inclusionAI/Ring-lite-rl-data huggingface dataset:
In a certain company, a third of the workers do not have a retirement plan. Some percentage of the workers who do not have a retirement plan are women, and 40% of the workers who do have a retirement plan are men. If 120 of the workers of that company are men, and there are 120 women workers, what is the percentage of women workers without a retirement plan?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
Greg and Sarah are going to split some apples evenly. Susan has twice as many apples as Greg will end up with. Mark has 5 fewer apples than Susan. Their mom needs 40 apples to make an apple pie and will have 9 apples left over. How many apples do Greg and Sarah have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Two trains of length 100 m and 220 m are running towards each other on parallel lines at a certain speed and 30 kmph respectively. They will be clear of each other in 15.99872010239181 seconds from the moment they meet. What is the speed of the first train in kmph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
$ m$  and  $ n$  are positive integers. In a  $ 8 \times 8$  chessboard,  $ (m,n)$  denotes the number of grids a Horse can jump in a chessboard ( $ m$  horizontal  $ n$  vertical or  $ n$  horizontal  $ m$  vertical ). If a  $ (m,n) \textbf{Horse}$  starts from one grid, passes every grid once and only once, then we call this kind of Horse jump route a  $ \textbf{H Route}$ . For example, the  $ (1,2) \textbf{Horse}$  has its  $ \textbf{H Route}$ . Find the smallest positive integer  $ t$ , such that from any grid of the chessboard, the  $ (t,t\plus{}1) \textbf{Horse}$  does not has any  $ \textbf{H Route}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ t = 2 }
In a cone with a height and base radius of 7, four identical spheres are placed. Each sphere touches two others (externally), the base, and the lateral surface of the cone. A fifth sphere touches the lateral surface of the cone and all four identical spheres (externally). Find the radius of the fifth sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{2} - 1}
The differentiable function \( F: \mathbb{R} \rightarrow \mathbb{R} \) satisfies \( F(0) = -1 \) and
$$
\frac{d}{d x} F(x) = \sin (\sin (\sin (\sin (x)))) \cdot \cos (\sin (\sin (x))) \cdot \cos (\sin (x)) \cdot \cos (x).
$$

Find \( F(x) \) as a function of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\cos(\sin(\sin(\sin(x))))}
In the arithmetic sequence $\{a\_n\}$, $4a_{12}=-3a_{23} > 0$. Let $b\_n= \frac {a\_n a_{n+1}}{a_{n+2}}$, $S\_n$ is the sum of the first $n$ terms of $\{b\_n\}$, and $S\_{n_0}$ is the maximum term of the sequence $\{S\_n\}$. Find $n\_0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Each face of a hexahedron and each face of a regular octahedron are squares with side length $a$. The ratio of the radii of the inscribed spheres of these two polyhedra is an irreducible fraction $\frac{m}{n}$. What is the product $m \cdot n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In $\mathrm{Rt} \triangle A B C$, $\angle A=90^{\circ}$, the inscribed square $P Q R S$ has one side $R S$ on the hypotenuse $B C$, $P$ is on $A B$, and $Q$ is on $A C$. Given $|A B|=a(a>0, a$ is a constant), $\angle B=\theta$, when $\theta$ varies, the minimum value of $S_{\triangle A B C}$ : $S_{P Q R S}$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{4}}
【Question 29】
8 different positive integers, among which there are at least: 6 multiples of 2, 5 multiples of 3, 3 multiples of 5, 1 multiple of 7. To make the maximum number among the 8 numbers as small as possible, then this maximum number is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Calculate the determinant of the matrix \( D_n = \begin{vmatrix} n & -1 & -3 & 0 & 0 & \cdots & 0 & 0 & 0 \\ n & 1 & 2 & -3 & 0 & \cdots & 0 & 0 & 0 \\ n & 0 & 1 & 2 & -3 & \cdots & 0 & 0 & 0 \\ n & 0 & 0 & 1 & 2 & \cdots & 0 & 0 & 0 \\ n & 0 & 0 & 0 & 1 & \cdots & 0 & 0 & 0 \\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots \\ n & 0 & 0 & 0 & 0 & \cdots & 1 & 2 & -3 \\ n & 0 & 0 & 0 & 0 & \cdots & 0 & 1 & 2 \\ n & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 1 \\ \end{vmatrix} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
$22$  football players took part in the football training. They were divided into teams of equal size for each game ( $11:11$ ). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the number of sequences consisting of 100 R's and 2011 S's that satisfy the property that among the first \( k \) letters, the number of S's is strictly more than 20 times the number of R's for all \( 1 \leq k \leq 2111 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{2111} \binom{2111}{100}}
By selling 20 pencils for a rupee, a man loses 30%. He wants to adjust the number of pencils he sells for a rupee to make a certain gain. He finds out that he should sell approximately 10.77 pencils for a rupee. What is the percentage of the gain he wants to make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30\%}
Rahim visited three different bookstores to purchase books for his collection. At Store A, he bought 25 books for a total of Rs. 1,500, and received a 15% discount on the entire purchase. At Store B, he bought 35 books for a total of Rs. 3,000, and got a "Buy 3, get 1 free" deal on the books, effectively getting 4 less books to pay for. At Store C, he bought 40 books for Rs. 3,500, and got a 10% discount on every 10th book he bought there. Calculate the average price Rahim paid per book considering the discounts and deals he got at each store.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77.40}
The sum \(\sum_{k=3}^{\infty} \frac{1}{k(k^4-5k^2+4)^2}\) is equal to \(\frac{m^2}{2n^2}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Find the largest five-digit positive integer such that it is not a multiple of 11, and any number obtained by deleting some of its digits is also not divisible by 11.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98765}
Rural School USA has 105 students enrolled. There are 60 boys and 45 girls. If $\frac{1}{10}$ of the boys and $\frac{1}{3}$ of the girls are absent on one day, what percent of the total student population is absent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20 \%}
Five people participate in a relay game, each randomly choosing between two options: left path and right path. If more than 2 people choose the same path, each of them gets 1 point; if fewer than 3 people choose the same path, each of them gets 0 points. Let $\xi$ denote the score of a player named Xiaoqiang. Calculate the expected value of $\xi$, $E\xi$. Express your answer as a fraction in the format $\boxed{[answer]}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{16}}
A moth starts at vertex $A$ of a certain cube and is trying to get to vertex $B$, which is opposite $A$, in five or fewer "steps," where a step consists in traveling along an edge from one vertex to another. The moth will stop as soon as it reaches $B$. How many ways can the moth achieve its objective?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
A right circular cone and a right cylinder with same height $20$ does not have same circular base but the circles are coplanar and their centers are same. If the cone and the cylinder are at the same side of the plane and their base radii are $20$ and $10$, respectively, what is the ratio of the volume of the part of the cone inside the cylinder over the volume of the part of the cone outside the cylinder?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1
}
In the set \(\{1, 2, 3, \cdots, 99, 100\}\), how many numbers \(n\) satisfy the condition that the tens digit of \(n^2\) is odd?
(45th American High School Mathematics Examination, 1994)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given the polynomial \( P(x) = x^{n+1} - 2x^n + 1 \) for all \( n \geq 2 \), and knowing that \( P(x) \) has a unique root \( a_n \in (1,2] \), find the limit \( L = \lim_{n\to\infty} 2^n(2-a_n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the genus of the curve $y^{2}+y=\frac{\alpha}{x^{2}+x}$ over a finite field with even characteristic?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In an isosceles right triangle \(AOB\), points \(P, Q, R,\) and \(S\) are chosen on the sides \(OB, OA,\) and \(AB\) respectively, such that \(PQRS\) forms a square. Given that the lengths of \(OP\) and \(OQ\) are \(a\) and \(b\) respectively, and the area of the square \(PQRS\) is \(\frac{2}{5}\) of the area of the triangle \(AOB\), find the value of \(\frac{a}{b}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given the function
\[ f(x) = \log_{2}(4^x + 1) + mx \quad (m > 0) \]
the equation regarding \( x \)
\[ f\left(8\left(\log_{4} x\right)^{2} + 2 \log_{2} \frac{1}{x} + \frac{4}{m} - 4\right) = 1 \]
has exactly two different real solutions in the interval \([1, 2\sqrt{2}]\). Then the set of all possible values of \( m \) is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{9} < m \leq 1}
Let $ABCDE$ be be a cyclic pentagon. Given that $\angle CEB = 17^o$, find $\angle CDE  + \angle EAB$, in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{197}
Determine the order of the quotient ring \( \mathbb{Z}_7[i]/\langle 2 \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine for which values of $n$ the improper integral $$\int_{0}^{2} \left(\log\left(\frac{2}{2-x}\right)\right)^n\,\mathrm dx$$ converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n > -1}
3. (CZS) A tetrahedron $A B C D$ is given. The lengths of the edges $A B$ and $C D$ are $a$ and $b$, respectively, the distance between the lines $A B$ and $C D$ is $d$, and the angle between them is equal to $w$. The tetrahedron is divided into two parts by the plane $\pi$ parallel to the lines $A B$ and $C D$. Calculate the ratio of the volumes of the parts if the ratio between the distances of the plane $\pi$ from $A B$ and $C D$ is equal to $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{k^{3}+3k^{2}}{3k+1}}
Given that in a class of 36 students, more than half purchased notebooks from a store where each notebook had the same price in cents greater than the number of notebooks bought by each student, and the total cost for the notebooks was 990 cents, calculate the price of each notebook in cents.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Given the function \( f(x) = 1 + \sum_{k=0}^{100}\frac{(-1)^{k+1}}{(k+1)!}x(x-1)(x-2)\cdots (x-k) \), find the value of \( S(f(x)) - S(f'(x)) \), where \( S \) denotes the sum of the real roots of the respective function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
集合 $U=\{1,2, \cdots, 366\}$ ，则 $U$ 的互不相交各元素之和为 $17$ 的倍数的二元子集最多有 $\qquad$个.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{179}
On January 1st, the day of the New Year celebration, an Elderly Sage was reflecting on his life. He noticed that over the past 5 years, all the days of the week occurred equally often, and 10 years ago, the New Year celebration fell on a Friday. What day of the week did the Sage's somber reflections fall on? It is known that the Sage's reflections occurred in the 21st century.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Thursday}
Determine which ordinal is larger: \(\tau_1\), the supremum of ordinals writable by iterated Infinite Time Turing Machines, or \(\tau_2\), the smallest \(\Sigma_2^1\)-reflecting ordinal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\tau_2}
Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes $4''$ or $10''$ or $19''$ to the total height of the tower. How many different tower heights can be achieved using all ninety-four of the bricks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{465}
10. Let real numbers $x, y$ be such that $x-y, x^{2}-y^{2}, x^{3}-y^{3}$ are all prime numbers, then the value of $x-y$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the maximum positive integer \( n \) such that
\[ 
n^{2} \leq 160 \times 170 \times 180 \times 190
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30499}
Let exp ( m , n ) = m to the power n . If exp ( 10 , some number ) = 25 exp ( 22 ) where m and n are integers. What is that number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $a_{1}, a_{2}, \ldots, a_{n}$ be a sequence of distinct positive integers such that $a_{1}+a_{2}+\cdots+a_{n}=2021$ and $a_{1} a_{2} \cdots a_{n}$ is maximized. If $M=a_{1} a_{2} \cdots a_{n}$, compute the largest positive integer $k$ such that $2^{k} \mid M$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
Light of a blue laser (wavelength  $\lambda=475 \, \text{nm}$ ) goes through a narrow slit which has width  $d$ . After the light emerges from the slit, it is visible on a screen that is  $ \text {2.013 m} $  away from the slit. The distance between the center of the screen and the first minimum band is  $ \text {765 mm} $ . Find the width of the slit  $d$ , in nanometers. 

*(Proposed by Ahaan Rungta)*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1250}
A train of a certain length is running at 60 kmph. It takes 21.598272138228943 seconds to pass a platform 240 m long. How long is the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
A town has a current population of 14,000 people. Over the next three years, the annual population growth rates and additional contributing factors change as follows:

- In the first year, the population increases by 12% due to a high birth rate, and an additional 150 new residents move to the town.
- In the second year, the population growth rate decreases to 8%, and only 100 new residents move to the town due to increased emigration.
- In the third year, the growth rate further declines to 6% due to economic challenges, and at the same time, 200 employees and their families move to the town for a new business, increasing the population by an additional 500 people.

What will the population be at the end of these three years?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18,728}
A Vandal and a Moderator are editing a Wikipedia article.  The article originally is error-free.  Each day, the Vandal introduces one new error into the Wikipedia article.  At the end of the day, the moderator checks the article and has a  $ 2/3$  chance of catching each individual error still in the article.  After  $ 3$  days, what is the probability that the article is error-free?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{416}{729}}
Place all terms of the arithmetic sequence \(2, 6, 10, 14, \cdots, 2006\) closely together to form a large number \(A = 261014 \cdots 2006\). Find the remainder when \(A\) is divided by 9.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The graph of the power function y=f(x) passes through the point (-2, -1/8). Find the value of x that satisfies f(x)=27. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
A store owner ordered different types of soda bottles in April, May, and June. The detailed order was as follows:

April:
- 100 cases of Soda A, with 24 bottles per case, priced at $1.50 per bottle.
- 50 cases of Soda B, with 30 bottles per case, priced at $2.00 per bottle.
April's orders had a discount of 15% applicable only to Soda A.

May:
- 80 cases of Soda A, with 24 bottles per case, priced at $1.50 per bottle.
- 40 cases of Soda B, with 30 bottles per case, priced at $2.00 per bottle.
May's orders had a 10% sales tax applicable on the total price.

June:
- 120 cases of Soda A, with 24 bottles per case, priced at $1.50 per bottle.
- 60 cases of Soda B, with 30 bottles per case, priced at $2.00 per bottle.
June's orders had a special offer: Buy one case of Soda B and get 10% off on one case of Soda A.

Find the net total expense of purchasing soda bottles for these three months, considering the varying discounts and taxes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$19572}
3 设X={0,1,,2,⋯,9},$F = \{ A _ { 1 } ,$,$A _ { 2 } , \cdots , A _ { k } \}$中的每个元素$A _ { i }$都是X 的非空子集，且对任何1≤i&lt;j≤k，有$\vert A _ { i } \cap A _ { j } \vert \leq 2$，求k的最大值.（第26届IMO备选题）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{175}
If $P = \{1, 2\}$, $Q = \{1, a^2\}$, and $P = Q$, then find the value of $a$. Express your answer as a mathematical expression, including the ± symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±\sqrt{2}}
Let \( ABC \) be a triangle with circumcenter \( O \) such that \( AC = 7 \). Suppose that the circumcircle of \( AOC \) is tangent to \( BC \) at \( C \) and intersects the line \( AB \) at \( A \) and \( F \). Let \( FO \) intersect \( BC \) at \( E \). Compute \( BE \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{2}}
There are two identical cups, A and B. Cup A is half-filled with pure water, and cup B is fully filled with a 50% alcohol solution. First, half of the alcohol solution from cup B is poured into cup A and mixed thoroughly. Then, half of the alcohol solution in cup A is poured back into cup B. How much of the solution in cup B is alcohol at this point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{8}}
For every integer n ≥ 3, the function g(n) is defined as the product of all the odd integers from 1 to n, inclusive. The difference between g(101) and g(some number) is 100. What is that some number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Determine the number of terms required in the Taylor series expansion of cos(x) to approximate cos(0.5) with a relative approximate error of less than 0.2%.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a sequence $\{a_n\}$ that satisfies: for any $n\in\mathbb{N}^*$, we have $a_{n+1}=ka_n+3k-3$, where $k$ is a constant not equal to $0$ or $1$. If $a_i \in \{-678, -78, -3, 22, 222, 2222\}$ for $i=2, 3, 4, 5$, then the sum of all possible values of $a_1$ that meet the conditions is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6023}{3}}
Given the function $f(x)=a^{x}+x^{2}-x\ln a-b$ ($a > 1,b\in\mathbb{R}$), where $e$ is the base of the natural logarithm. If there exist $x_{1}, x_{2}\in[-1,1]$ such that $|f(x_{1})-f(x_{2})|\geqslant e-1$, then the range of values for the real number $a$ is ______. (Reference formula: $(a^{x})'=a^{x}\ln a$)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[e,+\infty)}
In a global company with a diverse workforce, 60% are men, and 40% are women. Among the male employees, 40% speak only French, 20% speak only Spanish, 15% speak only German, 10% speak both French and Spanish, 5% speak both French and German, 5% speak both Spanish and German, and 5% speak all three languages. Among the female employees, 30% speak only French, 25% speak only Spanish, 20% speak only German, 10% speak both French and Spanish, 5% speak both French and German, 5% speak both Spanish and German, and 5% speak all three languages. What is the percentage of women in the company who do not speak French but speak one or more of the other two languages (Spanish and/or German)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50\%}
Fnargs are either red or blue and have 2, 3, or 4 heads. A group of six Fnargs consisting of one of each possible form (one red and one blue for each number of heads) is made to line up such that no immediate neighbors are the same color nor have the same number of heads. How many ways are there of lining them up from left to right?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points.
A natural number $m$ is called [i]good[/i] if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on.
Find the minimum value of a [i]good[/i] number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Twenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered 1 through 10 in some order. Each block in layers 2,3 and 4 is assigned the number which is the sum of numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{114}
32at8 ** Let $a, b, c \in \mathbf{R}^{+}, a+b+c=1$, find the minimum value of $\frac{a+3 c}{a+2 b+c}+\frac{4 b}{a+b+2 c}-$ $\frac{8 c}{a+b+3 c}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-17+12\sqrt{2}}
Two concentric circles have radii of $12$ meters and $24$ meters respectively. A rabbit starts at point X and runs along a sixth of the circumference of the larger circle, followed by a straight line to the smaller circle, then along a third of the circumference of the smaller circle, and finally across the diameter of the smaller circle back to the closest point on the larger circle's circumference. Calculate the total distance the rabbit runs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16\pi + 36}