inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the Markov chain with state space \(\{1, 2, 3\}\) and transition matrix \(P = \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{3} & \frac{1}{2} & \frac{1}{6} \\ 0 & 1 & 0 \end{pmatrix}\), where the initial state \(X_0 = 1\) almost surely, find the probability that the chain returns to state 1 before reaching state 3 for the first time. Let \(u(i)\) denote the probability of visiting state 1 before state 3 when starting from state \(i\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
The circles with centers \(O_{1}\) and \(O_{2}\) have a common chord \(AB\), and \(\angle AO_{1}B = 60^\circ\). The ratio of the length of the first circle to the length of the second circle is \(\sqrt{2}\). Find the angle \(\angle AO_{2}B\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing $6$ cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\text{Thursday}$}
Lance wants to finish reading a 100 page book in 3 days. Yesterday, he started to read 35 pages. Today, he read 5 fewer pages than yesterday. Lance wants to read an even number of pages tomorrow, in increments of 10 pages. How many pages should he read tomorrow to finish the book?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
设集合 $M=\{1,2, \cdots, 10\}, T$ 为 $M$ 的某些二元子集构成的集合，满足对 $T$ 中任意两个不同的元素 $\{a, b\} 、\{x, y\}$ ，均有  
$11 \nmid(a x+b y)(a y+b x)$ .  
求 $T$ 的元素个数的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
A set $S \subseteq \mathbb{N}$ satisfies the following conditions:

1. If $x, y \in S$ (not necessarily distinct), then $x + y \in S$.
2. If $x$ is an integer and $2x \in S$, then $x \in S$.

Find the number of pairs of integers $(a, b)$ with $1 \le a, b \le 50$ such that if $a, b \in S$ then $S = \mathbb{N}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2068}
A group $H$ of order $p^4$ is non-abelian. If it has $p^3+p^2+p+1$ subgroups of order $p$, how many subgroups of order $p^3$ does it have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p + 1}
Given real numbers \( a_1, a_2, \ldots, a_n \in [0, 2] \) with the condition that \( a_{n+1} = a_1 \), find the maximum value of the expression \(\frac{\sum_{i=1}^{n} a_{1}^{2} a_{i+1} + 8n}{\sum_{i=1}^{n} a_{i}^{2}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
11. (This question is worth 20 points) In the plane rectangular coordinate system $x O y$, let $A B$ be a chord of the parabola $y^{2}=4 x$ passing through the point $F(1,0)$, and the circumcircle of $\triangle A O B$ intersects the parabola at point $P$ (different from points $O, A, B$). If $P F$ bisects $\angle A P B$, find all possible values of $|P F|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{13}-1}
k是实数，令$f ( x ) = \frac { x ^ { 4 } + k x ^ { 2 } + 1 } { x ^ { 4 } + x ^ { 2 } + 1 }$.若对每三个实数a，b，c（可以相同），存在一个三角形，其三边长为f(a)、f(b)、f(c)，则k的取值范围是 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$- \frac { 1 } { 2 } < k < 4$ }
A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base $10$ with no leading zeros. A $6$-digit palindrome $n$ is chosen uniformly at random. Find the probability that \(\frac{n}{11}\) is also a palindrome. The original answer is in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
There are 3 boys, 3 girls, and 1 teacher taking a photo together. The teacher must be in the middle, and those adjacent to the teacher cannot be boys or girls at the same time. How many possible arrangements are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{432}
Determine the complex polynomial in the form \( p(z) = (z-(a+bi))^n + q \) that has roots at the vertices of a regular hexagon, with one root located at \( 2 + 0i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{z^6 - 64}
Compute the length of the segment tangent from the origin to the circle that passes through the points $(3,4),$ $(6,8),$ and $(5,13).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 \sqrt{2}}
Find the limit of the sequence \(x_n = \dfrac{\lfloor x \rfloor + \lfloor 3^2x \rfloor + \ldots + \lfloor (2n-1)^2x \rfloor}{n^3}\) as \(n\) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4x}{3}}
12. The parabolas $y=x^{2}-b x$, $y=-x^{2}+b x$ and the line $y=k x$ can divide the plane into at most
$\qquad$ parts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the integral \( \int_{0}^{\infty} \left(e^{-\frac{1}{x^2}}-e^{-\frac{4}{x^2}}\right) dx \) using appropriate substitution and integration techniques.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\pi}}
For what value of \(a\) are the graphs of \(y = a^x\) and \(y = \log_a x\) tangent to each other at exactly one point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{1/e}}
Five haikus each consist of three lines with syllable counts of five, seven, and five. If you randomly scramble the lines of these five haikus, what is the probability that you end up with five completely good haikus, each with the correct five, seven, five syllable structure? Express this probability as a fraction \( \frac{m}{n} \) in simplest form, where \( m \) and \( n \) are positive integers with \( \text{gcd}(m, n) = 1 \). Find the sum of \( m \) and \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3004}
在平面直角坐标系 $x O y$ 中, 椭圆 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ 的右焦点为 $F(c, 0)$,若存在经过点 $F$ 的一条直线 $l$ 交椭圆于 $A, B$ 两点，使得 $O A \perp O B$. 求该椭圆的离心率 $e=\frac{c}{a}$ 的取值范围。该取值范围两端点的和形如\frac{a\sqrt{b}+c}{d}。请求出a+b+c+d的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
At the "Lukomorye" station, tickets are sold for one, five, and twenty trips. All tickets cost an integer number of gold coins. Five single-trip tickets are more expensive than one five-trip ticket, and four five-trip tickets are more expensive than one twenty-trip ticket. It turned out that the cheapest way for 33 warriors to travel was to buy tickets for 35 trips, spending 33 gold coins. How much does a five-trip ticket cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $a$, $b$, and $c$ be real numbers satisfying the equations
$$a^3+abc=26$$
$$b^3+abc=78$$
$$c^3-abc=104.$$
Find $a^3+b^3+c^3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
184
}
Given real numbers $b$ and $c$, and the function $f(x) = x^2 + bx + c$, the equation $f(f(x)) = 0$ has exactly three different real roots. Find the maximum value of the sum of the roots of $f(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
For each of the nine natural numbers \( n, 2n, 3n, \ldots, 9n \), the first digit from the left in its decimal representation was written on the board. \( n \) was chosen so that the number of different digits among the nine written digits is as small as possible. What is this minimum number of different digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given seven natural numbers \(a, b, c, a+b-c, a+c-b, b+c-a, a+b+c\), it is known that all of them are distinct prime numbers. Find all possible values of the smallest of these seven numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The side \( AB \) of triangle \( ABC \) is equal to \( c \). Point \( M \) is taken on side \( AB \) such that \(\angle CMA = \varphi\).

Find the distance between the orthocenters of triangles \( AMC \) and \( BMC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c \cdot |\cot \varphi|}
Using three colors $\mathbf{R}$, $\mathbf{G}$, and $\mathbf{B}$ to color a $2 \times 5$ table in a way that two squares sharing a common edge must be colored differently. How many different coloring methods are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{486}
We consider 2018 lines on the plane, each pair of which are not parallel and no three of which are concurrent. Let $E$ be the set of their intersection points. We wish to assign a color to each point in $E$ such that any two points on the same line, whose connecting segment does not contain any other point from $E$, have different colors.

How many colors are needed at minimum to achieve such a coloring?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a 5x6 seating arrangement for 30 students, two students, A and B, from the same school cannot be seated adjacent to each other in any direction (front, back, left, right). Calculate the number of different seating arrangements for students A and B. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{772}
Vasya wrote consecutive natural numbers \( N \), \( N+1 \), \( N+2 \), and \( N+3 \) in rectangles. Under each rectangle, he wrote the sum of the digits of the corresponding number in a circle.

The sum of the numbers in the first two circles turned out to be 200, and the sum of the numbers in the third and fourth circles turned out to be 105. What is the sum of the numbers in the second and third circles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{103}
Determine if there exists a natural number that forms a perfect pair with 122, where both the sum and the product of the two numbers are perfect squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14762}
Let \( I \) be the augmentation ideal of the group ring \( \mathbb{Z}/p[\mathbb{Z}/p] \). For a given odd prime \( p \) and an integer \( n \) such that \( 1 \leq n \leq p-1 \), determine the dimension of \( I^n \) as a vector subspace of \( \mathbb{Z}/p[\mathbb{Z}/p] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p - n}
The numbers 1, 2, 3, 4, 5, 6, 7 are written in a circle in some order. A number is called "good" if it is equal to the sum of the two numbers written next to it. What is the maximum possible number of "good" numbers among those written?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a $25 \times n$ grid, each square is colored with a color chosen among $8$ different colors. Let $n$ be as minimal as possible such that, independently from the coloration used, it is always possible to select $4$ coloumns and $4$ rows such that the $16$ squares of the interesections are all of the same color. Find the remainder when $n$ is divided by $1000$. 

[i]Proposed by [b]FedeX333X[/b][/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{601}
Alex drives from point \( P \) to point \( Q \), then immediately returns to \( P \). Bob drives from point \( Q \) to point \( P \), then immediately returns to \( Q \). They travel on the same road, start at the same time, and each travels at a constant speed. Alex's speed is three times Bob's speed. They meet each other for the first time 15 minutes after the start. How many minutes after the start will they meet each other for the second time? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
In how many ways can we paint $16$ seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same colour is always odd?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1974}
Determine the interval of values for $x$ such that the inequality \((x^2-5x+4)(y^2+y+1)<2y\) holds for all real values of $y$. What is this interval?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 3)}
For a polynomial $f(x)$ of degree 3 with rational coefficients that touches the x-axis, determine how many of its roots are rational.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A cafe has 3 tables and 5 individual counter seats. People enter in groups of size between 1 and 4, inclusive, and groups never share a table. A group of more than 1 will always try to sit at a table, but will sit in counter seats if no tables are available. Conversely, a group of 1 will always try to sit at the counter ﬁrst. One morning, $M$ groups consisting of a total of $N$ people enter and sit down. Then, a single person walks in, and realizes that all the tables and counter seats are occupied by some person or group. What is the minimum possible value of $M + N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Find the sum of the coefficients of the terms containing $x^4t^2$ in the expansion of $(x^2 + y + 2t + 3k)^{12}$ using the multinomial expansion formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{778567680}
Evaluate the integral $$\int_0^\pi \log^2\left(\tan\frac{x}{2}\right)dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi^3}{8}}
Determine the number of digits that can be filled in the square to make the product of □4×32 a three-digit number. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
How many ordered triples of integers $ (a, b, c)$ satisfy\[|a + b| + c = 19\quad\text{and}\quad ab + |c| = 97?\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$12$}
Given the hyperbola x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0) with its right focus at F, if a line passing through point F with a slope angle of 60° intersects the right branch of the hyperbola at two points, calculate the range of the eccentricity of this hyperbola. Express your answer in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,2)}
12.217. Find the angle between the generatrix and the base of a truncated cone, the total surface area of which is twice the surface area of a sphere inscribed in it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arcsin\frac{2}{\sqrt{5}}}
Two natural numbers differ by 2, and the sum of their individual digits is divisible by 4. Find the smallest possible average of these two numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Calculate the area of the figure bounded by the closed curve \((x^{2}+y^{2})^{2}=4ay^{3}\), where \(a>0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\pi a^2}{2}}
Evaluate the limit: \[ \lim\limits_{n\to\infty} \frac{1}{n^2}\sum_{2\leq i < j \leq n} \log_i j \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find a function \( \beta \) such that \( \beta \circ f \), where \( f \) is the Dirichlet function, equals the characteristic function of the rationals, \( x_\mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\beta(y) = y}
Every time these two wheels are spun, two numbers are selected by the pointers. What is the probability that the sum of the two selected numbers is even? Express your answer as a fraction in simplest form. The original answer is in the format \frac{k}{m}, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The integer sequence \(\left\{a_{n}\right\}\) is defined as \(a_{1}=1\), \(a_{2}=2\), and \(a_{n+2}=5a_{n+1}+a_{n}\). Evaluate the expression \(\left[\frac{a_{2}}{a_{1}}\right] \left\{\left[\frac{a_{3}}{a_{2}}\right\} \left\{\frac{a_{4}}{a_{3}}\right\} \cdots \left\{\left[\left\{\frac{a_{2025}}{a_{2024}}\right\}\left[\frac{a_{2024}}{a_{2}}\right]\right.\right.\right.\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
6 / 9 of the population of the country of Venezia lives in Montague province, while the rest lives in Capulet province. In the upcoming election, 80% of Montague residents support Romeo, while 70% of Capulet residents support Juliet; each resident of Venezia supports exactly one of these two candidates. Rounded if necessary to the nearest percent, what is the probability that a Juliet supporter chosen at random resides in Capulet province?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64\%}
A and B run on a circular track at their respective constant speeds. If both start running from the same point in opposite directions, and after their first meeting, B takes another 8 minutes to reach the starting point, given that A takes 6 minutes to complete one lap, how many minutes does B take to complete one lap? Express your answer in minutes as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Petya has a 3x3 grid. He places pieces in the cells according to the following rules:

- No more than one piece can be placed in each cell;
- A piece can be placed in an empty cell if the total number of pieces already present in the corresponding row and column is even (0 is considered an even number).

What is the maximum number of pieces Petya can place?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Compute the limit: \[ \lim_{n\to\infty}\left(\frac{a_{n+1}}{\sqrt[n+1]{b_{n+1}}}-\frac{a_n}{\sqrt[n]{b_n}}\right), \] given that \((a_n)_{n\geq 1}\) and \((b_n)_{n\geq 1}\) are positive real sequences satisfying \[ \lim_{n\to\infty}\frac{a_{n+1}-a_n}{n}=a\in \mathbb{R}_{>0}, \quad \lim_{n\to\infty}\frac{b_{n+1}}{nb_n}=b\in \mathbb{R}_{>0}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{a e}{2 b}}
Person A and Person B start from points $A$ and $B$ respectively and walk towards each other. The first time they meet, it is 100 meters away from point $B$. After meeting, Person A doubles their speed. Person A reaches point $B$, turns around immediately, and catches up with Person B when Person B is 50 meters away from point $A$. How long is the distance between points $A$ and $B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250 \text{ meters}}
Consider the transformation $T$ given by the equations: $$x=u+v,\;\;\;\;y=v-u^2.$$ A triangle $Q$ in the $(u,v)$ plane has vertices at $(0,0)$, $(2,0)$, and $(0,2)$. The image of $Q$ under $T$ is the region $D$ in the $(x,y)$ plane. Compute the area of $D$ using a double integral over $Q$. The Jacobian of the transformation is $2u$. Calculate the area of $D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{14}{3}}
Evaluate the contour integral: $$I=-2i\oint_C\frac{dz}{z^2(2-i)+6z+(2+i)}$$ where $C$ is the unit circle in the complex plane. Use the residue theorem to find the value of $I$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
A fluid is poured down through the cone $z = \sqrt{x^2+y^2}$ for $x^2+y^2 \leq 1$, where the vector field describing the fluid flow is $f(x,y,z) = (0,0,-1)$. Determine the flux of the fluid through the cone.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi}
a man sells a car to his friend at 11 % loss . if the friend sells it for rs . 54000 and gains 20 % , the original c . p . of the car was :
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50,561.80}
What is the least possible sum of two positive integers $a$ and $b$ where $a \cdot b = 10!$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3810}
Two lines are perpendicular and intersect at point $O$. Points $A$ and $B$ move along these two lines at a constant speed. When $A$ is at point $O$, $B$ is 500 yards away from point $O$. After 2 minutes, both points $A$ and $B$ are equidistant from $O$. After another 8 minutes, they are still equidistant from $O$. Find the ratio of the speed of $A$ to the speed of $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
In rectangle $ABCD$, diagonal $AC$ is intersected by the angle bisector from $B$ at point $B'$ and by the angle bisector from $D$ at point $D'$. Similarly, diagonal $BD$ is intersected by the angle bisector from $A$ at point $A'$ and by the angle bisector from $C$ at point $C'$. The area of quadrilateral $A'B'C'D'$ is $\frac{9}{16}$ of the area of rectangle $ABCD$. Determine the ratio of the longer side to the shorter side of rectangle $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine if the sequence \( \{Y_n\} \) defined by \( Y_n = \frac{1}{n^3} \sum_{k=1}^n \frac{X_k^2}{k} \), where \( X_k \) are independent uniform random variables on the interval \((0,k)\), converges almost surely. If it does, find its limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the least possible value of \( f(1998) \), where \( f \) is a function from the set \(\mathbb{N}\) of positive integers into itself such that for all \( m, n \in \mathbb{N} \),
\[ f\left(n^{2} f(m)\right) = m [f(n)]^{2}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Determine the degree of the field extension $K=\mathbb{Q}(\alpha\zeta)$ over $\mathbb{Q}$, where $\alpha =2^{1/5}$ and $\zeta=e^{2i\pi/5}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Four people each input different positive integers into a series of three black boxes. Each black box follows these rules: if the input is odd, the output is \(4k+1\); if the input is even, the output is \(k \div 2\). All four final outputs are the same and minimized. If Sheep Daddy's input is the largest, what is his input value?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{680}
Determine the number of partial order relations for a set with 6 elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{130023}
Five points are chosen uniformly at random on a segment of length 1. What is the expected distance between the closest pair of points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{24}}
A merchant has two types of goods. For the first type of goods, which costs him $30 per unit, he marks up the price by 45% and then offers a discount of 15% on the marked price. For the second type of goods, which costs him $60 per unit, he marks them up by 75% and then offers a discount of 30% on the marked price. He sells 100 units of the first type and 50 units of the second type. Additionally, the merchant has to pay an 8% sales tax on the final selling price for each good. Calculate the overall percentage profit for the merchant after the discounts and taxes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32.71\%}