inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $s_1, s_2, s_3$ be the three roots of the polynomial $x^3 + x^2 + \frac{9}{2}x + 9$.

Find the value of $a + b + c$ if the expression $\prod_{i=1}^{3}(4s^4_i + 81)$ can be written as $2^a3^b5^c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
The original price of an article is increased by a certain percentage, and then the new price is increased by the same percentage. This is equivalent to increasing the original price by 16.64 percent. What is the percentage increase applied twice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\%}
Find the limit of the following expression as x approaches 2:
$$\lim _{ x\rightarrow 2 }{ \frac { f\left( x \right) -2 }{ \sqrt { x-1 } -1 } } $$
Given that
$$\lim_{x\to2}\frac{\ xf(x)-4}{x-2}=3.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Jenny leaves her house at 8:00 with some cupcakes for her daughter's birthday party at school. She jogs to the school at 15 miles per hour. Half an hour later, her wife Anna realizes that Jenny used peanut butter frosting because she forgot one of the kids in the class had a severe peanut allergy. Jenny doesn't have her cell phone, so Anna leaves the house driving at 45 miles her hour to catch Jenny and warn her. How long does Anna spend traveling in minutes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A plane is passed through the midpoint of an edge of a regular tetrahedron and is perpendicular to an adjacent edge. Find the area of the resulting cross-section if the edge of the tetrahedron is equal to \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^2 \sqrt{2}}{16}}
A company needs to transport at least 180 tons of goods every day. The company has 8 Type A trucks with a capacity of 6 tons each and 4 Type B trucks with a capacity of 10 tons each. Type A trucks can make 4 round trips per day, and Type B trucks can make 3 round trips per day. The daily cost for a Type A truck is 320 yuan, and for a Type B truck, it is 504 yuan. Determine the minimum daily expense for the company by reasonably allocating the vehicles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2560}
Given the vertices of triangle ABC as A(2,0), B(0,4), and C(4,1), where AE is the angle bisector of angle BAC, calculate the coordinates of a directional vector of line AE. Express your answer as a vector in the form (x, y).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,3)}
Let $A B C$ be a right triangle with $\angle A=90^{\circ}$. Let $D$ be the midpoint of $A B$ and let $E$ be a point on segment $A C$ such that $A D=A E$. Let $B E$ meet $C D$ at $F$. If $\angle B F C=135^{\circ}$, determine $B C / A B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{13}}{2}}
Given a sequence ${a_n}$ with first term $a_1 = a$, its sum of the first $n$ terms is $S_n$, and it satisfies the equation $S_n + S_{n-1} = 3n^2$ for $n \geq 2$. If $a_n < a_{n+1}$ holds true for any $n \in \mathbb{N}^*$, find the range of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\frac{9}{4}, \frac{15}{4})}
Let \( p \) be an odd prime. Find the number of \( p \)-element subsets \( A \) of \( \{1, 2, \ldots, 2p \} \) such that the sum of all elements of \( A \) is divisible by \( p \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{p}\left(\binom{2p}{p} - 2\right) + 2}
Unit square $ABCD$ is divided into four rectangles by lines $EF$ and $GH$. Line $EF$ is parallel to $AB$, and line $GH$ is parallel to $BC$. The point $P$ is the intersection of $EF$ and $GH$. It is given that $BF = \frac{1}{4}$ and $BF + DH = FH$. Calculate the nearest integer to the degree measure of $\angle FAH$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
4. An infinite sequence of integers, $a_{0}, a_{1}, a_{3} \ldots$, with $a_{0}>0$, has the property that for any $n \geq 0, a_{n+1}=a_{n}-b_{n}$, where $b_{n}$ is the number having the same sign as $a_{n}$, but having the digits written in the reverse order. For example if $a_{0}=1210$, $a_{1}=1089$ and $a_{2}=-8712$, etc. Find the smallest value of $a_{0}$ so that $a_{n} \neq 0$ for all $n \geq 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
Consider a 4x4 grid of points (equally spaced). How many rectangles, of any size, can be formed where each of its four vertices are points on this grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
A very large number of small particles forms a spherical cloud. Initially, they are at rest, have uniform mass density per unit volume $\rho_0$, and occupy a region of radius $r_0$. The cloud collapses due to gravitation; the particles do not interact with each other in any other way.

Find the time that passes until the cloud collapses fully, expressed in the form $\frac{k}{\sqrt{G \rho_0}}$. Please give the value of $10000k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5427}
If $ab \gt 0$, then the value of $\frac{a}{|a|}+\frac{b}{|b|}+\frac{ab}{{|{ab}|}}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Determine the convergence and find the limit of the sequence $b_n = \frac{n+2}{2^{n+2}}\left(\frac{2}{1}+\frac{2^2}{2}+...+\frac{2^{n+1}}{n+1}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $I$ be the incenter of $\triangle ABC$. If $BC = AC + AI$ and $\angle ABC - \angle ACB = 13^\circ$, find $\angle BAC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96.5^\circ}
For each of the $9$ positive integers $n, 2n, 3n, \ldots, 9n$, Alice takes the first decimal digit (from the left) and writes it onto a blackboard. She selected $n$ such that among the nine digits on the blackboard, there is the least possible number of different digits. What is this number of different digits equal to?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
1. Find the smallest 10-digit number, the sum of whose digits is not less than that of any smaller number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1899999999}
Solve the initial value problem: $$\dfrac{d^2y}{dx^2}=\left( \dfrac{dy}{dx} \right)^2 + 1$$ with initial conditions \( y(0)=0 \) and \( \dfrac{dy}{dx}(0)=0 \) on the interval \( \left( \dfrac{\pi}{2},-\dfrac{\pi}{2}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\ln(\cos x)}
In the sequence listing all positive rational numbers not exceeding \(\frac{1}{2}\), where fractions are ordered by increasing denominator and then by increasing numerator, find the sum of the numerator and denominator of the 2012th fraction. If the fraction is \(\frac{m}{n}\) with \(m\) and \(n\) relatively prime, compute \(m+n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
Determine the value of \( \lim\sup_{n \to \infty} \frac{E_n}{n!} \), where \( E_n \) is the number of permutations in \( S_n \) that can be obtained by using each transposition exactly once.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Solve the recurrence relation using the Divide and Conquer method: \[ T(n) = T\left(\frac{n}{7}\right) + T\left(\frac{11n}{14}\right) + n \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n)}
Calculate the work done by the force field \( F(x,y,z) = e^{y+2z}(i+xj+2xk) \) as a particle moves from \((0,0,0)\) to \((1,1,1)\) along the curve defined by \(x=t\), \(y=t^2\), and \(z=t^4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^3}
5 级方阵 A 满足: 第一行元素均为 1, 第二行元素从左至右依次为 $1,-1,1,-1,1$,且 $\operatorname{trace}\left(A A^{\prime}\right)=28$. 记 $\operatorname{det} A$ 的最大值为 M, 则 $[M]=\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Let  $ABC$  be an acute-angled triangle with altitude  $AT = h$ . The line passing through its circumcenter  $O$  and incenter  $I$  meets the sides  $AB$  and  $AC$  at points  $F$  and  $N$ , respectively. It is known that  $BFNC$  is a cyclic quadrilateral. Find the sum of the distances from the orthocenter of  $ABC$  to its vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2h}
In the linear model \( Y_i = \beta + U_i \) for \( i = 1, 2, 3 \), with conditions \( E(U_1) = E(U_2) = E(U_3) = 0 \), \( \text{Var}(U_1) = 1 \), \( \text{Var}(U_2) = 2 \), and \( \text{Var}(U_3) = 4 \), determine the best linear unbiased estimate (BLUE) of the variance of \( \beta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{7}}
Let  $\underline{xyz}$  represent the three-digit number with hundreds digit  $x$ , tens digit  $y$ , and units digit  $z$ , and similarly let  $\underline{yz}$  represent the two-digit number with tens digit  $y$  and units digit  $z$ .  How many three-digit numbers  $\underline{abc}$ , none of whose digits are 0, are there such that  $\underline{ab}>\underline{bc}>\underline{ca}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Given that the function f(x) defined on R satisfies: the graph of the function y=f(x-1) is symmetrical about the point (1,0), and f(x+2)=f(x) always holds when x≥0, and f(x)=e^x-1 when x∈[0,1], find the value of f(-2017)+f(2018)=____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1-e}
For a Brownian Motion starting at $0$, calculate the probability that $X(2T) > 0$ and $X(T) > 0$. Express your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{8}}
If the sufficient and necessary condition for the inequality $|x-m| < 1$ to hold is $\frac{1}{3} < x < \frac{1}{2}$, then the range of values for the real number $m$ is        .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2} \leq m \leq \frac{4}{3}}
Let $N=10^6$. For which integer $a$ with $0 \leq a \leq N-1$ is the value of \[\binom{N}{a+1}-\binom{N}{a}\] maximized?

[i]Proposed by Lewis Chen[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{499499}
The average height of 30 students in a class was calculated as a certain value. It was later found that the height of one of the students in the class was incorrectly written as 151 cm, whereas the actual height was 136 cm. The actual average height of the students in the class was 174.5 cm. What was the initially calculated average height?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{174}
Calculate the surface integral \( \iint_Y G \cdot N\, dS \) where \( Y \) is the surface defined by \((x+y)^2+(y+z)^2=4+z^2\) for \( 0 \leq z \leq 2 \), \( N \) is the outward-pointing normal vector, and \( G = (x^2y, y^2z, z^2x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-128\pi}
Evaluate the sum of the series: \[ \sum_{m=1}^\infty \frac{\left(\frac{1}{2}\right)^\underline{m}}{m!} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2} - 1}
Acute triangle $A B C$ has circumcenter $O$. The bisector of $\angle A B C$ and the altitude from $C$ to side $A B$ intersect at $X$. Suppose that there is a circle passing through $B, O, X$, and $C$. If $\angle B A C=n^{\circ}$, where $n$ is a positive integer, compute the largest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Calculate the value of $\cos 96^\circ \cos 24^\circ - \sin 96^\circ \sin 66^\circ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2}}
Given the sequence $\{a_n\}$ with the sum of its first $n$ terms $S_n = 6n - n^2$, find the sum of the first $20$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}}\right\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{4}{35}}
Evaluate the value of the infinite nested radical expression: \[ \sqrt{-1+1\sqrt{-2+2\sqrt{-3+3\sqrt{-4+4\sqrt{\ldots}}}}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given sets A = {2, 4, a} and B = {1, 2, 3}, if A ∪ B = {1, 2, 3, 4}, find the possible values of a. Express your answer as a list or set of values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{1, 3}}
已知正三棱锥 $P-A B C$ 的底面 $A B C$ 为正三角形，该正三棱锥的外接球的球心 $O$ 满足 $\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}=0$. 求二面角 $A-P B-C$ 的余弦值。答案的形式为\frac{k}{m}，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Each vertex of a regular dodecagon ($12$-gon) is to be colored either red or blue, and thus there are $2^{12}$ possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$928$}
In a population of drivers, 30% belong to Class I, where the number of accidents per driver follows a geometric distribution with a parameter uniformly distributed over (0, 0.5). The remaining drivers belong to Class II, where each driver has an expected number of claims of 0.4. Calculate the probability that a randomly selected driver from this population has at most three accidents.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.8832}
There are 5 apples, their weights are 125g, $a$, 121g, $b$, and 127g respectively. If the median and the average of this sample are both 124, then the standard deviation $s=$ ______ (Answer with a number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Ann and Bob each simultaneously toss a fair coin in each round. Let $X_n$ be the number of heads obtained in the $2n$ flips during the first $n$ rounds. For each integer $m > 0$, let $r_m$ be the probability that there exists an $n$ such that $X_n = m$. Determine the value of $\lim_{m \to \infty} r_m$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
Determine the number of ways to traverse an 8-cycle graph in 10 moves, starting from a given vertex and ending at the vertex directly opposite to it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Two parallel lines \(\ell_{1}\) and \(\ell_{2}\) are tangent to a circle \(\omega_{1}\) with center \(O_{1}\) at points \(A\) and \(B\) respectively. A second circle \(\omega_{2}\) with center \(O_{2}\) is tangent to line \(\ell_{1}\) at point \(D\), intersects line \(\ell_{2}\) at points \(B\) and \(E\), and intersects circle \(\omega_{1}\) again at point \(C\) (point \(O_{2}\) lies between lines \(\ell_{1}\) and \(\ell_{2}\)). Given that the ratio of the area of quadrilateral \(B O_{1} C O_{2}\) to the area of triangle \(O_{2} B E\) is \(\frac{6}{5}\), find the ratio of the radii of circles \(\omega_{2}\) and \(\omega_{1}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{6}}
Determine the norm of the functional $l: C[0,\pi] \rightarrow \mathbb{R}$ defined by \[ l(x) = x(0) - x(\pi/4) + \int_{0}^{\pi} x(s) \sin s \, ds. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given some $4$-tuples of positive real numbers $(a_1,b_1,c_1,d_1),\dots,(a_n,b_n,c_n,d_n)$ with all $a_i,b_i,c_i,d_i\leq 1$, determine the best possible value $k$ between $1$ and $3$ such that we can always partition $\{1,2,\dots,n\}$ into two subsets $X,Y$ satisfying:
\[1+\sum_{i \in X}a_i\geq \sum_{i \in Y}a_i, \quad 1+\sum_{i \in X}b_i\geq \sum_{i \in Y}b_i,\]
\[\sum_{i \in X}c_i\leq 1+\sum_{i \in Y}c_i, \quad \sum_{i \in X}d_i\leq 1+\sum_{i \in Y}d_i.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the values of \( p \) such that the series of functions \( f(x) = \sum_{n=1}^\infty \frac{\log(1+ n^2x^2)}{n^p} \) converges and can be differentiated term-by-term for all \( x \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > 2}
Find the greatest common divisor (GCD) of the complex numbers \(29-3i\) and \(11+10i\) in the Gaussian integers \(\mathbb{Z}[i]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 4i}
A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses equals the third, calculate the probability that at least one "2" is tossed. Express your answer as a fraction in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{15}}
What is the largest number of shells that cannot be delivered using whole boxes containing 15, 18, and 20 shells?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{97}
John goes for a jog every 3 days. He went for a jog on Monday, January 5, and his next jog was on January 8. What is the date of the next Monday on which he will go for a jog? Express your answer as a date in the format 'Month Day'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{January 26}
The non-isosceles triangle \(ABC\) with a perimeter of 12 is inscribed in a circle \(\omega\). Points \(P\) and \(Q\) are the midpoints of the arcs \(ABC\) and \(ACB\) respectively. The tangent drawn to the circle \(\omega\) at point \(A\) intersects the ray \(PQ\) at point \(R\). It turns out that the midpoint of the segment \(AR\) lies on the line \(BC\). Find the length of segment \(BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Compute the integral \( \int_M \omega \) where \( M = \{(x,y,z) : z = x^2 + y^2, z < 1\} \) is a smooth 2-manifold in \( \mathbb{R}^3 \) and \( \omega = x \, dy \wedge dz + y \, dz \wedge dx + z \, dx \wedge dy \in \Omega^2(\mathbb{R}^3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{2}}
Compute the integral \( \int_{|z|=2}\frac{f'(z)}{f(z)}dz \), where \( f(z) = 1 + \frac{1}{z}+\frac{1}{2!z^2}+\dots+\frac{1}{n!z^n} \). Given that \( f \) has no zeros with \( |z|\geq 2 \), apply the Argument Principle to determine the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that $\alpha$ is an obtuse angle and $\beta$ is also an obtuse angle with $\cos\alpha = -\frac{2\sqrt{5}}{5}$ and $\sin\beta = \frac{\sqrt{10}}{10}$, find the value of $\alpha + \beta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7\pi}{4}}
The hour and minute hands on a certain 12-hour analog clock are indistinguishable. If the hands of the clock move continuously, compute the number of times strictly between noon and midnight for which the information on the clock is not sufficient to determine the time.

[i]Proposed by Lewis Chen[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
How many integer values of x make the polynomial p = x^4 + 6x^3 + 11x^2 + 3x + 31 a perfect square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the surface integral \( \int\int_S z \, dx \, dy \) over the complete surface \( S \) of a circular cylinder with radius \( a \) and axis along the \( OZ \) direction, where the ends of the cylinder are in the planes \( z = 0 \) and \( z = 2a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi a^3}
Find the number of all trees planted at a five-foot distance from each other on a rectangular plot of land, the sides of which are 120 feet and 70 feet.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{375}
Use differentials to approximate the value of \( h(1.02, 1.99) \), where \( h = f \circ \vec{g} \), \( f(u,v) = 3u + v^2 \), \( \vec{g}(1,2) = (3,6) \), and the Jacobian of \( \vec{g} \) at \( (1,2) \) is \( \begin{pmatrix} 2 & 1 \\ 3 & 5 \end{pmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45.21}
Find the dimension of the null space of a linear transformation $T \in \wedge^{n-2}V^*$, where $V$ is a vector space with dimension $n$ and $T \neq 0$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the limit of the following expression as \( n \) approaches infinity, using the asymptotic expansion of the Barnes G-function:

\[\lim_{n\to\infty} \frac{G(n+\frac{11}{8})G(n+\frac{13}{8})^{2}G(n+\frac{15}{8})}{G(n+\frac{5}{2})^{2}G(n+\frac{7}{4})^{2}}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Semen has 20 numbers: \(1, 2, 3, \ldots, 19, 20\). He created 10 fractions by writing ten of these numbers in some order as the numerators and the remaining ten numbers in some order as the denominators. What is the maximum number of integers that Semen could obtain after reducing all the written fractions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the time complexity of the recurrence relation: $$T(n)=T\left(n-\frac{n}{\lg n}\right)+ O(n)$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n \log n)}
Let acute triangle $ABC$ have circumcenter $O$, and let $M$ be the midpoint of $BC$. Let $P$ be the unique point such that $\angle BAP=\angle CAM, \angle CAP=\angle BAM$, and $\angle APO=90^{\circ}$. If $AO=53, OM=28$, and $AM=75$, compute the perimeter of $\triangle BPC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{192}
Determine the number of pairwise non-isomorphic connected 3-regular graphs of order 14.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{509}
At the round table,  $10$  people are sitting, some of them are knights, and the rest are liars (knights always say pride, and liars always lie) .  It is clear thath I have at least one knight and at least one liar. 
What is the largest number of those sitting at the table can say: ''Both of my neighbors are knights '' ?
(A statement that is at least partially false is considered false.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Twelve students participated in a theater festival consisting of \( n \) different performances. Suppose there were six students in each performance, and each pair of performances had at most two students in common. Determine the largest possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the nearest prime number to 2304 that is divisible by 23.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2323}
A special clock in the museum has 20 positions on its dial. Every 7 minutes, the hand jumps once, and each jump covers 9 positions. This morning at 8:00 AM, the hand moved from position 0 to position 9. Where was the hand pointing at 8:00 PM yesterday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the expected value of the time at which the trajectory of the Wiener process, $W_t$, first crosses the line $y=t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}