inclusionAI/Ring-lite-rl-data huggingface dataset:
A little rascal ran a race with a horse mounted on a mechanical carousel. After \(a\) seconds, she found that the horse, having made a circle, had caught up to her. Instantly turning around, the little rascal ran towards the horse at the same speed and met it after \(\frac{a}{2}\) seconds. Determine how long it takes for the carousel to make a full rotation, assuming all movements are uniform.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3} a \text{ seconds}}
Suppose we have a convex polygon in which all interior angles are integers when measured in degrees, and the interior angles at every two consecutive vertices differ by exactly \(1^{\circ}\). If the greatest and least interior angles in the polygon are \(M^{\circ}\) and \(m^{\circ}\), what is the maximum possible value of \(M - m\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
1. On the cube $A B C D A_{1} B_{1} C_{1} D_{1}$, points $P$ and $Q$ are chosen, where $P$ is the midpoint of edge $B C$, and $Q$ is the intersection of the diagonals in the square $C C_{1} D D_{1}$. The plane $A P Q$ divides the cube into two parts. Find the ratio of the volumes of the resulting parts of the cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{29}}
Determine the minimum value of the expression $2x^4 - 2x^2y^2 + y^4 - 8x^2 + 18$ where $x, y \in \mathbb{R}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \(x\) and \(y\) be independent, uniformly distributed random variables over \([0,1]\). Let \(a\) and \(R\) be positive real numbers where \(R>a\). How can we find the probability of 
$$P(x>\frac{ya}{R-a})?$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1-\frac{a}{2(R-a)}}
Calculate the sum of the largest and smallest angles in a triangle with side lengths of 5, 7, and 8.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
A machine has a present value of $1200. Its value depletes based on a variable depreciation rate depending on the age of the machine (in years): 10% for the first year, 12% for the second year, and so on, increasing by 2% for each subsequent year. Additionally, the machine undergoes annual repairs and maintenance which add 3% of the previous year's value back to the machine. However, the machine requires major overhauls at irregular intervals, with the next major overhaul scheduled at the end of the second year, which will decrease its value by an additional 15%. Calculate the machine's value after the major overhaul at the end of the 2nd year, considering all depreciation, repairs, and the major overhaul costs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$863.23}
Let $\Omega$ be a circle with radius $18$ and let $\mathcal{S}$ be the region inside $\Omega$ that the centroid of $\triangle XYZ$ sweeps through as $X$ varies along all possible points lying outside of $\Omega$, $Y$ varies along all possible points lying on $\Omega$ and $XZ$ is tangent to the circle. Compute the greatest integer less than or equal to the area of $\mathcal{S}$.

[I]Proposed by [b]FedeX333X[/b][/I]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{904}
In the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($0 < b < a$), a line passing through the center intersects the ellipse at points $A$ and $B$, and the right focus is $F_2(c,0)$. Calculate the maximum area of $\triangle ABF_2$ and express your answer as a product of two variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{bc}
Evaluate the expression: \[ \lg_{2} \left( \prod\limits_{a=1}^{2015} \prod\limits_{b=1}^{2015} (1 + e^{\frac{2\pi iab}{2015}}) \right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13725}
In Plutarch Enterprises, 70% of the employees are marketers, some percentage are engineers, and the rest are managers. Marketers make an average salary of $50,000 a year, engineers make an average of $80,000, and managers make an average of $370,000. If the average for all employees is also $80,000, what percentage of the employees are engineers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22.76\%}
On a chess board ( $8*8$ ) there are written the numbers  $1$  to  $64$ : on the first line, from left to right, there are the numbers  $1, 2, 3, ... , 8$ ; on the second line, from left to right, there are the numbers  $9, 10, 11, ... , 16$ ;etc. The  $\"+\"$  and  $\"-\"$  signs are put to each number such that, in each line and in each column, there are  $4$   $\"+\"$  signs and  $4$   $\"-\"$  signs. Then, the  $64$  numbers are added. Find all the possible values of this sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In triangle \( ABC \), the angle bisectors intersect at point \( O \). The line \( AO \) intersects the circumcircle of triangle \( OBC \) at points \( O \) and \( M \). Find \( OM \), given that \( BC = 2 \) and angle \( A \) is \( 30^{\circ} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(\sqrt{6} - \sqrt{2})}
Daniel has a (mostly) standard deck of 54 cards, consisting of 4 suits each containing the ranks 1 to 13 as well as 2 jokers. Daniel plays the following game: He shuffles the deck uniformly randomly and then takes all of the cards that end up strictly between the two jokers. He then sums up the ranks of all the cards he has taken and calls that his score.
    
Let $p$ be the probability that his score is a multiple of 13. There exists relatively prime positive integers $a$ and $b,$ with $b$ as small as possible, such that $|p - a/b| < 10^{-10}.$ What is $a/b?$

[i]Proposed by Dilhan Salgado, Daniel Li[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{77}{689}}
In the convex quadrilateral $ABCD$, we have the following side lengths: $AB = 15$, $BC = 16$, $CD = 12$, $DA = 25$, and the diagonal $BD = 20$. Let $M$ be the circumcenter and $\gamma$ the circumcircle of $\triangle ABD$. Line $CB$ intersects $\gamma$ again at point $F$. Line $AF$ intersects $MC$ at point $G$, and line $GD$ intersects $\gamma$ again at point $E$. Determine the area of the pentagon $ABCDE$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{396}
Captain Billy the Pirate looted 1010 gold doubloons and set sail on his ship to a deserted island to bury his treasure. Each evening of their voyage, he paid each of his pirates one doubloon. On the eighth day of sailing, the pirates plundered a Spanish caravel, doubling Billy's treasure and halving the number of pirates. On the 48th day of sailing, the pirates reached the deserted island, and Billy buried all his treasure at the marked spot—exactly 1000 doubloons. How many pirates set off with Billy to the deserted island?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
设实数 $a$ 满足 $a<9 a^{3}-11 a<|a|$, 则 $a$ 的取值范围是一个区间，区间的两端点和形如\frac{a\sqrt{b}+c\sqrt{d}}{e}，请给出a+b+c+d+e
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Calculate the nearest integer to $(3+\sqrt{5})^6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20608}
Determine the number of permutations \( a_1, a_2, \ldots, a_{10} \) of the numbers \( 1, 2, \ldots, 10 \) such that \( a_i > a_{2i} \) for \( 1 \le i \le 5 \) and \( a_i > a_{2i+1} \) for \( 1 \le i \le 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3360}
Given that $x,y,z$ are reals such that $16 \sin^4(x+y) + 49 \cos^4(x+y) = \sin^2(2x + 2y)(8\sin^2(x+z) + 6 \cos^2(y+z))$, the largest possible value of $3\cos^2(x+y+z)$ can be expressed in the form $\frac{a+b\sqrt{c}}{d}$, where $a$ and $b$ are integers, $c$ is a positive integer not divisible by the square of any prime, and $d$ is a positive integer such that $\gcd(a,b,d)=1$. Find $a+b+c+d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135}
Jackson starts at position 1 on a number line. At each step, he stays in place with a probability of 85% and moves one position to the right with a probability of 15%. Let \(d_n\) represent his position after \(n\) steps, and let \(E_n\) be the expected value of \(\frac{1}{d_n}\). Determine the smallest \(n\) such that \(\frac{1}{E_n} > 2017\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13446}
Determine the asymptotic behavior \( \Theta(S_n) \) of the summation \( S_n = \sum_{k=1}^{n}\frac{n-k+1}{k} \), where \( n \in \mathbb{Z}^{+} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n \log n)}
Determine the number of $2 \times 2$ matrices $A$ with entries in $\mathbb{Z}/3\mathbb{Z}$ such that $A^{50} = I$, where $I$ is the identity matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Evaluate the outward flux of the vector field \( V = x(x^2 + y^2 + z^2)^{-3/2}\mathbf{i} + y(x^2 + y^2 + z^2)^{-3/2}\mathbf{j} + z(x^2 + y^2 + z^2)^{-3/2}\mathbf{k} \) across the boundary of the rectangular box defined by the points \((-1, -1, -1)\) to \((1, 1, 1)\). Note that \( V \) is not defined at the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Given four points on a plane and their displacement vectors after a rotation, determine the center of rotation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(h, k)}
Simon expands factored polynomials with his favorite AI, ChatSFFT. However, he has not paid for a premium ChatSFFT account, so when he goes to expand $(m - a)(n - b),$ where $a, b, m, n$ are integers, ChatSFFT returns the sum of the two factors instead of the product. However, when Simon plugs in certain pairs of integer values for $m$ and $n,$ he realizes that the value of ChatSFFTs result is the same as the real result in terms of $a$ and $b$. How many such pairs are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2
}
Consider natural numbers \(a\), \(b\), and \(c\) such that the fraction

\[ 
k = \frac{ab + c^2}{a + b} 
\]

is a natural number less than \(a\) and \(b\). What is the smallest number of natural divisors that \(a + b\) can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $X_1,X_2,..$ be independent random variables with the same distribution, and let $S_n=X_1+X_2+...+X_n, n=1,2,...$. For what real numbers $c$ is the following statement true:
$$P\left(\left| \frac{S_{2n}}{2n}- c \right| \leqslant \left| \frac{S_n}{n}-c\right| \right)\geqslant \frac{1}{2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ c \in \mathbb{R} }
Evaluate the integral $\int_0^{2\pi}e^{-\cos\theta} \cos⁡(n\theta+\sin\theta)~d\theta$ using contour integration for $n\in\mathbb{Z}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\pi(-1)^n}{n!}}
Find the value of the following integral:

$$\int _0^{\infty }\ln \left(\frac{e^x+1}{e^x-1}\right)dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$$\frac{\pi^2}{4}$$}
There are 2 sections, A and B, in a class, consisting of 26 and 34 students respectively. The average weight of section A is some kg and that of section B is 30 kg. The average weight of the whole class is 38.67 kg. What is the average weight of section A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50.01}
A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{8}}
2. Let $A B C D$ be a square of side 4. Determine the largest positive integer $k$ with the following property: For an arbitrary arrangement of $k$ points strictly inside square $A B C D$, one can always find a square of side 1 , entirely contained in square $A B C D$ (with sides not necessarily parallel to the sides of square $A B C D$ ), whose strict interior contains none of the $k$ given points.

(Bojan Bašić)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
For certain pairs $(m,n)$ of positive integers with $m\geq n$ there are exactly $50$ distinct positive integers $k$ such that $|\log m - \log k| < \log n$. Find the sum of all possible values of the product $mn$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{125}
Given that $A, B, C, D$ are points along a circle in that order, and $AC$ intersects $BD$ at point $X$. If $BC = 6$, $BX = 4$, $XD = 5$, and $AC = 11$, find the length of $AB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Evaluate the limit:
$$\lim_{x\to 0}\frac{x^{(\sin x)^{\textstyle x}}-(\textstyle \sin x)^{\textstyle x^{\textstyle \sin x}}}{\textstyle x^3}$$
without using Taylor's formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/6}
Given the sample data set $3$, $3$, $4$, $4$, $5$, $6$, $6$, $7$, $7$, calculate the standard deviation of the data set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{5}}{3}}
In tetrahedron $V-ABC$, it is known that $VA=VB=VC=\sqrt{3}$, $AB=\sqrt{2}$, and $∠ACB=\frac{π}{4}$. Find the surface area of the circumscribed sphere of this tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{2} \pi}
How many $r$-class variations can be formed from $n$ elements in which the first $s$ elements occur?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(n-s)! r!}{(r-s)! (n-r)!}}
9548 plus a certain number equals 3362 plus 13500. What is the second number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7314}
A cyclist is walking at a constant rate of 15 miles per hour along a path with a flat surface when she is passed by a car traveling in the same direction at 60 miles per hour. After passing the cyclist, the car continues for another mile before stopping to wait for her. Along this mile, there is a continuous uphill slope that starts right after the cyclist is passed by the car. As the cyclist enters the uphill slope, her speed decreases linearly from 15 miles per hour to 10 miles per hour by the time she reaches its end (the point where the car is waiting). Find how many minutes the car must wait for the cyclist to catch up.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.8}
a number when divided by 5 gives a number which is 8 more than the remainder obtained on dividing the same number by 34 . such a least possible number x is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Given the base of a triangle is $24$ inches, two lines are drawn parallel to the base, with one of the lines dividing the triangle exactly in half by area, and the other line dividing one of the resulting triangles further into equal areas. If the total number of areas the triangle is divided into is four, find the length of the parallel line closer to the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\sqrt{2}}
The graph of the function y=sin(2x+φ) is shifted to the left by π/6 units along the x-axis, resulting in an even function graph. Determine the value of φ such that the equation 2(x + π/6) + φ = -x + 2πk is satisfied for some integer k.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{6}}
A vendor sells 80 percent of the pears he had and throws away 50 percent of the remainder. The next day, the vendor sells 80 percent of the remaining pears and throws away the rest. In total, what percent of his pears does the vendor throw away?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\%}
How many 11 step paths are there from point A to point D, which pass through points B and C in that order? Assume the grid layout permits only right and down steps, where B is 2 right and 2 down steps from A, and C is 1 right and 3 down steps from B, and finally, D is 3 right and 1 down step from C.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Using the digits 1, 2, 3, 4 only once to form a 4-digit number, how many of them are divisible by 11?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
10. There is a sequence of numbers +1 and -1 of length $n$. It is known that the sum of every 10 neighbouring numbers in the sequence is 0 and that the sum of every 12 neighbouring numbers in the sequence is not zero.
What is the maximal value of $n$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Yatsenko I.V.

In the elections to the 100-seat parliament, 12 parties participated. Parties that received strictly more than $5\%$ of the voters' votes enter the parliament. Among the parties that entered the parliament, seats are distributed proportionally to the number of votes they received. After the elections, it turned out that each voter voted for exactly one party (there were no invalid ballots, votes "against all" and the like) and each party received an integer number of seats. At the same time, the Party of Mathematics Enthusiasts received 25% of the votes. What is the maximum number of seats in the parliament that it could have received?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Casey wants to decorate her toenails, fingernails, and her friend's toenails and fingernails with a more intricate design. First, she wants to do a base coat on each nail, then a coat of paint, followed by a coat of glitter on every nail, and finally adding a pattern which takes additional time. It takes 20 minutes to apply each coat, 20 minutes for each coat to dry before applying the next one, and an extra 40 minutes to meticulously create the pattern on all nails. Assuming Casey's nails and her friend's nails will be done when all coats have been applied and dried, and the pattern is completed, how many minutes will it take her to finish decorating her and her friend's fingernails and toenails?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{320}
How many four-digit integers $abcd$, with $a \neq 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
On a circle, points $A, B, C, D, E, F, G$ are arranged clockwise, as shown in the diagram. It is known that $AE$ is the diameter of the circle. Additionally, it is known that $\angle ABF = 81^\circ$ and $\angle EDG = 76^\circ$. How many degrees is $\angle FCG$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Alice has 8 keys strung on a ring. The keys are indistinguishable at a glance, and both sides of each key are identical. In order to differentiate the keys, Alice places a colored cap on each one. How many colors does she need?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $O$ be the center of the base $\triangle ABC$ of a regular triangular pyramid $P-ABC$. A moving plane passing through $O$ intersects the three lateral edges or their extensions of $P-ABC$ at points $Q, R, S$ respectively. Calculate the sum $\frac{1}{PQ}+\frac{1}{PR}+\frac{1}{PS}$. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the smallest real number $m$ such that for any positive integers $a, b, c$ satisfying $a + b + c = 1$, the inequality $m(a^3 + b^3 + c^3) \geq 6(a^2 + b^2 + c^2) + 1$ holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Let the natural number \( n \) have the following property: If you choose 50 different numbers from the natural numbers \( 1, 2, \cdots, n \), there must be two numbers whose difference is equal to 7. Find the maximum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
Point  $A,B$  are marked on the right branch of the hyperbola   $y=\frac{1}{x},x>0$ . The straight line  $l$  passing through the origin  $O$  is perpendicular to  $AB$  and meets   $AB$  and given branch of the hyperbola at points  $D$  and  $C$  respectively. The circle through  $A,B,C$  meets  $l$  at  $F$ . 
Find  $OD:CF$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2}
Let \( \triangle ABC \) be an acute triangle with \( BC = 5 \). \( E \) is a point on \( AC \) such that \( BE \perp AC \). \( F \) is a point on \( AB \) such that \( AF = BF \). Moreover, \( BE = CF = 4 \). Find the area of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8 \sqrt{3} - 6}
In triangle $ABC$, the angle $C$ is obtuse, $m(\angle A) = 2m(\angle B)$, and the side lengths are integers. Find the smallest possible perimeter of this triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77}
Lyndy conducted a survey of the 300 students in her school. She found that 60 students own dogs, while only 30 students own cats. What percent of the students own cats?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10\%}
【Question 14】Select 5 digits from $1 \sim 9$, to form a five-digit number, such that this five-digit number can be evenly divided by any of the 5 selected digits, but cannot be evenly divided by any of the 4 unselected digits. Then, the smallest value of this five-digit number is $\qquad$.

---

Note: The blank at the end of the question is intended for the answer to be filled in.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14728}
Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M, 32, 33, 34, 35, 36, 37, 38, 39,$ and $40.$ Find the value of $\frac{N}{M}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
The line $x+2ay-1=0$ is parallel to $\left(a-1\right)x+ay+1=0$, find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Calculate the curl of the cross product $v(x,y,z) = (x,y,z)^T \times \omega$, where $\omega \in \mathbb{R}^3$ and $v(x,y,z)$ is a vector function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\omega}
(10 Given that the three lateral edges of a triangular pyramid are pairwise perpendicular, and the base area is 1, find the maximum volume of this triangular pyramid.



保留源文本的换行和格式，翻译结果如下：

(10 Given that the three lateral edges of a triangular pyramid are pairwise perpendicular, and the base area is 1, find the maximum volume of this triangular pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}\sqrt[4]{\frac{4}{27}}}