inclusionAI/Ring-lite-rl-data huggingface dataset:
In a New Year's cultural evening of a senior high school class, there was a game involving a box containing 6 cards of the same size, each with a different idiom written on it. The idioms were: 意气风发 (full of vigor), 风平浪静 (calm and peaceful), 心猿意马 (restless), 信马由缰 (let things take their own course), 气壮山河 (majestic), 信口开河 (speak without thinking). If two cards drawn randomly from the box contain the same character, then it's a win. The probability of winning this game is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{5}}
Let $\alpha$ be an algebraic element over a field, $F$, with minimal polynomial $p(t) = t^3 + t + 1$. Determine a non-zero polynomial $q(t) \in F[t]$ such that $q(\alpha^2) = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t^3 + 2t^2 + t - 1}
Given point $A(a-1,2a-4)$, if the point is symmetric about the $y$-axis in the second quadrant, then the range of $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 2}
A teacher received letters on Monday through Friday with counts of $10$, $6$, $8$, $5$, $6$ respectively. Calculate the variance (${s^{2}} =$) of this data set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["3.2"]}
Square \( ABCD \) has a side length of 12 inches. A segment \( AE \) is drawn where \( E \) is on side \( DC \) and \( DE \) is 5 inches long. The perpendicular bisector of \( AE \) intersects \( AE, AD, \) and \( BC \) at points \( M, P, \) and \( Q \) respectively. The ratio of the segments \( PM \) to \( MQ \) is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5:19}
Let $n$ points be given arbitrarily in the plane, no three of them collinear. Let us draw segments between pairs of these points. What is the minimum number of segments that can be colored red in such a way that among any four points, three of them are connected by segments that form a red triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(n-1)(n-2)}{2}}
When a plane intersects a cube, what type of polygon cannot be formed as a result of the intersection? Express your answer as the name of the polygon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Heptagon}
Andrew and Blair are bored in class and decide to play a game. They pick a pair $(a, b)$ with $1 \le a, b \le 100$. Andrew says the next number in the geometric series that begins with $a,b$ and Blair says the next number in the arithmetic series that begins with $a,b$. For how many pairs $(a, b)$ is Andrew's number minus Blair's number a positive perfect square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{281}
Let $n$ be a positive integer such that $\lfloor\sqrt{n}\rfloor - 2$ divides $n - 4$ and $\lfloor\sqrt{n}\rfloor + 2$ divides $n + 4$. Find the greatest such $n$ less than $1000$. (Note: $\lfloor x \rfloor$ refers to the greatest integer less than or equal to $x$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{956}
A black bishop and a white king are placed randomly on a  $2000 \times 2000$  chessboard (in distinct squares). Let  $p$  be the probability that the bishop attacks the king (that is, the bishop and king lie on some common diagonal of the board). Then  $p$  can be expressed in the form  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Compute  $m$ .

*Proposed by Ahaan Rungta*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1333}
Consider the polynomial
\[ P(x)=\prod\limits_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024) \]
The coefficient of \( x^{2012} \) is equal to \( 2^a \). What is the value of \( a \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
For a Poisson process with rate $\lambda$, calculate $\mathbb{E}(T_2\mid N_1=1)$, where $T_2$ is the time of the second arrival.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \frac{1}{\lambda}}
If $f(x)$ is a differentiable function satisfying $f(x+y)=f(x)+f(y)-(e^{-x}-1)(e^{-y}-1)+1$ and $f(0)=-1$, find the value of $f'(0)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $n$ be a positive integer. In an $n\times n$ garden, a fountain is to be built with $1\times 1$ platforms covering the entire garden. Ana places all the platforms at a different height. Afterwards, Beto places water sources in some of the platforms. The water in each platform can flow to other platforms sharing a side only if they have a lower height. Beto wins if he fills all platforms with water. 
Find the least number of water sources that Beto needs to win no matter how Ana places the platforms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \lceil \frac{n^2}{2} \rceil }
Given $(2x-1)^{2015} = a_{0} + a_{1}x + a_{2}x^{2} + \ldots + a_{2015}x^{2015}$ ($x \in \mathbb{R}$), evaluate the expression $\frac {1}{2}+ \frac {a_{2}}{2^{2}a_{1}}+ \frac {a_{3}}{2^{3}a_{1}}+\ldots+ \frac {a_{2015}}{2^{2015}a_{1}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {1}{4030}}
Let $$f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)}),x\in R,n\in N$$ Let $$a_{n}=\max_{x\in R}{(f_{n}(x))}$$ Find this limit $$\lim_{n\to\infty}a_{n}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Al and Bill play a game involving a fair six-sided die. The die is rolled until either a number less than $5$ is rolled on consecutive tosses, or a number greater than $4$ is rolled on consecutive tosses. Al wins if the last roll is a $5$ or $6$. Bill wins if the last roll is a $2$ or lower. Let $m$ and $n$ be relatively prime positive integers such that $\frac{m}{n}$ is the probability that Bill wins. Find the value of $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy:
\[
(ab + 1)(bc + 1)(ca + 1) = 84.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A school held a soccer match, where each team played against every other team. The winning team of each match received $2$ points, the losing team received $0 points, and in case of a draw, both teams received $1$ point each. The team with the highest score would be crowned the champion. It is known that there is a team with the highest score (higher than all other teams), but this team has fewer wins than the other teams. The minimum number of teams participating in the match is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Ryan Alweiss storms into the Fine Hall common room with a gigantic eraser and erases all integers $n$ in the interval $[2, 728]$ such that $3^t$ doesn’t divide $n!$, where $t = \left\lceil \frac{n-3}{2} \right\rceil$. Find the sum of the leftover integers in that interval modulo $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Three cones are standing on their bases on a table, touching each other. The radii of their bases are 32, 48, and 48, and the apex angles are $\frac{\pi}{3}$, $\frac{2 \pi}{3}$, and $\frac{2 \pi}{3}$ respectively (the apex angle of a cone is the angle between its generatrices in an axial section). A sphere is suspended above the table, touching all the cones. It turns out that the center of the sphere is equidistant from the centers of the bases of all the cones. Find the radius of the sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13(\sqrt{3} + 1)}
Let the function \( f(x) \) have a derivative \( f'(x) \) on \( \mathbf{R} \), such that for any \( x \in \mathbf{R} \), \( f(x) + f(-x) = x^2 \). In the interval \( (0, +\infty) \), \( f'(x) > x \). If \( f(1+a) - f(1-a) \geq 2a \), then the range of the real number \( a \) is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0,+\infty)}
A fault occurred in the transmission line from the substation to the Wuyang Hospital, which cannot be detected from the outside. It requires sending people to climb to the top of the utility poles for inspection. Each inspection can determine whether the fault is at the beginning, middle, or end of the line segment being inspected. If there are a total of 240 utility poles in the transmission line, and each time 2 people are sent to climb the poles, what is the maximum number of inspections needed to ensure the fault location is identified under the optimal plan?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
James has 20 pairs of red socks and half as many black socks. He has a certain number of white socks as red and black combined. James has a total of 90 socks combined. What is the ratio of the number of white socks to the number of red and black socks combined?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2}
What is the least positive integer m such that, given m integers between 1 and 2023 inclusive, there must exist two of them a, b such that 1 < a/b <= 2? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In $\triangle ABC$, $\angle A = 30^{\circ}$, $BC = 13$. Given two circles $\gamma_1$ and $\gamma_2$ with radii $r_1$ and $r_2$, respectively, contain point $A$ and touch line $BC$ at points $B$ and $C$. Find the product $r_1r_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{169}
Emilia wishes to create a basic solution with $7 \%$ hydroxide $(\mathrm{OH})$ ions. She has three solutions of different bases available: $10 \%$ rubidium hydroxide $(\mathrm{Rb}(\mathrm{OH})$ ), $8 \%$ cesium hydroxide $(\mathrm{Cs}(\mathrm{OH})$ ), and $5 \%$ francium hydroxide $(\operatorname{Fr}(\mathrm{OH})$ ). ( $\mathrm{The} \mathrm{Rb}(\mathrm{OH})$ solution has both $10 \% \mathrm{Rb}$ ions and $10 \% \mathrm{OH}$ ions, and similar for the other solutions.) Since francium is highly radioactive, its concentration in the final solution should not exceed $2 \%$. What is the highest possible concentration of rubidium in her solution?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 \%}
Phil has a coin collection he enjoys adding to. He began with 1,000 international coins his grandparents gave him. Over the next five years, his collection habits varied:

- In the first year, he managed to quadruple his collection.
- In the second year, he collected 7 coins each week.
- In the third year, he collected 3 coins every second day.
- In the fourth year, he found two rare coins every week and added them to his collection.
- However, during the fifth year, a portion of his collection was stolen, resulting in the loss of 40% of his coins.

Phil was not discouraged and continued to grow his collection over the next four years:

- In the sixth year, he added 5 coins to his collection every fourth day.
- In the seventh year, he got a job at a coin shop and added 20 coins to his collection each month.
- In the eighth year, he collected 10 coins per week.
- In the ninth year, unfortunately, Phil misplaced a third of his collection during a move to a new city.

How many coins did Phil have left after losing some in the ninth year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2,816}
Given the vectors $\overrightarrow{a} = (\cos 25^\circ, \sin 25^\circ)$, $\overrightarrow{b} = (\sin 20^\circ, \cos 20^\circ)$, and $\overrightarrow{u} = \overrightarrow{a} + t\overrightarrow{b}$, where $t\in\mathbb{R}$, find the minimum value of $|\overrightarrow{u}|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{2}}
The function f(x) is defined as f(x) = $$\frac{1}{3} \sin^4 x + \frac{1}{4} \cos^4 x$$. If f(θ) = $$\frac{1}{7}$$, find the value of tanθ.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{± \frac{\sqrt{3}}{2}}
The sets \( A \) and \( B \) are bounded subsets of the real line. The set \( A \) is the union of \( n \) disjoint and open intervals, and the set \( B \) is the union of \( m \) disjoint and open intervals. What is the maximum number of disjoint intervals that the set \( A \cap B \) can consist of?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + m - 1}
现安排 7 名同学去参加 5 个运动项目, 要求甲、乙两同学不能参加同一个项目，每个项目都有人参加，每人只参加一个项目，则满足上述要求的不同安排方案数为 $\qquad$。（用数字作答）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15000}
Six children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that no two children who stood next to each other in the line end up sitting next to each other in the circle. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
# 8. Problem 8.10

Pentagon $A B C D E$ is circumscribed around a circle.

Angles $\angle A B C, \angle B A E, \angle C D E$ are each $104^{\circ}$. Find $\angle A D B$. Write the answer in degrees (the number only, without specifying the unit of measurement).

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
34. In a triangular prism $A B C A_{1} B_{1} C_{1}{ }^{*}$, two planes are drawn: one passes through the vertices $A, B$ and $C_{1}$, and the other through the vertices $A_{1}, B_{1}$ and $C$. These planes divide the prism into four parts. The volume of the smallest of these parts is $V$. Find the volume of the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12V}
A local college is offering German lessons and currently has 8 students enrolled. Through advertising, 12 more became interested but a fourth of these dropped out within a day. Meanwhile, 3 more students from a neighboring school joined the class, but 2 got frustrated and left within a week. The class then rallied to demonstrate how simple the course actually is and increased enrollment by exactly 1.5 times the students already enrolled. However, 4 of them eventually dropped out due to scheduling conflicts, and one week later, 7 more people enrolled.

In the following weeks, 8 students formed a study group outside of class, and as a result, tripled their enrollment from their current study group size. But, 15% of these students got overwhelmed by their workload, and decided to drop the German lessons. By the end of the session, one-third of all remaining students graduated. How many are still enrolled?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
A furniture shop owner charges his customers a 20% markup on the cost price of the items. However, the customers are also given a special discount. If a customer purchases a computer table and a chair together, they will receive a 10% discount on the total price. A customer paid Rs. 8400 for a computer table and a chair after the discount. Determine the cost price of both the computer table and the chair before the discount was applied.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7777.78}
Factorize: $2a^{5}-8a=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2a(a^{2}+2)(a+\sqrt{2})(a-\sqrt{2})}
In a chess tournament, 12 people participated. After the tournament concluded, each participant created 12 lists. The first list includes only themselves, the second one includes themselves and those they defeated, the third includes everyone from the second list and those they defeated, and so on. The 12th list includes everyone from the 11th list and those they defeated. It is known that for any participant in the tournament, there is a person in their 12th list who was not in their 11th list. How many drawn games were played in the tournament?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Two players are playing a game; one of them thinks of a set of integers \( x_{1}, x_{2}, \ldots, x_{n} \), where each \( x_i \) can be a single-digit positive or negative number. The second player is allowed to ask for the sum \( a_{1} x_{1} + \ldots + a_{n} x_{n} \), with any chosen set of coefficients \( \left(a_{1}, \ldots, a_{n}\right) \). What is the minimum number of questions needed for the guesser to find out the thought-of set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In an exotic country, the National Bank issues coins that can take any value in the interval  $[0, 1]$ . Find the smallest constant  $c > 0$  such that the following holds, no matter the situation in that country:

*Any citizen of the exotic country that has a finite number of coins, with a total value of no more than  $1000$ , can split those coins into  $100$  boxes, such that the total value inside each box is at most  $c$ .*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{1000}{91} }
Three planes intersect pairwise, and their three lines of intersection are parallel to each other. How many distinct parts does this divide the space into? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Hay's Linens sells hand towels in sets of 23 and bath towels in sets of 29. If the store sold the same number of each set this morning and the total number of towels sold was between 500 and 700, what is the smallest number of each type of towel that the store could have sold?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{290}
13. (BUL) ${ }^{\mathrm{IMO} 6}$ Determine the least possible value of $f(1998)$, where $f$ is a function from the set $\mathbb{N}$ of positive integers into itself such that for all $m, n \in \mathbb{N}$,  
$$ f\left(n^{2} f(m)\right)=m[f(n)]^{2} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
carl is facing very difficult financial times and can only pay the interest on a $ 10,000 loan he has taken . the bank charges him a quarterly compound rate of 5 % . what is the approximate interest he pays annually ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$500}
设a为实数，n为任意给定的自然数，且n≥2，若不等式$1 + 2 ^ { x }$ $+ 3 ^ { x } + \cdots + ( n - 1 ) ^ { x } + n ^ { x } a > 0$.对(-∞,1]上的一切x都成立，则a的取值范围为·
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$( \frac { 1 - n } { 2 } , \infty )$}
Sam invested $10,000 and earned a certain interest compounded for 3 years. He then invested more until he had three times as much invested. The next year, he got a 15% return on his investment. Sam now has $59,616. What was the interest rate on his initial investment?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. Find the probability that after they are painted, the cubes can be rotated to be identical in appearance. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1171}
There are 5 pairs of socks in various colors and patterns: 2 red striped, 2 green polka-dotted, 2 blue checked, 2 yellow floral, and 2 purple plaid. A person must wear socks from different pairs, and the mismatched pair must not have the same color or pattern. How many unique mismatched pairs can be formed in this situation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A number, when divided by 44, gives 432 as quotient and a certain remainder. When dividing the same number by 39, the remainder is 15. What is the remainder when the number is divided by 44?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In the Cartesian coordinate system $xOy$, given a circle $C$: $x^{2}+(y-1)^{2}=4$, if one side $AB$ of the equilateral triangle $\triangle PAB$ is a chord of circle $C$, then the maximum value of $PC$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the smallest positive integer  $n$  with the property that in the set  $\{70, 71, 72,... 70 + n\}$  you can choose two different numbers whose product is the square of an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
7. Determine the number of 8-digit positive integers such that after deleting any one digit, the remaining 7-digit number is divisible by 7 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
The monkey has 100 bananas and its home is 50 meters away. The monkey can carry at most 50 bananas at a time and eats one banana for every meter walked. Calculate the maximum number of bananas the monkey can bring home.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
The edges of \( K_{2017} \) are each labeled with 1, 2, or 3 such that any triangle has a sum of labels of at least 5. Determine the minimum possible average of all \( \binom{2017}{2} \) labels.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 - \frac{1}{2017}}
A boy rides his bicycle for a total of 60 km. He travels 15 km on flat ground with an average speed of 12 km/hr, then goes uphill for 20 km at an average speed of 8 km/hr. After reaching the top, he travels 10 km downhill at an average speed of 25 km/hr, and finally rides 15 km on flat ground again at an average speed of 18 km/hr. What is his overall average speed for the entire journey?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.04}
In a circle of radius $R$, a chord $AB$ and a diameter $AC$ are drawn. Chord $PQ$, which is perpendicular to the diameter $AC$, intersects chord $AB$ at point $M$. Given that $AB = a$ and the ratio $PM : MQ = 3$, find $AM$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4R^2 a}{16R^2 - 3a^2}}
Given a bounded sequence \(\left\{a_{n}\right\}\), with 
\[ 
a_{1}=a, \quad a_{n+1}=3 a_{n}^{3}-7 a_{n}^{2}+5 a_{n} \quad (n \in \mathbf{Z}_{+}),
\]
determine the range of values for \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 \leqslant a \leqslant \frac{4}{3}}
In a novel that consists of 530 pages, each page number is printed once, starting from page 1 up to page 530. How many more 3's are printed than 7's throughout the book?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Lyka wants to buy a smartphone worth $800 but she only has $200 at the moment. She also has to pay for her monthly $50 gym membership. She plans to save money for the remaining amount she needs for the smartphone and the gym membership over the next four months. However, Lyka expects a raise in her income during this period. 

For the first two months, she can save $50 per week, and after the raise, for the next two months, she can save $80 per week. 

Given the varying weekly savings and additional expenses, how much will she have saved by the end of the four months?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1040}