inclusionAI/Ring-lite-rl-data huggingface dataset:
In the chart, each number below the top row is the positive difference of the two numbers to the right and left in the row immediately above it. What is the value of x? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), the left and right foci coincide with the symmetric points about the two asymptotes, respectively. Then, the eccentricity of the hyperbola is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\sqrt{2}"]}
In the diagram, triangle \(ABC\) lies between two parallel lines as shown. If segment \(AC\) has length \(5 \ \text{cm}\), what is the length (in \(\text{cm}\)) of segment \(AB\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In the given grid, all circles share the same center. What is the ratio of the area of the highlighted gray region to the area of the largest circle? Express your answer as a fraction or ratio.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Rectangle \(PQRS\) is divided into 60 identical squares, as shown. The length of the diagonal of each of these squares is 2. Calculate the length of \(QS\), rounded to 2 decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18.38}
In the given graph of the function \( y = kx + b \), compare \( |k| \) and \( |b| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|k| < |b|}
A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt{5}-1$}
For positive integers $a$ and $N$, let $r(a, N) \in \{0, 1, \dots, N - 1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \le 1000000$ for which
\[r(n, 1000) > r(n, 1001).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{499500}
Regular hexagon $ABCDEF$ has its center at $G$. Each of the vertices and the center are to be associated with one of the digits $1$ through $7$, with each digit used once, in such a way that the sums of the numbers on the lines $AGC$, $BGD$, and $CGE$ are all equal. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Alice and Bob play a game on a circle with 8 marked points. Alice places an apple beneath one of the points, then picks five of the other seven points and reveals that none of them are hiding the apple. Bob then drops a bomb on any of the points and destroys the apple if he drops the bomb either on the point containing the apple or on an adjacent point. Bob wins if he destroys the apple, and Alice wins if he fails. If both players play optimally, what is the probability that Bob destroys the apple?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
In the diagram, $\triangle ABC$ is isosceles with $AB = AC$ and $BC = 30 \mathrm{~cm}$. Square $EFGH$, which has a side length of $12 \mathrm{~cm}$, is inscribed in $\triangle ABC$, as shown. Calculate the area of $\triangle AEF$ in square centimeters (cm^2).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Two trains of length 120 m and 300 m are running towards each other on parallel lines at 42 kmph and a certain speed respectively. They will be clear of each other from the moment they meet in 20.99832013438925 seconds. What is the speed of the second train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Each of the three cutlets needs to be fried on a pan on both sides for five minutes each side. The pan can hold only two cutlets at a time. Is it possible to fry all three cutlets in less than 20 minutes (ignoring the time for flipping and transferring the cutlets)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15 \text{ minutes}}
Find the x-coordinate of the point of intersection of the functions \( y = 2^x \) and \( y = \frac{1}{x} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.641}
In how many ways can the number 210 be factored into a product of four natural numbers? The order of the factors does not matter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\mathcal{T}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2520}
Find the mass percentage of N in NH4Br
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.30\%}
What is the molecular weight of a compound having 2 H, 3 C, 1 N, 1 Cl, and 3 O?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135.51}
Determine the values of the parameter \(a\) for which the function \(f(x) = 8ax - a\sin 6x - 7x - \sin 5x\) is increasing and has no critical points for all real \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(6, \infty)}
Vasya solved problems for 10 days, solving at least one problem each day. Each day (except for the first day), if the weather was cloudy, he would solve one more problem than on the previous day, and if it was sunny, he would solve one less problem. In the first 9 days, Vasya solved a total of 13 problems. What was the weather on the tenth day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Cloudy}
Arrange the numbers 1 through 9 in a 3x3 grid such that consecutive numbers are adjacent (share an edge), and the sum of the numbers in the four corners is 18. What is the number in the center of the grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
For those real numbers $x_1 , x_2 , \ldots , x_{2011}$ where each of which satisfies $0 \le x_1 \le 1$ ($i = 1 , 2 , \ldots , 2011$), find the maximum of
\[ x_1^3+x_2^3+ \cdots + x_{2011}^3 - \left( x_1x_2x_3 + x_2x_3x_4 + \cdots + x_{2011}x_1x_2 \right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1340}
Triangle $ABC$ has $AB = 5$, $BC = 13$, and $AC = 12$. Square $BCDE$ is constructed outside of the triangle. The perpendicular line from $A$ to side $DE$ cuts the square into two parts. What is the positive difference in their areas?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{119}
Jones is waiting for one of two telephone booths occupied by A and B. Let the duration of their calls be an exponential random variable with $\lambda=1/8$. What is the probability that of the three of them, Jones is not the last to finish their call?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
In an isosceles triangle \(ABC\) with \(AB = BC = 25\) cm and \(AC = 14\) cm, find the radius of a circle that touches \(BC\) at point \(D\), which is the foot of the altitude \(AD\), and passes through the midpoint of \(AC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{175}{48} \, \text{cm}}
Mena listed the numbers from 1 to 30 one by one. Emily copied these numbers and substituted every digit 2 with digit 1. By how much is the sum that Mena calculated greater than the sum that Emily calculated?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{103}
Bill picked 50 apples from the orchard with his wife and two children. He sends each of his kids to school with 3 apples for their two favorite teachers. His wife Jill bakes two apple pies, using a certain number of apples per pie. Bill has 24 apples left. How many apples does Jill use per pie?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
On the Island of Misfortune, there are knights, who always tell the truth, and liars, who always lie. One day, $n$ islanders gathered in a room.

The first person said: "Exactly 1 percent of the people present in this room are liars."

The second person said: "Exactly 2 percent of the people present in this room are liars."

and so on.

The person with number $n$ said: "Exactly $n$ percent of the people present in this room are liars."

How many people could be in the room, given that it is known that at least one of them is a knight?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Find the value of \(a\) such that \(\lim_{x\to\infty} \frac{(ax)!}{x^x} = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
The radii of five concentric circles \(\omega_{0}, \omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}\) form a geometric progression with common ratio \(q\). What is the largest value of \(q\) for which it is possible to draw an open broken line \(A_{0} A_{1} A_{2} A_{3} A_{4}\), consisting of four segments of equal length, where \(A_{i}\) lies on \(\omega_{i}\) for all \(i = 0, 1, 2, 3, 4\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5}+1}{2}}
A taxi driver passes through six traffic checkpoints on the way from the restaurant to the train station. Assuming that the events of encountering a red light at each checkpoint are independent of each other and the probability is $\frac{1}{3}$ at each checkpoint, calculate the probability that the driver has passed two checkpoints before encountering a red light.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{27}}
Find the expectation \( E[e^{iu(X_{4/5}-\frac{1}{2} X_{3/5})}|X_{3/5}] \) where \( X_t \) is a Brownian bridge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-u^2/20}}
Suppose $15\%$ of $x$ equals $20\%$ of $y$. What percentage of $x$ is $y$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
75
}
Two small weights are attached to the ends of a vertically oriented uniform spring. The weight with mass $m_{1}$ is on the top, and the weight with mass $m_{2}$ is at the bottom. A person grasps the middle of the spring and holds it vertically in the air. As a result, the upper half of the spring is deformed by $x_{1} = 8 \, \text{cm}$, and the lower half by $x_{2} = 15 \, \text{cm}$. Then, without inverting it, the person places the spring on a horizontal surface and releases it. Determine the deformation of the spring in this case. Neglect the size of the person's hand compared to the length of the spring.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16\, \text{cm}}
In an  ${8}$ × ${8}$  squares chart , we dig out  $n$  squares , then we cannot cut a "T"shaped-5-squares out of the surplus chart .
Then find the mininum value of  $n$  .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
In a 150 member association consisting of men and women, exactly 10% of men and exactly 20% of women are homeowners. What is the least number of members who are homeowners?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In the country of Draconia, there are red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Furthermore, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:

- 1st head: "To my left is a green dragon."
- 2nd head: "To my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."

What is the maximum number of red dragons that could have been at the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{176}
Cagney can frost a cupcake in 15 seconds and Lacey can frost a cupcake in 25 seconds. After a 1-minute warm-up, their frosting speeds increase by 20%. Calculate the total number of cupcakes they can frost together in 10 minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76}
If $V = gt + V_0$ and $S = \frac{1}{2}gt^2 + V_0t$, then $t$ equals:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2S}{V+V_0}}
Given a positive sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, for $n \geq 2$, $(a_n - S_{n-1})^2 = S_nS_{n-1}$, and $a_1 = 1$. Let $b_n = \log_2 \frac{a_{n+1}}{6}$. Derive an expression for $b_n$ in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n-3}
A cylindrical glass weighs 200 pounds and has a center of gravity 4.8 cm high. The base area of the glass is 20 cm². Determine the height of the water at which the center of gravity of the glass and water system is the lowest.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
2. (10 points) In each cell of a $50 \times 50$ square, a number is written that is equal to the number of $1 \times 16$ rectangles (both vertical and horizontal) in which this cell is an end cell. In how many cells are numbers greater than or equal to 3 written?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1600}
Two swimmers, A and B, start swimming towards each other from opposite ends of a 100-meter pool at the same time. Swimmer A takes 72 seconds to swim 100 meters, and Swimmer B takes 60 seconds to swim 100 meters. Ignoring the time it takes to turn around, calculate how many times they meet in 12 minutes. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color you pulled out. What is the probability that all beads in the bag are red after three such replacements? Express your answer in the form of \frac{k}{m}, where k and m are coprime integers. Please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
You have a  $3 \times 2021$  chessboard from which one corner square has been removed. You also have a set of  $3031$  identical dominoes, each of which can cover two adjacent chessboard squares. Let  $m$  be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps.
What is the remainder when  $m$  is divided by  $19$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let triangle $ABC$ be a right triangle with right angle at $C.$ Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C.$ If $\frac{DE}{BE} = \frac{8}{15},$ then find $\tan B.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4 \sqrt{3}}{11}}
Given that $| \overrightarrow{a}|=12$, $| \overrightarrow{b}|=9$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-54 \sqrt {2}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\pi}{4}}
The circle C is tangent to the y-axis and the line l: y = (√3/3)x, and the circle C passes through the point P(2, √3). Determine the diameter of circle C.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{14}{3}}
For certain ordered pairs $(a,b)\,$ of real numbers, the system of equations

$ax+by=1\,$
$x^2+y^2=50\,$
has at least one solution, and each solution is an ordered pair $(x,y)\,$ of integers.  How many such ordered pairs $(a,b)\,$ are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
In a group of people, 40% of the total number of people plus 60 more people constitutes the number of people wearing a specific type of hat. If 25% of the total number of people are wearing shoes of a specific brand, find the difference between the number of people wearing that specific type of hat and those wearing the specific brand of shoes, given that the number of people wearing both the hat and the shoes is 20% of the total population size. What is the total number of people in the group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1200}
What is the molecular weight of a compound having 1 Aluminium, 3 Bromine, 2 Oxygen, and 1 Chlorine in its molecular structure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{334.13}
Given three independent, identically distributed continuous random variables $X_1$, $X_2$, and $X_3$, each following an Exponential Distribution, calculate the probability that the median of these variables is closer to the smallest value $X_1$ than it is to the largest value $X_3$. Provide your answer as a probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^{\circ}$. Thus the three polygons form a new polygon with $A$ as an interior point. What is the largest possible perimeter that this polygon can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Evaluate the integral: \[ \int_0^{2\pi} x \frac{\cos x}{2-\cos^2 x} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A piece of alloy weighing 6 kg contains copper. Another piece of alloy weighing 8 kg contains copper in a different percentage than the first piece. A certain part was separated from the first piece, and a part twice as heavy was separated from the second piece. Each of the separated parts was then alloyed with the remainder of the other piece, resulting in two new alloys with the same percentage of copper. What is the mass of each part separated from the original alloy pieces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.4 \text{ kg}}
What is the molecular weight of a compound having 2 Al, 4 O, 6 H, 3 N, and 1 P?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{197.02}
Given that the domains of functions f(x) and g(x) are both R, and f(x) + g(2-x) = 5, g(x) - f(x-4) = 7. If the graph of y = g(x) is symmetric about the line x = 2, g(2) = 4, determine the value of \sum _{k=1}^{22}f(k).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-24}
The numbers $2^0, 2^1, \dots , 2{}^1{}^5, 2{}^1{}^6 = 65536$ are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one form the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number that can remain on the blackboard when there is only one number left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{131069}
设$T$为$2004^{100}$的所有正约数的集合，$T$的子集$S$满足$S$中的元素不是$S$中其他任意元素的整数倍.求$S$的元素个数的最大可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10201}
Given the function \( f(x) = \left(\frac{1}{a^{x}-1} + \frac{1}{2}\right) x^{2} + b x + 6 \) (where \(a\) and \(b\) are constants and \(a > 1\)), and \( f(\log \log_{8} 1000) = 8 \), find the value of \( f(\log \log 2) \). Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the two smallest consecutive natural numbers, each of which has a digit sum divisible by 7.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{69999, 70000}
Principal Skinner is thinking of two integers  $m$  and  $n$  and bets Superintendent Chalmers that he will not be able to determine these integers with a single piece of information.  Chalmers asks Skinner the numerical value of  $mn+13m+13n-m^2-n^2$ .  From the value of this expression alone, he miraculously determines both  $m$  and  $n$ .  What is the value of the above expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{169}
Let $S = \{1, 2, 3, \ldots, 280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{217}
Right triangle \(ABC\) has side lengths \(BC=6\), \(AC=8\), and \(AB=10\). A circle centered at \(O\) is tangent to line \(BC\) at \(B\) and passes through \(A\). A circle centered at \(P\) is tangent to line \(AC\) at \(A\) and passes through \(B\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
Given a triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. If $2b\cos B = a\cos C + c\cos A$, then $B = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3? Express your answer as a fraction in simplest form, 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{265}
Determine the limit of the sequence $\left(\frac{N_{2r}}{N_r}\right)_{r\in\mathbb{N}}$, where $N_r$ is the number of unit squares with integral rectangular coordinates that intersect a circle of radius $r$ centered at the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The decimal representation of \(\dfrac{1}{25^{10}}\) consists of a string of zeros after the decimal point, followed by non-zero digits. Find the number of zeros in that initial string of zeros after the decimal point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Mr. Rose, Mr. Stein, and Mr. Schwartz start at the same point around a circular track and run clockwise. Mr. Stein completes each lap in $6$ minutes, Mr. Rose in $10$ minutes, and Mr. Schwartz in $18$ minutes. How many minutes after the start of the race are the runners at identical points around the track (that is, they are aligned and are on the same side of the track) for the first time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Find the number of positive integers $n$ for which there exists a sequence $x_1, x_2, \cdots, x_n$ of integers with the following property: if indices $1 \le i \le j \le n$ satisfy $i+j \le n$ and $x_i - x_j$ is divisible by $3$, then $x_{i+j} + x_i + x_j + 1$ is divisible by $3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
For all real numbers $x$ such that $x \neq \pm1, \pm2, \dots, \pm10$, find the constant $k$ that satisfies the equation:

$$\sum_{j=1}^{10} \frac{4j}{x^2 - j^2} = k \sum_{j=1}^{10} \frac{1}{(x - j)(x + 11 - j)}$$

Calculator usage is not allowed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
$在\triangle ABC中，a=3，b-c=2，\cos  B=-\frac{1}{2}.$
$求\sin (B-C)的值.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{4\sqrt{3}}{7}$}