inclusionAI/Ring-lite-rl-data huggingface dataset:
A train of certain length traveling at 72 kmph takes 13.998880089592832 seconds to cross a bridge of 170 m in length. How long is the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{109.98}
For certain real values of $p, q, r,$ and $s,$ the equation $x^4+px^3+qx^2+rx+s=0$ has four non-real roots. The product of two of these roots is $17 + 2i$ and the sum of the other two roots is $2 + 5i,$ where $i^2 = -1.$ Find $q.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
$3$名男同学、$3$名女学生和$2$位老师站成一排拍照合影，要求$2$位老师必须站正中间，队伍左右两端不能同时是一男学生与一女学生，则总共有__________种排法．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$576$}
Given the universal set $I=\{(x,y)|x\in\mathbb{R},\ y\in\mathbb{R}\}$, set $A=\{(x,y)|\frac{y-3}{x-2}=1\}$, and set $B=\{(x,y)|y=x+1\}$, find $(C_{I}A)\cap B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2,3)}
Find the mean number of records set by the first $n$ jumpers, where each jumper's height is an independent and identically distributed continuous random variable. What is the behavior of this mean as $n$ approaches infinity?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln n}
What is the correct ordering of the three numbers, $10^8$, $5^{12}$, and $2^{24}$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2^{24}<10^{8}<5^{12} $

}
Let $G$ be the centroid of $\triangle ABC$, and $\frac{\sqrt{7}}{\overrightarrow{GA}}\sin A + 3\overrightarrow{GB}\sin B + 3\sqrt{7}\overrightarrow{GC}\sin C = \overrightarrow{0}$, then the magnitude of angle $B$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^{\circ}}
Dr. Riley, an archaeologist studying ancient artifacts, has discovered a collection of 120 bronze coins buried in an old excavation site. Over the years, corrosion has affected 45% of these coins, making them difficult to analyze and categorize. Dr. Riley plans to clean the corroded coins to restore them for further study. 

After cleaning each corroded coin, Dr. Riley finds that 60% of them can be fully restored, while the rest are too damaged to be of any use. 

How many of the original 120 bronze coins are successfully restored and usable for further study after the cleaning process?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Points $D$ and $E$ are chosen on the exterior of $\vartriangle ABC$ such that $\angle  ADC = \angle  BEC = 90^o$. If $\angle ACB = 40^o$, $AD = 7$, $CD = 24$, $CE = 15$, and $BE = 20$, what is the measure of $\angle ABC $ in,degrees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70^\circ}
Compute the residues of all singularities of $\tanh(z)$ and evaluate the integral $\int_C \tanh(z) \, dz$, where $C$ is the circle of radius $12$ centered at $z_0=0$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16\pi i}
Marco visits Ristorante Machiavelli every Thursday for dinner. He has been a regular customer for 5 years. Each visit, he orders his favorite pasta dish for $12 and a dessert for $5. Every third visit, he also orders a special appetizer for an additional $8. How much money has Marco spent on average at the restaurant each year, considering he visits every week without missing a Thursday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1020}
A train passenger arrives at a busy railway station and chooses a taxi stand. At stand A, taxis arrive according to a Poisson process with a rate of 2 taxis per minute, and passengers choose this stand with a probability of 0.7. At stand B, taxis arrive according to a Poisson process with a rate of 1 taxi per minute, and passengers choose this stand with a probability of 0.3. After waiting for 3 minutes at the front of the queue without a taxi, what is the probability that the passenger is waiting at taxi stand A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{7 + 3e^3}}
Given a circle of radius $3$, there are multiple line segments of length $6$ that are tangent to the circle at their midpoints. Calculate the area of the region occupied by all such line segments.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9\pi}
If the inequality $2ae^{2x}-\ln x+\ln a\geqslant 0$ holds for any real number $x \gt 0$, then the minimum value of the real number $a$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2e}}
Let $R$ be a square region and $n\ge4$ an integer. A point $X$ in the interior of $R$ is called $n\text{-}ray$ partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2320
}
Evaluate the limit: \[ \lim_{(x, y) \to (0, 0)} (1+x^2+y^2)^{\frac{1}{x^2+y^2+x y^2}}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
A magician has one hundred cards numbered $1$ to $100$. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card.

A member of the audience selects two of the three boxes, chooses one card from each and announces the sum of the numbers on the chosen cards. Given this sum, the magician identifies the box from which no card has been chosen.

How many ways are there to put all the cards into the boxes so that this trick always works? (Two ways are considered different if at least one card is put into a different box.)


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12 
}
Given $a<0$ and $-1<b<0$, then the order from smallest to largest of $a$, $ab$, $ab^2$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a < ab^2 < ab}
Let $X$ be a set such that for every metric space $Y$ containing $X$, the set $X$ is open in $Y$. Must $X$ be the empty set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
What is the maximum number of zeros for the Laplace transform of a function that is the difference of two sums, each containing $k$ exponential terms with positive exponents?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2k - 1}
How many structural isomers have the formula ${C_3H_6Cl_2}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4
}
From the positive integers,  $m,m+1,\dots,m+n$ , only the sum of digits of  $m$  and the sum of digits of  $m+n$  are divisible by  $8$ . Find the maximum value of  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A criminal must pass 10 lie detector tests successfully to escape from jail. The probabilities of each lie detector detecting a lie are as follows: 0, \( \frac{1}{2} \), \( \frac{2}{3} \), \( \frac{3}{4} \), \( \frac{4}{5} \), \( \frac{5}{6} \), \( \frac{6}{7} \), \( \frac{7}{8} \), \( \frac{8}{9} \), \( \frac{9}{10} \). If a detector indicates a lie, the criminal must start over from the first detector. Each test takes 1 second. Calculate the expected time in seconds for the criminal to pass all tests and escape.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9864100}
Ten spheres of the same radius are stacked in the form of a triangular pyramid so that each sphere touches at least three others. Find the radius of the circumscribed sphere for the pyramid of spheres, given that the radius of the sphere inscribed in the center of the pyramid of spheres, touching six identical spheres, is $\sqrt{6}-1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5(\sqrt{2}+1)}
A fair die is rolled repeatedly. Let $X$ be the number of rolls needed to obtain a 5, and $Y$ the number of rolls needed to obtain a 6. Calculate the expected value $E[X \mid Y=2]$, where $Y=2$ indicates that a 6 is rolled on the second roll.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{33}{5}}
Given the parabola y = x^2 - 2x + 1 and points A(0, y1), B(1, y2), and C(-2, y3) on the parabola, determine the relationship among y1, y2, and y3. Express your answer using inequality symbols in the format yX > yY > yZ, where X, Y, and Z are the corresponding point indices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y3 > y1 > y2}
In a group of 10 chickens, 9 are real and each weighs 2kg, while 1 is a false chicken that weighs either 1kg or 3kg, each with a probability of \(\frac{1}{2}\). A machine randomly selects each chicken with a probability of \(\frac{1}{2}\) for weighing, and after weighing, all chickens are returned. Three weightings are performed, resulting in sets weighing 11kg, 12kg, and 13kg, respectively. What is the probability that the false chicken weighs 3kg?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{5}}
Evaluate the second-order partial derivative \( \frac{\partial^2}{\partial \bar{z}\,\partial z}\log|z|^2 \) for the function \( z \mapsto \log|z|^2 \) defined on \( \mathbb{C} - \{0\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Walter has exactly one penny, one nickel, one dime and one quarter in his pocket.  What percent of one dollar is in his pocket?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41\%}
On the side  $ AB$  of a cyclic quadrilateral  $ ABCD$  there is a point  $ X$  such that diagonal  $ BD$  bisects  $ CX$  and diagonal  $ AC$  bisects  $ DX$ . What is the minimum possible value of  $ AB\over CD$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the real number set R, set A = {x|0 ≤ x ≤ 6}, and set B = {x|x > 5}, describe the resulting set from the intersection of the complement of B in R and set A. Express your answer in set notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{ x | 0 ≤ x ≤ 5 }}
Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$. Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$. Does there exist a natural number $d = d(\overline{X})$ such that for any $k$-form $X$ of $\overline{X}$, the variety $X$ has a $K$-point over some finite field extension $K$ of $k$ with degree $[K:k] \le d$? Answer YES or NO.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{NO}
The sequence starts with 12, 13, 15, 17, 111, 113, 117, 119, followed by a number, then 129, and 131. What is the ninth number in the sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
Let \(f(z)= \frac{z+a}{z+b}\) and \(g(z)=f(f(z))\), where \(a\) and \(b\) are complex numbers. Suppose that \(\left| a \right| = 1\) and \(g(g(z))=z\) for all \(z\) for which \(g(g(z))\) is defined. Find the difference between the largest and smallest possible values of \(\left| b \right|\). The original answer is in the form \(\sqrt{m} - n\). Please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine whether the Stieltjes integral \( \int_{0} ^{4} x^{2} d([x^{2}]) \) exists. If it exists, calculate its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{136}
Consider the real sequence $x_0=a$ and $x_{n+1}=\frac{1}{4}+x_n-(x_n)^2$ for $n\geq 1$. Determine the values of $a$ for which the sequence $x_n$ converges to $-\frac{1}{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2}}
Evaluate the limit: $$\lim_{x\to0}\left(\frac {\csc(x)}{x^3} - \frac{\sinh(x)}{x^5}\right)$$ using L'Hopital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{90}}
A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers  $a$  and  $b$ . A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between  $a$  and  $b$  be for this to happen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{a}{b} }
Find all pairs of positive integers \((m, n)\) such that \( \frac{m}{n} = n.m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(5, 2)}
Given that point $P$ lies on the line $y=2x$, if there exist two points $A$ and $B$ on the circle $C(x-3)^2+y^2=4$ such that $PA \perp PB$, then the range of the abscissa $x_{0}$ of point $P$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[ \frac{1}{5},1\right]}
Fran writes the numbers \(1,2,3, \ldots, 20\) on a chalkboard. Then she erases all the numbers by making a series of moves; in each move, she chooses a number \(n\) uniformly at random from the set of all numbers still on the chalkboard, and then erases all of the divisors of \(n\) that are still on the chalkboard (including \(n\) itself). What is the expected number of moves that Fran must make to erase all the numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{131}{10}}
Find the largest prime number that divides the quantity $0! + (1!) \times 1 + (2!) \times 2 + (3!) \times 3 + \cdots + (50!) \times 50$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
Given a 10cm×10cm×10cm cube cut into 1cm×1cm×1cm small cubes, determine the maximum number of small cubes that can be left unused when reassembling the small cubes into a larger hollow cube with no surface voids.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{134}
a, b, c, and d have a total of Rs. 2000 between them. a and c together have Rs. 900, b and c together have Rs. 1100, and a and d together have Rs. 700. How much money does c have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Determine the number of isomorphism classes of associative rings with identity that have 35 elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
One day, a beverage warehouse sold 252 cans of soda to 100 customers, and each customer bought at least one can of soda. On this day, what is the maximum possible median number of cans of soda purchased by each customer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.5}
A bag contains 2 white balls and 2 black balls. Each time a ball is taken from the bag, if it's white it's returned to the bag and if it's black it's replaced by a white ball. The game ends when no black balls remain in the bag. Let Y be the number of times the game is played; calculate Var(Y).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
For all real numbers $x$ and $y$, define the mathematical operation $\diamond$ such that $x \diamond 0 = 2x, x \diamond y = y \diamond x$, and $(x + 1) \diamond y = (x \diamond y) \cdot (y + 2)$. What is the value of $6 \diamond 3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{93750}
Evaluate the limit \( \lim_{n\to\infty}\dfrac{1}{n^{2+\beta}}\sum_{i=1}^{n}i^{\beta} \) using the Riemann integral of suitably chosen functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Elmo and Elmo\u0027s clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player\u0027s turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo\u0027s clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
Elmo's clone}
In an equilateral triangle \(ABC\) with side length 2011, a point \(P\) is located inside the triangle. Line segments \(DE\), \(FG\), and \(HI\) are drawn through \(P\) such that \(DE \parallel BC\), \(FG \parallel CA\), and \(HI \parallel AB\). Given that the ratio of the lengths of these segments is \(DE : FG : HI = 8 : 7 : 10\), calculate the sum \(DE + FG + HI\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4022}
给定正整数 $n 、 k$, 其中, $n$ 为偶数, $k \geqslant 2$,且 $n>4 k$. 在圆周上有 $n$ 个点, 若 \frac{n}{2} 条在圆内互不相交的弦的端点恰为这 $n$ 个点，则称这些弦为一个"匹配"。求最大的整数 $m$ ，满足对任意一个匹配，均存在圆周上连续 $k$ 个点，使得这个匹配中至少有 $m$ 条弦的所有端点均在这 $k$ 个点中. 请给出当$ k = 7$ 时 $m$ 的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine whether the following limit is true: \( \lim_{a\to -\infty}\int_{-\infty}^a f(x)\, dx=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $f: X \rightarrow S^1$ be a continuous map from a path-connected topological space $X$, and let $p: \mathbb{R} \rightarrow S^1$ be the universal covering map. Determine the condition under which there exists a continuous map $h: X \rightarrow \mathbb{R}$ such that $p \circ h = f$. Provide the condition in terms of the fundamental group of $X$ and the map $f$. \( \pi_1(X) \rightarrow \pi_1(S^1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f_*(\pi_1(X)) = 0}
How many three-digit positive numbers \( x \) are there that are divisible by 3 and satisfy the equation \( \gcd(15, \gcd(x, 20)) = 5 \)? Find the largest of them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{990}
In a convex 12-sided polygon, all angles are equal. It is known that the lengths of ten of its sides are 1, and the length of one more side is 2. What could be the area of this 12-sided polygon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8 + 4\sqrt{3}}
Evaluate the integral \( \oint_C \frac{\sinh(z)\cos(z)}{\cos(z)}dz \) using the Residue Theorem, where \( C \) is the contour defined by \( |z|=(n+1/2)\pi \) with \( n \in \mathbb{N}_0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a school chat, a vote was held: "On which day to hold the disco: October 22 or October 29?"

The graph shows how the votes were distributed an hour after the start of the vote.

Then, 80 more people participated in the vote, voting only for October 22. After that, the voting finished. The final distribution of votes is also shown on the graph.

How many people participated in the vote?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{260}
Given a certain sunflower cell has 34 chromosomes at the late stage of the second meiotic division when forming pollen grains, determine the number of tetrads that can be produced by this cell during meiosis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
The hyperbola $C$: $\dfrac{x^2}{4} - y^2 = 1$ has its left and right foci as $F_1$ and $F_2$, respectively. A line passing through $F_1$ intersects the left branch of the hyperbola at points $A$ and $B$. The minimum perimeter of $\Delta F_2AB$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the constant $k$ so that the graphs of the parabolas $y = x^2 + k$ and $x = y^2 + k$ are tangent to each other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
When taking the Fast Fourier Transform (FFT) of a 2D Gaussian Dispersal Kernel given by the function \( G(x,y) = \frac{1}{4\pi \mu} e^{-\frac{x^2 + y^2}{4\mu}} \), where the spatial domain is a 7x7 grid with 500m cell length, and using a grid of 8 intervals (\(N = 8\)) for efficiency, the kernel is scaled by a factor of 4 during the calculation \(fK2D = dx \cdot dy \cdot \text{fft2}(K2D)\). Determine the normalization factor required for the kernel to ensure correct population values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
The focus of the parabola $y^{2}=-8x$ coincides with the left focus of the hyperbola $\frac {x^{2}}{a^{2}}-y^{2}=1$. Find the angle between the two asymptotes of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
The competition with $n (n \geqslant 2)$ contestants lasts for $k$ days. Each day, the scores of the contestants are exactly $1, 2, \cdots, n$ (with no two contestants having the same score). After $k$ days, the total score of each contestant is 26. Find all possible values of $(n, k)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(25,2),(12,4),(3,13)}
Let the vertices of a graph $G$ be the integers $1,2,\dots,99$. The numbers $i \neq j$ are connected if their greatest common divisor is at least $3$. Find the chromatic number of $G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
We divide the height of a cone into three equal parts, and through the division points, we lay planes parallel to the base. How do the volumes of the resulting solids compare to each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:7:19}
Call a number unremarkable if, when written in base $10$, no two adjacent digits are equal. For example, $123$ is unremarkable, but $122$ is not. Find the sum of all unremarkable $3$-digit numbers. Note that numbers like $012$ and $007$ are not considered $3$-digit numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400185}
The last digit in the natural number is 2016 times less than the number itself. Find all such numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4032, 8064, 12096, 16128}
Determine the number of triples $0 \le k,m,n \le 100$ of integers such that \[ 2^mn - 2^nm = 2^k. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
例5 若自然数n满足这样的条件：存在由n个实数组成的数列，使得任何连续17个项之和为正，而任何连续10个项之和为负，求n的最大值．（改编题）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Evaluate the sum $\sum_{n=-\infty}^\infty \frac{\sin nx}{1+n^2}$ for $x\in[-\pi,\pi]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In how many ways can we place two bishops on a chessboard in different rows, columns, and squares of different colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{768 \text{ ways}}
3.3. Two identical cylindrical vessels are connected at the bottom by a small-section pipe with a valve. While the valve was closed, water was poured into the first vessel, and oil into the second, so that the level of the liquids was the same and equal to \( h = 40 \, \text{cm} \). At what level will the water in the first vessel stabilize if the valve is opened? The density of water is 1000 kg \(/ \text{m}^3\), and the density of oil is 700 kg \(/ \text{m}^3\). Neglect the volume of the connecting pipe. Provide the answer in centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
The product of three prime numbers equals five times their sum. Find these three numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2, 5, 7}