inclusionAI/Ring-lite-rl-data huggingface dataset:
In a table tennis match, each game is played in an $11$-point system, where the player who reaches $11$ points first wins. However, if the score reaches a tie at $10:10$, the winner must be the first player to gain a lead of $2$ points. After reaching a tie at $10:10$, the players alternate serving, with each player serving only one ball at a time. In a match between players A and B, when player A serves, the probability of A scoring is $\frac{2}{3}$, and when player B serves, the probability of A scoring is $\frac{1}{2}$. The outcomes of each ball are independent. If the score is tied at $10:10$ and player A serves first, the probability of both players playing until a score of $13:11$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
In a regular quadrilateral pyramid, two identical spheres of radius $r$ are placed, with their centers located on the axis of symmetry of the pyramid. One of the spheres touches all the lateral faces of the pyramid, while the second touches the base of the pyramid and the first sphere. Find the height of the pyramid for which the volume of the pyramid is the smallest.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["(6+2\\sqrt{3})r"]}
Find a function $y=f(x)$ such that the arc length of the curve from $x=0$ to $x=a$ is equal to $\sqrt{2}a$ for any constant $a$. Describe the function $y=f(x)$ that satisfies the condition: $$\int_{0}^{a}\sqrt{1+\left(f'(x)\right)^2}dx=\sqrt{2}a.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = \pm x + C}
Determine the condition under which the integral \( \int \frac{ax^2+bx+c}{x^3(x-1)^2} \, dx \) represents a rational function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a + 2b + 3c = 0}
Jay is given $99$ stacks of blocks, such that the $i$th stack has $i^2$ blocks. Jay must choose a positive integer $N$ such that from each stack, he may take either $0$ blocks or exactly $N$ blocks. Compute the value Jay should choose for $N$ in order to maximize the number of blocks he may take from the $99$ stacks.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
67
}
How many different ways can 2017 be expressed as a sum of distinct elements from the sequence defined by: \( a_{1}=1, a_{2}=2 \), and for \( k \in \mathbb{N}, a_{k+2}=a_{k+1}+a_{k} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given a $(0,1)$-matrix $B$ with $n$ columns and $n^3$ rows, where each row contains exactly four $1$'s, determine the minimum number of columns that must be deleted so that each row contains fewer than four $1$'s.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - 3}
From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{6}{35}"]}
Evaluate the improper integral \(\int_{-\infty}^{\infty}\frac{\sin(x)}{x+i}dx\) using the hint \(\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{e}}
Al, Betty, and Clara are in the same class of 50 students total, but are not friends with each other. Al is friends with 24 students, Betty is friends with 39, and Clara is friends with 20. What is the greatest number of students that could be friends with all 3 of them?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
18
}
Determine the value of \( x \) that satisfies the equation:

\[ \pi(xn) = \frac{xn}{\frac{n}{\pi(n)} + \ln(x)} \]

where \( \pi(n) \) is the prime-counting function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Compute the dimension of the space of global sections of the tangent bundle of a genus $g$ smooth projective curve $C$, denoted as $h^0(T_C)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Suppose $n = 18$ and $(n_1, n_2, n_3)$ follow a multinomial distribution with probabilities $p_1 = 0.6$, $p_2 = 0.3$, and $p_3 = 0.1$. Let $v = n_1 + 3n_2 + 10n_3$. Calculate the variance of $v$ given that $n_3 = 2$, i.e., find $\text{Var}(v \mid n_3 = 2)$. \( \text{Var}(v \mid n_3 = 2) = \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{128}{9}}
Given independent and identically distributed (i.i.d.) random variables $X_1, X_2, X_3$ with a normal distribution $N(\mu,\sigma^2)$, find the expected value of $2X_1+3X_2$ conditioned on the event $X_1+3X_2-X_3=4$. Provide your answer in terms of $\mu$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\mu + 4}
Determine the maximum number of critical points a solution of the differential equation \(y''(t) = a(t)y(t)\) can have, where \(a(t)\) is continuous and positive for all real numbers \(t\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of real solutions for \(x\) in the equation \[ \left[\frac{2x+1}{3}\right] + \left[\frac{4x+5}{6}\right] = \frac{3x-1}{2} \] where \([x]\) denotes the greatest integer less than or equal to \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Consider a grid of points \((x, y)\) where both \(x\) and \(y\) are positive integers less than or equal to 6. A *magic grid* is an assignment of an integer to each point in this grid such that for every square with horizontal and vertical sides and all four vertices in the grid, the sum of the integers at these vertices is the same for any such square. Determine the smallest possible product of all 36 integers in the grid that is greater than 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{36}}
Determine the asymptotic behavior of the integral \(\int_0^1 (\log t)^k e^{-t} \, {\rm d}t\) as \(k\) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^k k!}
Let $f$ be a function with $f'$ continuous on $[0,1]$ such that \[f(1)=0,\quad\int_{0}^{1}{{\left[ f'(x) \right]}^{2}}\,dx=7,\quad\int_{0}^{1}{x^{2}f(x)\,dx=\frac{1}{3}.}\] Calculate \[\int_{0}^{1}{f(x)\,dx}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{5}}
Consider a Markov Chain with the transition matrix:

$$M = \left( \begin{array}{ccc} b_0 & b_1 & b_2 \\ 0 & b_0+b_1 & b_2 \\ b_2 & b_1 & b_0 \\ \end{array} \right)$$

where all $b_i >0$ and $b_0+b_1+b_2 = 1$. If the Markov chain starts from state $0$, calculate the expected time until the chain first reaches state $2$. Express your answer in terms of $b_0$, $b_1$, and $b_2$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{b_2}}
Evaluate the limit: \[ \lim_{x\to\infty} \left(x-\sqrt{x^2-x-1}\frac{\ln(e^x-x)}{x}\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Petya chooses non-negative numbers \(x_{1}, x_{2}, \ldots, x_{11}\) such that their sum is 1. Vasya arranges them in a row to minimize the maximum product of adjacent numbers. What is the largest possible value of the maximum product that can be achieved with optimal strategies from both Petya and Vasya?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{40}}
A pedestrian and a cyclist are moving in one direction along a highway, while a cart and a car are moving in the opposite direction. All participants move at constant speeds (each with their own speed). The cyclist first overtakes the pedestrian, then after some time meets the cart, and then after the same amount of time meets the car. The car first meets the cyclist, then after some time meets the pedestrian, and then after the same amount of time overtakes the cart. The cyclist overtook the pedestrian at 10 o'clock, and the pedestrian met the car at 11 o'clock. When did the pedestrian meet the cart?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10:40 \text{ am}}
Let \( X = \{1, 2, \ldots, 100\} \). How many functions \( f: X \rightarrow X \) satisfy \( f(b) < f(a) + (b - a) \) for all \( 1 \leq a < b \leq 100 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{199}{100}}
A number x is mistakenly divided by 10 instead of being processed by a certain operation. The percentage error in the result is 99%. What operation should have been applied to the number x?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a row, there are 99 people - knights and liars (knights always tell the truth, and liars always lie). Each of them said one of two phrases: "To my left, there are twice as many knights as liars" or "To my left, there are as many knights as liars." In reality, there were more knights than liars, and more than 50 people said the first phrase. How many liars said the first phrase?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
What is the mass percentage of O in Bromic acid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37.24\%}
The graph of the function $y=a^{x+2}-2$ ($a>0$, $a\neq1$) always passes through a fixed point A. If point A lies on the line $mx+ny+1=0$, where $m$, $n>0$, then the minimum value of $\frac{1}{m} + \frac{2}{n}$ is \_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $a > 3$. Determine the value of $a$ given that $f(g(a)) = 16$, where $f(x) = x^2 + 10$ and $g(x) = x^2 - 6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\sqrt{6} + 6}}
In triangle  $ABC$ , find the smallest possible value of   $$ |(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)| $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8\sqrt{3}}{9}}
At a certain fruit stand, the price of each apple is 40 cents and the price of each orange is some cents. Mary selects a total of 10 apples and oranges from the fruit stand, and the average (arithmetic mean) price of the 10 pieces of fruit is 48 cents. Mary must put back 2 oranges so that the average price of the pieces of fruit that she keeps is 45 cents. What is the price of each orange?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Let \( ABCD \) and \( WXYZ \) be two squares that share the same center such that \( WX \parallel AB \) and \( WX < AB \). Lines \( CX \) and \( AB \) intersect at \( P \), and lines \( CZ \) and \( AD \) intersect at \( Q \). If points \( P, W \), and \( Q \) are collinear, compute the ratio \( AB / WX \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2} + 1}
Given the proposition "There exists an $x \in \mathbb{R}$, such that $x^2+1<0$", what is the direct statement of its negation? Express your answer as a complete sentence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{For all $x \in \mathbb{R}$, $x^2+1 \geq 0$.}
There are numbers from 1 to 2013 on the blackboard. Each time, two numbers can be erased and replaced with the sum of their digits. This process continues until there are four numbers left, whose product is 27. What is the sum of these four numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
In a similar setup, square $PQRS$ is constructed along diameter $PQ$ of a semicircle. The semicircle and square $PQRS$ are coplanar. Line segment $PQ$ has a length of 8 centimeters. If point $N$ is the midpoint of arc $PQ$, what is the length of segment $NS$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{10}}
Eight distinct points, $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, $P_6$, $P_7$, and $P_8$, are evenly spaced around a circle. If four points are chosen at random from these eight points to form two chords, what is the probability that the chord formed by the first two points chosen intersects the chord formed by the last two points chosen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Given real numbers $x$ and $y$ satisfy $|x-4|+(y-8)^{2}=0$, then the perimeter of an isosceles triangle with sides of length $x$ and $y$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
The Fibonacci sequence is defined as follows: $F_{0}=0, F_{1}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for all integers $n \geq 2$. Find the smallest positive integer $m$ such that $F_{m} \equiv 0(\bmod 127)$ and $F_{m+1} \equiv 1(\bmod 127)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
The tower clock strikes three times in 12 seconds. For how long will it strike six times?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30 \, \text{seconds}}
A building has installed 5 colored lights to welcome the 2010 Asian Games in Guangzhou. The sequence in which they shine is not fixed, and each light can only shine in one of the colors: red, orange, yellow, green, or blue, with each of the 5 lights shining in a different color. Let's define the ordered shining of these 5 lights once as one "flash". In each flash, only one light shines per second, and the interval between two consecutive flashes is 5 seconds. Calculate the minimum time required to achieve all different flashes. Express your answer in seconds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1195}
Evaluate $2000^3 - 1999 \cdot 2000^2 - 1999^2 \cdot 2000 + 1999^3$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3999}
Find the total number of cards in a stack where cards are numbered consecutively from 1 through $2n$ and rearranged such that, after a similar process of splitting into two piles and restacking alternately (starting with pile B), card number 252 retains its original position.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{504}
Find the maximum value of \( x_0 \) for which there exists a sequence \( x_0, x_1, \ldots, x_{1995} \) of positive reals with \( x_0 = x_{1995} \) such that for \( i = 1, \ldots, 1995 \): 
\[ x_{i-1} + \frac{2}{x_{i-1}} = 2x_i + \frac{1}{x_i}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{997}}
11 groups perform at a festival. Each day any groups not performing watch the others (but groups performing that day do not watch the others). What is the smallest number of days for which the festival can last if every group watches every other group at least once during the festival?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
对一切映射 f:\{0,1,2,\cdots,99\}\rightarrow\{0,1,2,\cdots,99\}，求下式的最小可能值：S=\ds{x=0}{99}|f(f(x))-((x+1) \text{mod} 100)|.这里的记号 a \text{mod} 100 表示正整数 a 除以 100 所得的余数.（董子超、吴茁供题）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a redesign of his company's logo, Wei decided to use a larger square and more circles. Each circle is still tangent to two sides of the square and its adjacent circles, but now there are nine circles arranged in a 3x3 grid instead of a 2x2 grid. If each side of the new square measures 36 inches, calculate the total shaded area in square inches.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1296 - 324\pi}
Let $\Delta ABC$ be an equilateral triangle. How many squares in the same plane as $\Delta ABC$ share two vertices with the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A train traveling at an initial speed of 90 kmph and accelerating at a constant rate of 0.5 m/s² overtakes a motorbike traveling at a constant speed of 72 kmph. The train takes 50 seconds to completely pass the motorbike. What is the length of the train in meters, given that the motorbike's length is 2 meters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{877}
Teams A and B each have 7 players who will compete in a Go tournament in a predetermined order. The match starts with player 1 from each team competing against each other. The loser is eliminated, and the winner next competes against the loser’s teammate. This process continues until all players of one team are eliminated, and the other team wins. Determine the total number of possible sequences of matches.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3432}
What is the expected time it takes for a monkey typing random capital letters to produce the sequence "ABBA"? The problem is modeled as a Markov Process. While $26^4$ is a rough estimate, what is the precise expected time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{456976}
将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+\operatorname{def}$ 是奇数的概率为 $\qquad$.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{11}
A high school bowling team's 5 members scored a total of 2000 points in their first match. The first bowler scored 1/4 as many points as the second, the second bowler scored 5/3 times as high as the third bowler who didn't exceed 500 points, and the third bowler scored 3/5 as many points as the fourth bowler. The fourth bowler scored 9/10 of what the fifth bowler scored. How many points did the fifth bowler score?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{561}
A farmer bought 749 sheep. He sold 700 of them for the price paid for the 749 sheep. The remaining 49 sheep were sold at the same price per head as the other 700. Calculate the percent gain on the entire transaction. Express your answer as a percentage, rounded to two decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
9.5. The cells of a chessboard are painted in 3 colors - white, gray, and black - in such a way that adjacent cells, sharing a side, must differ in color, but a sharp change in color (i.e., a white cell adjacent to a black cell) is prohibited. Find the number of such colorings of the chessboard (colorings that coincide when the board is rotated by 90 and 180 degrees are considered different).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{33}}
Place the numbers $0, 1, 2, 3, 4, 5, 6, 7$ on the eight vertices of a cube (each vertex having one number, with each number used exactly once) in such a way that the sum of the two numbers at the ends of each edge is always a prime number. What is the maximum sum of the four numbers on one face of the cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Suppose that a positive integer $N$ can be expressed as
the sum of $k$ consecutive positive integers
$$N=a+(a+1)+(a+2)+\cdots+(a+k-1)$$
for $k=2017$ but for no other values of $k>1.$ Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of these expressions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
An icosahedron is a regular polyhedron with 20 equilateral triangular faces. If the faces are colored either blue or white, what is the maximum number of blue faces that can be colored such that no two blue faces are adjacent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given that $f \in C^{(1)} [0,1]$, $f(0) = a$, and $\int_{0}^{1} f(x) \, dx = 0$, find the minimum value of $\int_{0}^{1} (f'(x))^2 \, dx$. Provide your answer in terms of $a$ if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3a^2}
Determine the boundary value of the positive integer \(n\) for which the integral \(\int_{a}^{\infty} \frac{1}{x (\ln x)^n} \, dx\) converges, where \(a > 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the conditional expectation \( \mathsf{E}(N_3 \mid N_5 = 5) \), where \( N_t \) is a Poisson process with rate 1, and \( X_t \) and \( Y_t \) are independent Poisson processes that contribute to \( N_t \) with probability \( P = 0.5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the limit: $$\lim_{x\to\infty}\left(\int_0^xe^{t^2}dt\right)^{\frac{1}{x^2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Evaluate the series \( \sum_{n=1}^\infty \frac{1}{2n-1} \sin((2n-1)x) \) for \( x \in (0, \pi) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Determine the correct limit for \( r \) when transforming the integral \( \iiint_{E}z\sqrt{x^2+y^2}dV \) to cylindrical coordinates, where \( E \) is the region bounded by the cylinder \( x^2+y^2=2x \) and the planes \( y=0 \), \( z=0 \), and \( z=a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 \leq r \leq 2\cos\theta}
Given that \( \det(A) = \frac{1}{24} \), solve for the determinant of the matrix expression:
\[
\det\left(\left(\frac{1}{3}A\right)^{-1} - 120 A^*\right)
\] where \( A^* \) is the adjugate of \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-192}
Determine the smallest integer n such that there exists a Pythagorean n-tuple \((X_1, X_2, \ldots, X_n)\) where each \(X_i\) is a prime number and satisfies the equation:
\[ X_1^2 + X_2^2 + \cdots + X_{n-1}^2 = X_n^2 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
The roots of the polynomial \( f\left( x\right)  = {x}^{8} + {x}^{7} - {x}^{5} - {x}^{4} - {x}^{3} + x + 1 \) are all roots of unity. We say that a real numbe
 \( r \in  \lbrack 0,1) \) is nice if \( {e}^{2i\pi r} = \cos {2\pi r} + i\sin {2\pi r} \) is a root of the polynomial \( f \) and if \( {e}^{2i\pi r} \) has positive imaginary part
 be the sum of the values of nice real numbers \( r \) . If \( S = \frac{p}{q} \) for relatively prime positive integers \( p,q \) ,find \( p + q \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
31
}
Find the number of subgroups in the dihedral group $D_{16}$ that contain the stabilizer of a vertex in the action of $D_{16}$ on the vertices of an octagon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the expected value $E[S(t)]$ for the expression $S(t) = \sum_{i=1}^{N(t)} (X_i + T_i)^2$, where $X_i$ are independent and identically distributed random variables with density $f(x) = \frac{1}{2}$ for $0 \leq x \leq 2$, and $T_i$ are the arrival times of a homogeneous Poisson process $N(t)$ with intensity $\lambda = 3$. Assume $N(t)$ is independent of $X_i$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t^3 + 3t^2 + 4t}
Find the limit of the following expression as \( n \to +\infty \):
\[ \lim_{n \to +\infty} \sqrt[n]{\frac{1}{3^n}+\frac{n^5}{5^n}+\frac{1}{2^{nn^{10^{10}}}}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}