inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the equation $(m^2-4)x^2+(2-m)x+1=0$ is a quadratic equation in terms of $x$, the range of values for $m$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \neq \pm 2}
The sales volume of a shopping mall in February this year was 4 million yuan, and the sales volume in March increased by 10% compared to February. The sales volume in May reached 6.336 million yuan. Calculate the average growth rate of the sales volume from March to May is \_\_\_\_\_ \%.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
4. (3 points) From the set of three-digit numbers that do not contain the digits 0, 6, 7, 8, 9 in their notation, several numbers were written on paper in such a way that no two numbers can be obtained from each other by swapping two adjacent digits. What is the maximum number of such numbers that could have been written?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
63. A number that can be expressed as the difference of squares of two non-zero natural numbers is called a "wise number", for example, $16=5^{2}-3^{2}, 16$ is a "wise number". In the sequence of natural numbers starting from 1, the 2008th "wise number" is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2680}
Let  $S$  denote the sum of the 2011th powers of the roots of the polynomial  $(x-2^0)(x-2^1) \cdots (x-2^{2010}) - 1$ . How many ones are in the binary expansion of  $S$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2011}
Given the sets $M=\left\{ x\mid \log_{10} x < 1 \right\}$ and $N=\left\{ x\mid -4 < x < 6 \right\}$, determine the set $M\cap N$. Express your answer in set notation, including all necessary conditions and bounds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\{ x\mid 0 < x < 6 \right\}}
Given: \(\sum_{n=0}^{\infty} a_n \frac{t^n}{n!} = \left(\sum_{k=0}^{\infty} \frac{t^{2k}}{(2k)!}\right)^2 \left(\sum_{j=0}^{\infty} \frac{t^j}{j!}\right)^3\), find \(a_n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5^{n} + 2 \cdot 3^{n} + 1}{4}}
Sebastian works everyday and gets his salary on a weekly basis. If Sebastian didn't go to work on 2 days this week, his deducted salary was 745 dollars. What is his usual weekly salary?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$1862.50}
Evdokimov M.A.

By the river, there lives a tribe of Mumbo-Jumbo. One day, with urgent news to the neighboring tribe, a young warrior Mumbo and a wise shaman Yumbo set out simultaneously. Mumbo ran at a speed of 11 km/h to the nearest raft storage and then sailed on a raft to the neighboring tribe. Yumbo, on the other hand, walked leisurely at a speed of 6 km/h to another raft storage and then sailed to the neighboring tribe from there. In the end, Yumbo arrived before Mumbo. The river is straight, and the rafts move at the speed of the current, which is the same everywhere and is expressed as an integer number of km/h, not less than 6. What is the greatest possible value of this speed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
A 220 m long train running at a certain speed crosses another train running in opposite direction at the speed of 80 km/hr in 9 sec. The length of the other train is 280.04 m. What is the speed of the first train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120.016}
Select three different numbers from the set {1,2,3,...,10} such that these three numbers form a geometric sequence. Count the number of such geometric sequences. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a differentiable function $y=f(x)$ defined on $\mathbb{R}$, its derivative is denoted as $f'(x)$, satisfying $f'(x) < f(x)$. Also, $y=f(x+2)$ is an even function, and $f(4)=1$. Then, the solution set of the inequality $f(x) < e^x$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,+\infty)}
Given the tower function $T(n)$ defined by $T(1) = 3$ and $T(n + 1) = 3^{T(n)}$ for $n \geq 1$, calculate the largest integer $k$ for which $\underbrace{\log_3\log_3\log_3\ldots\log_3B}_{k\text{ times}}$ is defined, where $B = (T(2005))^A$ and $A = (T(2005))^{T(2005)}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2005}
The area of a quadrilateral formed by the midpoints of the bases and diagonals of a trapezoid is four times smaller than the area of the trapezoid itself. Find the ratio of the lengths of the bases of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:1}
There are \( n \) distinct lines in the plane. One of these lines intersects exactly 5 of the \( n \) lines, another one intersects exactly 9 of the \( n \) lines, and yet another one intersects exactly 11 of them. Find the smallest possible value of \( n \). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given a pyramid S-ABCD with a height of $\sqrt{2}$ and a square base ABCD with side length 1, where points S, A, B, C, D all lie on the same spherical surface with a radius of 1, calculate the distance between the center of the base ABCD and the vertex S. Express your answer as a simplified radical.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{10}}{2}}
In $\triangle ABC$, angle bisectors $BD$ and $CE$ intersect at $I$, with $D$ and $E$ located on $AC$ and $AB$ respectively. A perpendicular from $I$ to $DE$ intersects $DE$ at $P$, and the extension of $PI$ intersects $BC$ at $Q$. If $IQ = 2 IP$, find $\angle A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Stan has a stack of 100 blocks and starts with a score of 0, and plays a game in which he iterates the following two-step procedure: (a) Stan picks a stack of blocks and splits it into 2 smaller stacks each with a positive number of blocks, say $a$ and $b$. (The order in which the new piles are placed does not matter.) (b) Stan adds the product of the two piles' sizes, $a b$, to his score. The game ends when there are only 1-block stacks left. What is the expected value of Stan's score at the end of the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4950}
A triangle $ABC$ with $AC = 20$ is inscribed in a circle $\omega$. A tangent $t$ to $\omega$ is drawn through $B$. The distance from $A$ to $t$ is $25$ and the distance from $C$ to $t$ is $16$. If $S$ denotes the area of triangle $ABC$, find the largest integer not exceeding $\frac{S}{20}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The diameters of balls for bearings are controlled as follows: if the ball does not pass through a hole with diameter $d_{1}$ but passes through a hole with diameter $d_{2} > d_{1}$, its size is considered acceptable. If any of these conditions are not met, the ball is rejected. It is known that the diameter of the ball is a random variable $X$ distributed according to a normal law, with the mathematical expectation of this quantity $a = \left(d_{1} + d_{2}\right) / 2$ and the standard deviation $\sigma = \left(d_{2} - d_{1}\right) / 4$. Find the probability that the tested ball will be rejected.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0456}
Two circles $x^2+y^2=9$ and $x^2+y^2+8x-6y+25-r^2=0$ ($r>0$) intersect. Find the range of $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 < r < 8}
In triangle \( ABC \), angle bisectors \( BB_1 \) and \( CC_1 \) are drawn. It is known that the center of the circumcircle of triangle \( BB_1C_1 \) lies on line \( AC \). Find the angle \( C \) of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
In trapezoid \(ABCD\), the base \(BC\) is equal to 13, and the angle \(BAD\) is acute and twice the angle \(ADC\). A circle with its center on the line \(BC\) is tangent to the lines \(AC\), \(AD\), and segment \(CD\). Find the area of trapezoid \(ABCD\), given that the radius of the circle is 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{157.5}
Solve the equation \(\log _{5}(3x - 4) \cdot \log _{5}(7x - 16) \cdot \left(3 - \log _{5} \left(21x^{2} - 76x + 64\right) \right) = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the limit: $$\lim_{x \rightarrow 0} \frac{1}{x^4} \int_{\sin{x}}^{x} \arctan{t}dt$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{6}$}
Given the digits 0, 1, 2, 3, 4, 5, how many unique six-digit numbers greater than 300,000 can be formed where the digit in the thousand's place is less than 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216}
If the range of the function $y=\log_{2}(x^{2}-ax-a)$ is $\mathbb{R}$, and it is a decreasing function on $(-\infty, 1- \sqrt{3})$, find the range of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0\leq a<2}
Andy is a lawyer who's working on four lawsuits. The first lawsuit has a 30% chance of paying out $5,000,000 upon a win and $0 if he loses it. The second lawsuit has a 50% chance of paying out $1,000,000 if Andy loses and a 50% chance of paying out $2,000,000 if he wins. The third lawsuit has a 40% chance of paying out $3,000,000 if Andy wins and a 60% chance of paying out $0 if he loses. The fourth lawsuit has a 25% chance of paying out $6,000,000 if Andy wins and a 75% chance of paying out $0 if he loses. Expressed as a percentage, how much more likely is it that Andy loses all four lawsuits compared to winning all of them?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{950\%}
Consider the points $A_1$ and $A_2$ on the side $AB$ of the square $ABCD$ such that $|AB| = 3 |AA_1|$ and $|AB| = 4 |A_2B|$. Similarly, consider points $B_1$ and $B_2$ on $BC$, $C_1$ and $C_2$ on $CD$, and $D_1$ and $D_2$ on $DA$.

The intersection point of lines $D_2A_1$ and $A_2B_1$ is $E$.
The intersection point of lines $A_2B_1$ and $B_2C_1$ is $F$.
The intersection point of lines $B_2C_1$ and $C_2D_1$ is $G$.
The intersection point of lines $C_2D_1$ and $D_2A_1$ is $H$.

Find the area of the square $EFGH$, given that the area of $ABCD$ is $1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Team A and Team B each send out 7 players to compete in a chess relay match. Players from both teams compete in a predetermined order. The first players from both teams compete; the loser is eliminated, and the winner goes on to compete against the second player from the losing team. This continues until one team has all their players eliminated, at which point the other team wins, forming a sequence of match outcomes. It is known that Team A used only their first 5 players to consecutively defeat all 7 players from Team B. How many different sequences of matches are possible in such a contest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
29 boys and 15 girls attended a ball. Some boys danced with some girls (no more than once in each pair). After the ball, each child told their parents how many times they danced. What is the maximum number of distinct numbers the children could have reported?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Given a connected graph $G=(V,E)$ with $n$ vertices and $n-2$ 3-cycles, where every vertex is contained in at least one 3-cycle, what is the maximum number of vertices of degree $3$ if the minimum degree of the graph is $2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $a_1, a_2, \ldots, a_{100}$ be non-negative integers such that:

1. There exists a positive integer $k \leq 100$ such that $a_1 \leq a_2 \leq \cdots \leq a_k$ and $a_i = 0$ for $i > k$.
2. $a_1 + a_2 + a_3 + \cdots + a_{100} = 100$.
3. $a_1 + 2a_2 + 3a_3 + \cdots + 100a_{100} = 2022$.

Find the minimum value of $a_1 + 2^2a_2 + 3^2a_3 + \cdots + 100^2a_{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40940}
The function $f\left(x\right) = {\left(\frac{1}{e}\right)}^{2x-1} - 1$ always passes through a fixed point $P$. Determine the exact coordinates of point $P$ as an ordered pair $(x, y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{1}{2}, 0\right)}
Given that the polynomial $2a x^4 + 5a x^3 - 13 x^2 - x^4 + 2021 + 2x + bx^3 - bx^4 - 13x^3$ is a quadratic polynomial, find the value of $a^2 + b^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
At a school, 3-fifths of the male students and two-thirds of the female students speak a foreign language. If the number of males has a certain relationship to the number of females, 0.6333333333333333 fraction of the students speak a foreign language. What is the relationship between the number of males and the number of females?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:1}
A middle school holds a sports meeting, participating by class. There are three events: high jump, long jump, and 100m sprint. The top three in each event will be awarded points: 5 points for first place, 3 points for second place, and 1 point for third place. It is known that Class 1 and Class 2 have the same total score and are tied for first place. Moreover, the number of people from Class 2 who placed in the top three is twice that of Class 1. What is the total score of Class 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The decimal expression of a 100-digit number consists of 1’s and 2’s. The Rule allows one to select arbitrarily 10 consecutive digits of which the first five may change places with the second five. Two numbers will be called similar if one can be obtained from the other one in several such operations. What greatest number of such 100-digit numbers can be selected so no two of them are similar? Please provide the integer value of the original answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4084101}
There are three machines A, B, and C working on a large production order. Machine A can complete the order in 6 hours, Machine B can complete it in 7 hours, and Machine C can complete it in 9 hours. How many hours would it take all three machines, working simultaneously at their respective constant rates, to complete the order?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{126}{53}}
Let $x$ be chosen at random from the interval $(0,1)$. Find the probability that $\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0$. Here $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$. The answer is in the form of $\frac{k}{m}$, please provide the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given a hyperbola with eccentricity $e$ satisfying $-\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1$, its right focus coincides with the focus of the parabola $y^2 = 16x$. Find the value of $e$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the smallest positive integer $a$ such that $x^4 + a^2$ is not prime for any integer $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In the expansion of $(x+2)^6$, which term has the largest binomial coefficient?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the maximum number of pieces of birthday cake of size 4 ” by 4 ” that can be cut from a cake of 100 ” by 100 ”? Also, you must consider a condition that after each cut, the size of the remaining piece of cake must be even; if not, the last slice needs to be discarded.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{625}
An odd person did not get lazy and wrote 2023 numbers in a circle in such a way that each number is equal to the product of its two neighbors. What is the maximum number of different numbers that could be used?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
$已知椭圆C：\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)过点\left(1,\frac{\sqrt{6}}{3}\right)，过其右焦点F_2且垂直于x轴的直线交椭圆C于A，B两点，且|AB|=\frac{2\sqrt{3}}{3}．$
$若直线l：y=kx-\frac{1}{2}与椭圆C交于E，F两点，线段EF的中点为Q，在y轴上是否存在定点P，使得\angle EQP＝2\angle EFP恒成立？若存在，求出点P的坐标；$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(0,1)$}
The LCM of two numbers is 25,974 and their HCF is 107. If one of the numbers is 4,951, then what is the other number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{561}
From the digits \(1, 2, 3, 4, 5, 6, 7, 8, 9\), nine nine-digit numbers (not necessarily distinct) are formed; each digit is used exactly once in each number. What is the maximum number of zeros by which the sum of these nine numbers may end?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the number of integers $2 \leq n \leq 2016$ such that $n^{n}-1$ is divisible by $2,3,5,7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
If \( f(n, k) \) is the number of ways to divide the set \(\{1,2, \ldots, n\}\) into \(k\) nonempty subsets and \(m\) is a positive integer, find a formula for \(\sum_{k=1}^{n} f(n, k) m(m-1)(m-2) \cdots (m-k+1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m^n}
4. In the country of Landia, which breeds an elite breed of horses, an annual festival is held to test their speed, in which only one-year-old, two-year-old, three-year-old, and four-year-old horses can participate. For each horse that meets the speed standard, the festival organizers pay a fixed amount of money to the stud farm where the horse was raised: 1 landric for a one-year-old, 2 landrics for a two-year-old, 3 landrics for a three-year-old, and 4 landrics for a four-year-old. Each stud farm participating in the festival annually presents four new horses (of any age combination as they wish) for testing, which have not previously participated in the tests, as well as personally all horses (not older than four years) that previously participated in the tests at a younger age and met the standard. What is the maximum amount of money a stud farm can earn in the first six years of its participation in the festival?

(12 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their spot to the spectator (there were no draws). It turned out that Egor participated in 13 games, and Nikita in 27 games. How many games did Innokentiy play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Find the area of the triangle formed by the axis of the parabola $y^{2}=8x$ and the two asymptotes of the hyperbola $(C)$: $\frac{x^{2}}{8}-\frac{y^{2}}{4}=1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Fresh grapes contain a certain percentage of water by weight, while dried grapes contain 10% water by weight. The weight of dry grapes available from 400 kg of fresh grapes is 155.56 kg. What is the percentage of water in fresh grapes by weight?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65\%}
Call a positive integer $N \geq 2$ "special" if for every $k$ such that $2 \leq k \leq N, N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). How many special integers are there less than $100$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
In each of the following 7-digit natural numbers:
1001011, 5550000, 3838383, 7777777,
every digit in the number appears at least 3 times. Find the number of such 7-digit natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2844}
My car gets 20 miles per gallon of gas in the city and 25 miles per gallon on the highway. If Grandma's house is 100 miles away, with 40 miles of the trip being in the city and 60 miles on the highway, and I also need to stop at a grocery store that is 10 miles off the highway, how many total gallons of gas would it take to get to her house?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.2}
A vampire drains five people a week and leads a group of three vampires. His best friend is an alpha werewolf who eats seven people a week and commands a pack of four other werewolves, each eating five people a week. Additionally, a ravenous ghost haunts the village and feeds on the energy of two people per week. A coven of four witches requires a sacrifice of three people per week for their rituals. Lastly, a horde of 20 zombies descends upon the village, with each zombie craving one person's brains per week. How many weeks will a village of 500 people last them all?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $a$ be a positive real number such that $\frac{a^2}{a^4-a^2+1}=\frac{4}{37}$. Then $\frac{a^3}{a^6-a^3+1}=\frac{m}{n}$, 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{259}
2. Each cell of a $100 \times 100$ board is painted blue or white. We will call a cell balanced if among its neighbors there are an equal number of blue and white cells. What is the maximum number of balanced cells that can be on the board? (Cells are considered neighbors if they share a side.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9608}
Let  $O$  be the origin.  $y = c$  intersects the curve  $y = 2x - 3x^3$  at  $P$  and  $Q$  in the first quadrant and cuts the y-axis at  $R$ . Find  $c$  so that the region  $OPR$  bounded by the y-axis, the line  $y = c$  and the curve has the same area as the region between  $P$  and  $Q$  under the curve and above the line  $y = c$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{9}}
There are 10 numbers:
$$
21, 22, 34, 39, 44, 45, 65, 76, 133, \text{ and } 153
$$

Group them into two groups of 5 numbers each, such that the product of the numbers in both groups is the same. What is this product?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{349188840}
Calculate directly the line integral $\int_C y^2 dx+x \, dy+z \, dz$, where $C$ is the intersection line of surfaces $x^2+y^2=x+y$ and $2(x^2+y^2)=z$, oriented in the positive direction viewed from the point $(0;0;2R)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A man purchased 1 blanket @ Rs. 100 each, 5 blankets @ Rs. 150 each and two blankets at a certain rate which is now slipped off from his memory. But he remembers that the average price of the blankets was a certain amount. The unknown rate of the two blankets was Rs. 650. What was the average price of the blankets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{187.50}