inclusionAI/Ring-lite-rl-data huggingface dataset:
A point $P(\frac{\pi}{12}, m)$ on the graph of the function $y = \sin 2x$ can be obtained by shifting a point $Q$ on the graph of the function $y = \cos (2x - \frac{\pi}{4})$ to the left by $n (n > 0)$ units. Determine the minimum value of $mn$. Express your answer as a fraction of π.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\pi}{48}}
Janice works at a convenience store where her schedule fluctuates. She has to work 5 to 6 days per week with varying hours per day and can earn $30 per day if she works 5 hours. When she works more than 5 hours, she earns an additional hourly rate: $10 per additional hour on weekdays, and $15 per additional hour on weekends. This week, she worked 5 days, including one weekend day. Her total work hours are as follows: Monday - 5 hours, Tuesday - 6 hours, Wednesday - 7 hours, Thursday - 5 hours, and Sunday - 6 hours. Janice earned $15 in tips this week, but she needs to pay 10% tax on her basic earnings and a 5% tax on her overtime and tips. How much will Janice earn this week after deductions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$192}
Let $\vec a, \vec b,\vec c$ be three unit vectors such that $\vec a \cdot \vec b+\vec b \cdot \vec c-\vec a \cdot \vec c=\frac{3}{2}$. What is the value of $\vec a \cdot \vec b+\vec b \cdot \vec c+\vec a \cdot \vec c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{2}\)}
Place 5 points in a rectangle of size $17 \times 32$ such that the minimum distance between any two points is maximized. What is the highest possible minimum distance between any two points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
How many functions \( f : \{1,2, \ldots, 2013\} \rightarrow \{1,2, \ldots, 2013\} \) satisfy \( f(j) < f(i) + j - i \) for all integers \( i, j \) such that \( 1 \leq i < j \leq 2013 \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{4025}{2013}}
Consider the initial value problem for the classical Burgers equation given by \( u_t + uu_x = 0 \) with the initial condition:
\[
 u(x,0) = \phi(x) = \begin{cases} 
 2, & x \leq \pi/2 \\
 \sin x + 1, & \pi/2 < x \leq 3\pi/2 \\
 0, & x > 3\pi/2 
\end{cases}
\]
Determine the breaking time \( t_B \) for this problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( S \) be the set of functions \( f \) defined on the interval \([0, 1]\) with non-negative real values such that \( f(1) = 1 \) and \( f(x) + f(y) \leq f(x + y) \) for all \( x, y \) with \( x + y \leq 1 \). Determine the smallest constant \( k \) such that \( f(x) \leq kx \) for all functions \( f \) in \( S \) and for all \( x \) in \([0, 1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the cardinality of \(|\mathbb{R}| + |\mathbb{N}|\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Amy has divided a square into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes down its width divided by its height. Within each red rectangle, she writes down its height divided by its width. Finally, she calculates $x$, the sum of these numbers. If the total area of white equals the total area of red, determine the minimum of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.5}
Egor wrote a number on the board and encrypted it using a letter puzzle, where different letters correspond to different digits, and the same letters correspond to the same digits. The word "GVATEMALA" was obtained. How many different numbers could Egor have originally written if his number was divisible by 8?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67200}
Find the maximum distance between two parallel normals of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), where \(a > b\). Express your answer in terms of \(a\) and \(b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2a}
Let $ABC$ be a triangle with inradius $r$ and circumradius $R$. Construct triangle $A'B'C'$ such that $A'B'$ is perpendicular to $OC$ through $C$, and similarly for the other sides. Let $r_1$, $r_2$, and $r_3$ be the inradii of triangles $A'BC$, $AB'C$, and $ABC'$ respectively. Find the sum $r_1 + r_2 + r_3$ in terms of $r$ and $R$. Express your answer as a single expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2R - r}
Find the form of the polynomial $Q(x)$ that satisfies the equation $Q(x+y) + Q(x-y) = 2Q(x) + 2Q(y)$. Provide your answer in terms of the degree and coefficients of $Q(x)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Q(x) = ax^2}
Find the sum of the shifted sinc function given by:
\[ g(y) = \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find a closed formula for the power series $A(x) = \sum_{n\geq0} a_nx^n$ that satisfies the differential equation $A'(x) = 2xA(x)$. Express your answer in terms of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{x^2}}
For a Riemannian 3-manifold diffeomorphic to \(\mathbb{R}^3\), determine the number of conditions of the spherical coordinate metric that can be satisfied by choosing a suitable coordinate system, considering the curvature of the manifold.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given independent random samples \((X_{1},X_{2},X_{3},...,X_{8}) \sim N(8,16)\), \((Y_{1},Y_{2},Y_{3}) \sim N(1,9)\), and \((Z_{1},Z_{2},Z_{3}) \sim N(6,10)\), find the probability \(P(2\bar{X}-\bar{Z}>3\bar{Y})\). The sample means are distributed as \(\bar{X} \sim N(8,2)\), \(\bar{Y} \sim N(1,3)\), and \(\bar{Z} \sim N(6,\frac{10}{3})\). Calculate \(P(2\bar{X}-\bar{Z}>3\bar{Y})\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.8708}
if \( a,b \in  \mathrm{R} + ,{a}^{\log b} = 2,{a}^{\log a}{b}^{\log b} = 5 \) ,find out \( {\left( ab\right) }^{\log {ab}} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
20

}
Determine the number of three-digit numbers with the following property:
The number formed by the first two digits is prime and the number formed by the last two digits is prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
52
}
In a Poisson process with intensity \( \lambda = 2 \), let \( S_{n} \) denote the time of the \( n^{th} \) event. Given that \( N(2) = 4 \), find \( E(S_{6}|N(2)=4) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=0}^n\binom{n}{k}\frac{3k}{2^n(n+3k)} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{5}}
Evaluate the limit: \[ \lim_{x\to 0}\frac{10^x-2^x-5^x+1}{x\tan x} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\ln 2)(\ln 5)}
Determine the value of \( \theta \) in the interval \( 0 < \theta < \pi \) that satisfies the equation \( \int_{\pi/2}^\theta \sin x\,dx = \sin 2\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Find the number of ways to arrange 6 people at three identical round tables such that each table is occupied.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225}
Several students are seated at a large circular table. They pass
 around a bag containing 100 pieces of candy. Each person
 receives the bag, takes one piece of candy and then passes the
 bag to the next person. If Chris takes the first and last piece of
 candy, then the number of students at the table could be
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11
}
In a country with 110 cities, each pair of cities may or may not have a road connecting them. A driver starts in a city with exactly one road leading out of it. He travels along this road to a second city, which has exactly two roads leading out of it. Continuing this pattern, he travels to a third city with exactly three roads, and so on. Eventually, he reaches the $N$-th city, which has exactly $N$ roads leading out of it, and stops his journey. What is the maximum possible value of $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Solve in real numbers the system
$$
\begin{cases} x^3 + y = 3x + 4 \\ 2y^3 + z = 6y + 6 \\ 3z^3 + x = 9z + 8\end{cases}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x=y=z=2}
Determine the values of $i$ for which the isomorphism $H_i(K^{(p)}) \simeq H_i(K)$ holds, where $K$ is a topological space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i < p}
Determine the value of $p$ for which the series \( \sum\limits_{n=1}^\infty\left(\dfrac1{n^p}\sum\limits_{k=1}^nk^{3/2}\right) \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{2}}
Determine the number of zero elements in the inverse of the $n \times n$ matrix $A=\begin{bmatrix} 1&1&1&1&\cdots&1\\ 1&2&2&2&\cdots&2\\ 1&2&1&1&\cdots&1\\ 1&2&1&2&\cdots&2\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ 1&2&1&2&\cdots&\cdots \end{bmatrix}$. \(n \geq 2\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(n-2)}
Given a set $\mathcal{A}$ of $K$ unique integers, $\mathcal{A} = (X_{1}, \ldots, X_{K})$, we randomly pick an integer $X_{j}$ from the set. We then draw a random subset $\mathcal{B} \subset \mathcal{A} \setminus (X_{j})$ of $M < K-1$ integers (excluding $X_{j}$ from the draw). What is the probability that $X_{j}$ is less than the minimum integer in subset $\mathcal{B}$? That is, find/estimate the probability: $\mathbb{P}(X_{j} < \min(X_{i}): X_{i} \in \mathcal{B})$.

Example: Let $\mathcal{A} = (2,3,4,5,6,7)$ and $X_{j} = 4$. Let also $\mathcal{B1} = (3,6,7)$ and $\mathcal{B2} = (6,7,5)$. In $\mathcal{B1}$, $X_{j} = 4$ is not less than 3 = min(3,6,7), thus, does not satisfy the criterion, while in $\mathcal{B2}$, $X_{j} = 4$ is less than 5 = min(5,6,7), which satisfies the criterion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{m+1}\)}
Let \( \{U_n\} \) be a sequence of i.i.d Uniform\((0,1)\) random variables and \( \{a_k\} \) be a sequence of i.i.d random variables such that \( P(a_k = \pm1) = \frac{1}{2} \) and are independent of the \( \{U_n\} \). Define the quantities:
\[ X_n = \sum\limits_{k=1}^nU_ka_k, \quad Y_n = \sum\limits_{k=1}^n(U_k^2 - 1/2)a_k \]
Find the limit:
\[ \lim\limits_{n\rightarrow\infty}P\left(|X_n| > Y_n\sqrt{\frac{20}{7}}\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
In three-dimensional space, determine the maximum number of points that can be arranged such that the Euclidean distance between every pair of points is an integer, and no four points are coplanar.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the smallest natural number \( n \) such that for any finite subset \( A \) of \( \mathbb{R}^2 \), if for every \( n \) points in \( A \) there exist two lines that include all these \( n \) points, then all points in \( A \) lie on two lines.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Consider two independent random variables, $X \sim \mathrm{Exp}(4)$ and $Y \sim \mathrm{Exp}(12)$. Calculate the expected value of the minimum of $4X$ and $Y$, i.e., find $E[\min(4X, Y)]$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{13}}
Given the set $S = \{1, 2, 3, 4, 5\}$ and non-empty subsets $A, B, C \subseteq S$, consider functions $f: A \rightarrow B$ and $g: B \rightarrow C$ with their composite function $h: A \rightarrow C$ defined as $h(x) = g(f(x))$. If $h$ is a bijective function, determine the number of possible functions $h(x)$. Note that $n(X)$ denotes the number of elements in set $X$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1545}
Identify the finite abelian group of order 8 that has three elements of order two and four elements of order four, up to isomorphism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_4 \times \mathbb{Z}_2}
Determine the range of values of \( k \) for which the function \( f(x) = \frac{x^2 - 1}{(x-2)(x+k)} \) takes all real values, where \( x \) is a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
Determine the theoretical number of iterations required for the BFGS method to minimize a quadratic function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Determine the smallest integer \( n \geq 3 \) such that for any partition of the set \( I = \{1, 2, \ldots, n\} \) into two non-empty disjoint subsets \( A \) and \( B \), there exist two numbers in either \( A \) or \( B \) whose sum is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
What is the maximum amount a player with an initial wealth of 2 would pay to participate in the St. Petersburg paradox, assuming a logarithmic utility function?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the area under the curve $x^2 + y^2 = 3\sqrt{2} x - 5\sqrt{2} y + 2xy$ and above the x-axis between $x = 0$ and $x = 3\sqrt{2}$. Provide your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{3}}
Given the graph $G$ represented by the adjacency matrix:

$$A=\begin{pmatrix} 0 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 &1 &0 \\ 0 &1 &1 &0 & 1\\ 1&1 &0&1&0 \end{pmatrix}$$

Determine the number of minimum spanning trees of the graph $G$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Determine the condition on \( \alpha \) such that the function \( f(x) = |x|^\alpha \) with \( f(0) = 0 \) is weakly differentiable on the unit ball \( A = B_1(0) \subset \mathbb{R}^n \). Assume \( \alpha \neq 0 \) and \( n \geq 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > 1 - n}
In a regular triangular prism, a plane passes through a side of the lower base and the opposite vertex of the upper base, forming a $45^\circ$ angle with the plane of the lower base. The area of the cross-section is $S$. Find the volume of the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{S \sqrt{S} \cdot \sqrt[4]{6}}{2}}
Let $S$ be a regular oriented surface such that for any simple, closed, and positively oriented curve in $S$, the integral of the geodesic curvature along this curve is always the same. What can be concluded about the Gaussian curvature of $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the residue of the function \( f(z) = z \sin\left(\frac{1}{1-z}\right) \) at \( z = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let $X$ be as in problem 13. Let $Y$ be the number of ways to order $X$ crimson flowers, $X$ scarlet flowers, and $X$ vermillion flowers in a row so that no two flowers of the same hue are adjacent. (Flowers of the same hue are mutually indistinguishable.) Find $Y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the rank of the matrix \( A \) given by:
\[
A=\begin{bmatrix}
a & b & c& d& e& f& g\\
a+1& b-1& 0 & 0 & e& 0 & g \\
a+1& 0 & c-1& 0 & e& f& 0\\
a+1& 0 & 0& d-1& 0& f & g\\
a+1& b& c& 0&  e-1& 0& 0\\
a+1& 0 & c& d& 0& f-1& 0\\
a+1& b& 0 & d& 0 &0 & g-1
\end{bmatrix}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A sequence $\{a_n\}$ is defined such that $a_i=i$ for $i=1,2,3\ldots,2020$ and for $i>2020$, $a_i$ is the average of the previous $2020$ terms. What is the largest integer less than or equal to $\displaystyle\lim_{n\to\infty}a_n$?

2020 CCA Math Bonanza Lightning Round #4.4
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$1347$
}
Let $X = (x_1, x_2, \ldots, x_9)$ be a permutation of the set $\{1, 2, \ldots, 9\}$. Let $A$ be the set of all such permutations $X$. For any $X \in A$, define the function $f(X) = x_1 + 2x_2 + \cdots + 9x_9$. Let $M = \{f(X) \mid X \in A\}$. Find $|M|$, where $|S|$ denotes the number of elements in the set $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
Consider a focal chord PQ in the parabola \( y^2 = 4ax \), where \( a > 0 \). The tangents to the parabola at points P and Q intersect at a point lying on the line \( y = 2x + a \). If chord PQ makes an angle \( \theta \) with the vertex of the parabola, find the value of \( \tan\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Calculate the following limit superior:
\[ \limsup_{n\to\infty}\left(\frac{A^n}{n}+\frac{B^n}{n^2}\right)^{\frac{1}{n}} \]
where \( A, B > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\max\{A, B\}}
The numbers 123456789 and 999999999 are multiplied. How many of the digits in the final result are 9's?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of distinct monic quadratic polynomials with integer coefficients such that they have at least one root, all roots are powers of 5 with non-negative integer exponents, and all coefficients in absolute value do not exceed $122^{85}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16256}
A triangle is inscribed in a circle with a unit radius. Determine the possible distance between the centers of two excircles of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the minimum number of cells that need to be marked on a 5 by 5 grid so that among the marked cells, there are no neighbors (sharing a common side or a common vertex), and adding any one cell to these cells would violate this condition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given rectangle ABCD with AD = 5 and CD = 3, draw three circles with centers at A, B, and C and radii of 1, 2, and 3, respectively. Calculate the approximate area of the shaded region. Express your answer as a single number with one decimal place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.0}
Consider the linear transformation \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) defined by \( T\left(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\right) = \begin{bmatrix} xy \\ xz \end{bmatrix} \). Determine the dimension of the kernel of \( T \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $\xi, \eta$ be independent random variables, both following the same exponential distribution. Find $\mathrm{P}\left(\log _{2}(\eta /(\xi-1))<0\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{e^{-\lambda}}{2}}
Calculate the derivative of the cross-entropy loss function \( J = -\sum_i p_i \log{q_i} \) with respect to \( z_j \), where \( q_j \) is defined using the softmax function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{q_j - p_j}
Ali Baba arrived at a cave where there is gold, diamonds, and a chest in which they can be carried. A full chest of gold weighs 200 kg, a full chest of diamonds weighs 40 kg, and an empty chest weighs nothing. A kilogram of gold costs 20 dinars in the market, and a kilogram of diamonds costs 60 dinars. Ali Baba can lift and carry no more than 100 kg. What is the maximum amount of money he can get for the treasures he brings from the cave in one trip?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3000 \text{ dinars}}
Calculate the limit \(\lim\limits_{n\to\infty}\left(\frac{a_1}{a_0S_1}+\frac{a_2}{S_1S_2}+\ldots+\frac{a_n}{S_{n-1}S_n}\right)\) where the sequence is defined by \(a_n=2015^n\) and \(S_n=\frac{2015^{n+1}-1}{2014}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Xiao Ming needs to do the following things after school in the evening: review lessons for 30 minutes, rest for 30 minutes, boil water for 15 minutes, and do homework for 25 minutes. The minimum time Xiao Ming needs to spend to complete these tasks is ___ minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
We write the equation on the board:

$$
(x-1)(x-2) \ldots (x-2016) = (x-1)(x-2) \ldots (x-2016) .
$$

We want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimal number of factors that need to be erased to achieve this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that the sum of the first $n$ terms of a geometric sequence ${a_{n}}$ is $S_{n}$, if $a_{3}=4$, $S_{3}=12$, find the common ratio.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2}}
求最大的正实数 $\alpha$, 对于满足 $a+b+c \mid a^{2}+b^{2}+c^{2}$ 的不全相等的正整数 $a, b, c$, 都有方程  
$$  
(x+b)^{2}=(x+c)(x+a)  
$$  
在区间 $(0, \alpha)$ 内没有解。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{3}
Calculate the integral \( \int_{-\infty}^\infty |f(x)|^2 \mathrm{d}x \) using Plancherel's formula, where \( f(\xi) = \int_0^1 \sqrt{x} \sin(\xi x) \mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Evaluate the limit: \[ \lim_{\beta\to+\infty}\frac{\int_{0}^2 I_0(\sqrt{t(2-t)})e^{-\beta t}(t-1)dt}{\int_{0}^2 I_0(\sqrt{t(2-t)})e^{-\beta t}dt} \] where \( I_0(x) \) is the modified Bessel function of the first kind.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given a Markov chain with the following transition matrix and time weights for each state:

\[ \begin{bmatrix} \frac{3}{4} & \frac{1}{4} & 0 & 0 \\
\frac{4}{5} & 0 & \frac{1}{5} & 0 \\
\frac{6}{7} & 0 & 0 & \frac{1}{7} \\
0 & 0 & 0 & 1 \end{bmatrix} \]

and time weights:
- \( t_0 = 5 \)
- \( t_1 = 10 \)
- \( t_2 = 15 \)
- \( t_3 = 20 \)

Calculate the expected time to reach the absorbing state \( t_3 \) starting from \( t_0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1155}
Arnel had ten boxes of pencils with the same number of pencils in each box. He kept some pencils and shared the remaining pencils equally with his five friends. His friends got eight pencils each, and there were 5 pencils in each box. How many pencils did Arnel keep?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}