are similar right-angled triangles with acute angles at vertices \(A\) and \(B\). The edge \(AB\) is equal to 1. Find the length of the shortest edge of the tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{\sqrt{5}-1}{2}\right)^{\frac{3}{2}}}
Calculate the pullback of the 1-form \( \omega = -\frac{y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy \) under the transformation \( f: U \to \mathbb{R}^{2} \) given by \( f(r,\theta) = (r\cos\theta, r\sin\theta) \), where \( U = \{(r,\theta) \mid r>0,0<\theta<2\pi\} \). Express the result in terms of \( r \) and \( \theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d\theta}
In a basket, there are 17 red peaches, 16 green peaches, and 12 blue peaches. How many more red peaches are there than the combined number of green and blue peaches in the basket?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let \( P(x) \) be a polynomial of degree \( n \) such that for \( k = 1, 2, 4, \ldots, 2^{n} \), we have \( P(k) = \frac{1}{k} \). Find \( P(0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 - \dfrac{1}{2^n}}
Given that Raashan, Sylvia, and Ted each start with $1, a bell rings every 15 seconds, and each player who has money gives $1 to another player chosen at random, what is the probability that after the bell has rung 2019 times, each player will have $1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Students arrive at a help centre according to a Poisson process with rate $r$. When there are $n \geq 2$ students in the centre, the first two students to leave do so at random $Exp (4r)$ times. Currently, there is only one student in the centre. What is the expected time until there are three students in the centre?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{10}{r}}
If $f(x) = \frac{2^x}{2^x+1},$ find the value of $\sum \limits_{i=1}^{2023} f\left(\frac{i}{2023}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1011 + 1/2}
Evaluate the limit: $\lim_{x \to 0^+} x^{x-c}$, where $c$ is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a triangle $ABC$ with an area of $1$. Let $A'$, $B'$, and $C'$ be the midpoints of the sides $[BC]$, $[CA]$, and $[AB]$ respectively. What is the minimal possible area of the common part of the two triangles $A'B'C'$ and $KLM$, if the points $K$, $L$, and $M$ are lying on the segments $[AB']$, $[CA']$, and $[BC']$ respectively?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that the four vertices A, B, C, D of the tetrahedron A-BCD are all on the surface of the sphere O, AC ⊥ the plane BCD, and AC = 2√2, BC = CD = 2, calculate the surface area of the sphere O.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16\pi}
Let $S$ and $T$ be Fredholm operators on a separable complex Hilbert space $\mathcal{H}$ such that the tensor product $S \otimes T$ is also a Fredholm operator on $\mathcal{H} \otimes \mathcal{H}$. Determine the index of $S \otimes T$, given by:
\[ j(S \otimes T) = ? \]
where the index $j(T)$ is defined as $\operatorname{dim}(\ker(T)) - \operatorname{dim}(\ker(T^{*}))$. Provide your answer in terms of $j(S)$ and $j(T)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{j(S) \cdot j(T)}
Let $D = \{(x, y, z) \mid x^2 + y^2 \leq 1, z = 0\}$ be a subset of $\mathbb{R}^3$. Consider the linear transformation $T : \mathbb{R}^3 \to \mathbb{R}^3$ given by $T(x, y, z) = (2x - y + z, x - y + 10z, 3x - 7y + 2z)$. Determine the area of the set $T(D)$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \sqrt{138}}
A list of five positive integers has mean $12$ and range $18$. The mode and median are both $8$. How many different values are possible for the second largest element of the list?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$6$}
Given a Noetherian domain $A$ of dimension 1 and a non-zero prime ideal $p$, determine the maximal ideal of the Noetherian local domain $A/p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0)}
Given a cube \(ABCD - A_1B_1C_1D_1\) with edge length 1, a sphere is constructed with center at vertex \(A\) and radius \(\frac{2 \sqrt{3}}{3}\). Find the length of the curve formed by the intersection of the sphere's surface with the cube's surface.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\sqrt{3}\pi}{6}}
Find the limit of the expression \( \frac{\sin(ax^2 + bx + c)}{(x-\beta)^2} \) as \( x \) approaches \( \beta \), where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a}