inclusionAI/Ring-lite-rl-data huggingface dataset:
Joe is given a permutation $p = (a_1, a_2, a_3, a_4, a_{5})$ of $(1, 2, 3, 4, 5)$. A swap is an ordered pair $(i, j)$ with $1 \le i < j \le 5$, and this allows Joe to swap the positions $i$ and $j$ in the permutation. For example, if Joe starts with the permutation $(1, 2, 3, 4, 5)$, and uses the swaps $(1, 2)$ and $(1, 3)$, the permutation becomes
\[(1, 2, 3, 4, 5) \rightarrow (2, 1, 3, 4, 5) \rightarrow (3, 1, 2, 4, 5). \]Out of all $\tbinom{5}{2} = 10$ swaps, Joe chooses $4$ of them to be in a set of swaps $S$. Joe notices that from $p$ he could reach any permutation of $(1, 2, 3, 4, 5)$ using only the swaps in $S$. How many different sets are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
125
}
Given six thin wooden sticks, the two longer ones are $\sqrt{3}a$ and $\sqrt{2}a$, while the remaining four are of length $a$. If they are used to form a triangular prism, find the cosine of the angle between the lines containing the two longer edges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6}}{3}}
In the first pile, there were 7 nuts, and in the rest, there were 12 nuts each. How many piles were there in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
4. On the table, there are 100 weights of different masses. A weight is called successful if its mass is equal to the sum of the masses of some two other weights on the table. What is the smallest number of successful weights that would allow us to assert with certainty that the masses of some two weights differ by at least a factor of three?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{87}
In a variant of SHORT BINGO, a $5\times5$ card is modified such that the middle square is marked as FREE, and the remaining 24 squares are populated with distinct numbers. The modification includes:
- First column contains 5 distinct numbers from the set $1-15$.
- Second column contains 5 distinct numbers from the set $16-30$.
- Third column (excluding the FREE middle square) contains 4 distinct numbers from $31-45$.
- Fourth column contains 5 distinct numbers from the set $46-60$.
- Fifth column contains 5 distinct numbers from the set $61-75$.

How many distinct possibilities are there for the values in the first column of this modified SHORT BINGO card?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360360}
Given the differential equation \( \frac{dx}{dt} = -x + x^7 \) with the initial condition \( x(0) = \lambda \), where \( x = x(t, \lambda) \), find \( \frac{ \partial x(t, \lambda)}{\partial \lambda} \bigg| _{\lambda = 0} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-t}}
For every integers  $ a,b,c$  whose greatest common divisor is  $n$ , if
\[ \begin{array}{l} {x \plus{} 2y \plus{} 3z \equal{} a} 
{2x \plus{} y \minus{} 2z \equal{} b} 
{3x \plus{} y \plus{} 5z \equal{} c} \end{array}
\]
has a solution in integers, what is the smallest possible value of positive number  $ n$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
In triangle \( \triangle ABC \), \( AB = AC \), \( AD \) and \( BE \) are the angle bisectors of \( \angle A \) and \( \angle B \) respectively, and \( BE = 2 AD \). What is the measure of \( \angle BAC \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Two players play alternately on a $ 5 \times 5$ board. The first player always enters a $ 1$ into an empty square and the second player always enters a $ 0$ into an empty square. When the board is full, the sum of the numbers in each of the nine $ 3 \times 3$ squares is calculated and the first player's score is the largest such sum. What is the largest score the first player can make, regardless of the responses of the second player?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate 469111111 x 99999999 = ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46911111053088889}
Jon's laundry machine can do 8 pounds of laundry at a time. The weights of different types of clothes are as follows:

- 4 shirts weigh 1 pound
- 2 pairs of pants weigh 1 pound
- 6 pairs of socks weigh 1 pound
- 1 jacket weighs 2 pounds

Jon needs to wash 20 shirts, 20 pairs of pants, 18 pairs of socks, and 6 jackets. If the machine works at 100% efficiency with shirts and pants but loses 10% efficiency for each jacket in the load, how many loads of laundry does Jon have to do to complete the washing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the smallest possible value of the moment generating function $m_X(2)$ for a random variable $X$ with an expected value of $0$ and a moment generating function $m_X(1) = z \geq 1$. Provide your answer in terms of $z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2z^2 - 1}
5. Given 9 points in space, where no four points are coplanar, a line segment is drawn between every pair of points. These line segments can be colored blue or red, or left uncolored. Find the minimum value of $n$ such that if any $n$ of these line segments are arbitrarily colored red or blue, the set of these $n$ line segments will necessarily contain a triangle with all sides of the same color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Given conditions $p$: $|x+1| > 2$, and $q$: $x > a$, where $\neg p$ is a sufficient but not necessary condition for $\neg q$, determine the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \geq 1}
9. A positive integer that can be written in the form $20a + 8b + 27c$ $(a, b, c \in \mathbf{N})$ is called a "good number." The number of good numbers in the set $\{1, 2, \cdots, 200\}$ is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{153}
2. We will call a set of distinct natural numbers from 1 to 9 good if the sum of all numbers in it is even. How many good sets are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^8}
Consider the following sequence of sets of natural numbers. The first set \( I_{0} \) consists of two ones, 1,1. Then, between these numbers, we insert their sum \( 1+1=2 \); we obtain the set \( I_{1}: 1,2,1 \). Next, between each pair of numbers in \( I_{1} \) we insert their sum; we obtain the set \( I_{2}: 1,3,2,3,1 \). Proceeding in the same way with the set \( I_{2} \), we obtain the set \( I_{3}: 1,4,3,5,2,5,3,4,1 \), and so on. How many times will the number 1973 appear in the set \( I_{1000000} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1972}
Given the joint density function \( f(y_1,y_2) = \frac{1}{8}y_1 e^{-(y_1 + y_2)/2} \) for \( y_1 > 0 \) and \( y_2 > 0 \), and \( f(y_1,y_2) = 0 \) elsewhere, find the expected value of \( D = 50 + 2Y_1 + 4Y_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Determine the radius of convergence of the function \( f(z) = \frac{\sin(z)}{z} \) for any point \( z_0 \in \mathbb{C} \setminus \{0\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the kernel of the homomorphism \( \phi: R/I \oplus R/J \rightarrow R/(I+J) \) defined by \( (a+I, b+J) \mapsto (a-b) + I + J \), where \( I \) and \( J \) are ideals in a Noetherian ring \( R \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{R/(I \cap J)}
Determine the highest power of two that divides the expression \(32! + 33! + 34! + \dots + 90!\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Calculate the probability $\mathbb{P}\{X_{(1)}+X_{(2)} > X_{(3)}\}$, where $X_1, X_2, X_3$ are independent uniform $(0, 1)$ random variables, and $(X_{(1)}, X_{(2)}, X_{(3)})$ denotes their corresponding order statistics.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5}
A permutation $a_1, a_2, \ldots, a_{13}$ of the numbers from $1$ to $13$ is given such that $a_i > 5$ for $i=1,2,3,4,5$. Determine the maximum possible value of 
\[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Evaluate the limit: $$\lim_{n \to \infty} \int_{0}^{1} \frac{n^4u^2e^{-nu}}{1 + n^2u}du$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
How do you solve the following limit?

$$\lim_{n\to \infty }\left[\lim_{x\to \:0}\left(1+\sin^2x+\sin^22x+\ldots+\sin^2nx\right)^{1/(n^3x^2)}\right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(e^{\frac{1}{3}}\)}
Find the approximate interval in which the root of the equation $\ln \left( x+1 ight)-\frac{2}{x}=0$ ($x > 0$) exists. Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,2)}
The area of trapezoid \(ABCD\) is 30. Point \(P\) is the midpoint of the lateral side \(AB\). Point \(R\) on side \(CD\) is chosen such that \(2CD = 3RD\). Lines \(AR\) and \(PD\) intersect at point \(Q\). Find the area of triangle \(APQ\) if \(AD = 2BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{3}}
Determine all values of the parameter \(a\) such that the integral \(\int_0^{+\infty} \frac{dx}{1+x^a \sin^2x}\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 2}
Let $\triangle ABC$ be a right triangle with $B$ as the right angle. A circle with diameter $AC$ intersects side $BC$ at point $D$. If $AB = 18$ and $AC = 30$, find the length of $BD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.4}
Kelvin the Frog is hopping on a number line (extending to infinity in both directions). Kelvin starts at $0$. Every minute, he has a $\frac{1}{3}$ chance of moving $1$ unit left, a $\frac{1}{3}$ chance of moving $1$ unit right, and $\frac{1}{3}$ chance of getting eaten. Find the expected number of times Kelvin returns to $0$ (not including the start) before he gets eaten.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{5} - 5}{5}}
Troy's and Emily's homes are 75 meters and 98 meters away from school, respectively. However, there is a park, a river, a construction site, and a closed road along their routes, which force them to take different detours each day.

On Monday, Troy takes a detour adding 15 meters to his route due to the park, while Emily takes a detour adding 10 meters because of the river. On Tuesday, Troy takes another detour adding 20 meters due to the construction site, while Emily takes one adding 25 meters because of the closed road. On Wednesday, both take a detour around the park, adding 10 meters each. On Thursday, Troy has no detours while Emily takes one adding 15 meters to avoid the construction site. On Friday, Troy takes a detour adding 5 meters due to the closed road, and Emily takes one adding 10 meters because of the river.

Additionally, on Thursday, both Troy and Emily encounter an unexpected road closure that adds another 10 meters to each of their routes.

Given the varying detours and distances added on each day due to the obstacles and sudden changes, find out how much farther Emily walks to school and back in five days.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{270}
Mr. Harmon is a teacher known for his calm and patient nature. Every day, he takes a peaceful 25-minute walk in the morning to start his day. His partner, Alex, on the other hand, loves to run and completes the same distance in just 15 minutes. Over the course of a week, Mr. Harmon walks every morning and Alex runs every other day. How many minutes do they spend exercising together in one week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Determine the center of the division ring \(D\), where \(D\) consists of all Laurent series \(\sum_{m=-k}^{\infty} f_iy^i\) in \(y\) over the field \(F\) of rational functions in \(x\) over \(\mathbb{C}\), satisfying \(yr(x)y^{-1}=r(2x)\) for any \(r(x) \in F\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{C}}
Evaluate the limit: \[ \lim_{n\to\infty}n^2\left(\frac{1^k+2^k+\cdots+n^k}{n^{k+1}}-\frac{1}{k+1}-\frac{1}{2n}\right) \] given that \[ \lim_{n\to\infty}\left(\frac{1^k+2^k+\cdots+n^k}{n^k}-\frac{n}{k+1}\right)=\frac{1}{2}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{k}{12}}
Consider the graphs $y=Ax^2$ and $y^2+3=x^2+4y$, where $A$ is a positive constant and $x$ and $y$ are real variables. In how many points do the two graphs intersect?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function satisfying the conditions \( f(10-x) = f(x) \) and \( f(2-x) = f(2+x) \) for all \( x \in \mathbb{R} \). Given that \( f(0) = 101 \), determine the minimum possible number of values of \( x \) in the interval \([0, 30]\) such that \( f(x) = 101 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
If one side of square \( ABCD \) lies on the line \( y = 2x - 17 \) and the other two vertices are on the parabola \( y = x^2 \), what is the minimum possible area of the square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Let $ABCD$ and $AEFG$ be two faces of a cube with edge $AB=10$. A beam of light emanates from vertex $A$ and reflects off face $AEFG$ at point $Q$, which is 3 units from $\overline{EF}$ and 6 units from $\overline{AG}$. The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube is given by $p\sqrt{q}$, where $p$ and $q$ are integers and $q$ is not divisible by the square of any prime. Find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{155}
Find an upper bound for the argument of the complex function \( f(z)=\frac{z(z-1)}{2} \) where \( z=x+iy \) with \( 0<x<1 \) and \( 3<y<\infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Given a sequence $\{a_{n}\}$ where ${a_1}=1,{a_2}=2$, and $\frac{n}{2}{a_{n+2}}$ is an arithmetic mean of $({n+2}){a_n},λ({{n^2}+2n})$, if ${a_{2n+1}}>{a_{2n}}$ ($n∈{N^*}$), then the range of real number $\lambda$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[0,+\infty \right)}
Andy is attempting to solve the quadratic equation $$64x^2 - 96x - 48 = 0$$ by completing the square. He aims to rewrite the equation in the form $$(ax + b)^2 = c,$$ where \(a\), \(b\), and \(c\) are integers and \(a > 0\). Determine the value of \(a + b + c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86}
Jane and Jena sit at non-adjacent chairs of a four-chair circular table. In a turn, one person can move to an adjacent chair without a person. Jane moves in the first turn, and alternates with Jena afterwards. After nine moves, in how many ways can Jena be adjacent to Jane?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Determine the number of points of intersection between the graphs of the functions \( y = x^{12} \) and \( y = 2^x \) in the xy-plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
It is known that 10% of people own at least 90% of all the money in the world. What is the minimum percentage of all people that can be guaranteed to own 95% of all the money?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55\%}
If the equation $x^{2}-2\sqrt{a}x+2a-1=0$ has two real roots with respect to $x$, then the range of values for $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 \leq a \leq 1}
Two families of three, with a total of $4$ adults and $2$ children, agree to go on a countryside trip together on Sunday using two cars, an "Audi" and a "Jetta". Each car can accommodate a maximum of $4$ people, and the two children cannot be left alone in one car. Determine the number of different seating arrangements possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Dima and Tsiren are playing a game. At the beginning of the game, the number 0 is written on the board. On a turn, a player adds any natural number no greater than 10 to the number written and records the result on the board in place of the original number. The winner is the first person to reach a three-digit number. Dima starts the game. Who will win with a perfect play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Dima}
Task 1. Let $n$ be a positive integer. Given are circular discs with radii $1, 2, \ldots, n$. For each size, we have two discs: one transparent and one opaque. Each disc has a hole in the middle, allowing us to stack the discs on a vertical rod. We want to create stacks that meet the following conditions:

- For each size, there is exactly one disc on the stack.
- When viewed from directly above, the outer edges of all $n$ discs on the stack can be seen. (This means that if there is an opaque disc on the stack, then no smaller discs may lie below it.)

Determine the number of different stacks that can be made under these conditions. (Two stacks are different if they do not use exactly the same set of discs, but also if they use exactly the same set of discs but not in the same order.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n+1)!}
At vertex $A$ of a unit square $ABCD$, an ant begins its journey. It needs to reach point $C$, where the entrance to an anthill is located. Points $A$ and $C$ are separated by a vertical wall in the shape of an isosceles right triangle with hypotenuse $BD$. Find the length of the shortest path the ant must take to reach the anthill.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let  $x_1=1/20$ ,  $x_2=1/13$ , and \[x_{n+2}=\dfrac{2x_nx_{n+1}(x_n+x_{n+1})}{x_n^2+x_{n+1}^2}\] for all integers  $n\geq 1$ .  Evaluate  $\textstyle\sum_{n=1}^\infty(1/(x_n+x_{n+1}))$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
A box contains $3$ shiny pennies and $4$ dull pennies. One by one, pennies are drawn at random from the box and not replaced. If the probability is $\frac{a}{b}$ that it will take more than four draws until the third shiny penny appears and $\frac{a}{b}$ is in lowest terms, find the value of $a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
If $a,b$ are two unequal positive numbers, then connect $\dfrac{a+b}{2}$ and $\dfrac{2ab}{a+b}$ using $>$, $<$, or $=$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{>}
Determine the positive integers that are one and a half times the product of their digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Let the product of the digits of a positive integer \( n \) be denoted by \( a(n) \). Find the positive integer solution to the equation \( n^2 - 17n + 56 = a(n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Convex quadrilateral \(ABCD\) has right angles \(\angle A\) and \(\angle C\) and is such that \(AB = BC\) and \(AD = CD\). The diagonals \(AC\) and \(BD\) intersect at point \(M\). Points \(P\) and \(Q\) lie on the circumcircle of triangle \(AMB\) and segment \(CD\), respectively, such that points \(P\), \(M\), and \(Q\) are collinear. Suppose that \(\angle ABC = 160^\circ\) and \(\angle QMC = 40^\circ\). Find \(MP \cdot MQ\), given that \(MC = 6\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Given an  $n \times n \times n$  grid of unit cubes, a cube is *good* if it is a sub-cube of the grid and has side length at least two. If a good cube contains another good cube and their faces do not intersect, the first good cube is said to *properly* contain the second. What is the size of the largest possible set of good cubes such that no cube in the set properly contains another cube in the set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ (n-1)^3 + (n-2)^3 }
In the right triangle ABC, side AC is divided into 8 equal parts. From each division point, a segment parallel to BC is drawn, resulting in a total of 7 segments. Given that BC = 10, find the sum of the lengths of these 7 segments. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Xena is trying to outrun a dragon to get to the safety of a cave it's too big to fit into. Xena has a 600 foot head start, but the dragon can burn her if it gets within 120 feet of her. If Xena runs 15 feet per second and the dragon flies at a certain speed, Xena has 32 seconds to get to the cave. How fast does the dragon fly in feet per second?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the limit of the following expression using Stolz's theorem:
$$\lim _{n \rightarrow \infty}\left(\frac{2}{2^{2}-1}\right)^{\frac{1}{2^{n-1}}} \left (\frac{2^{2}}{2^{3}-1}\right)^{\frac{1}{2^{n-2}}} \cdots\left(\frac{2^{n-1}}{2^{n}-1}\right)^{\frac{1}{2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
The train system consists of two trains A and B. Train A's speed excluding stoppages is 45 kmph, while including stoppages it is 30 kmph. Train B's speed excluding stoppages is 60 kmph, and including stoppages it is 40 kmph. Additionally, Train A stops at each station for varying durations of 5, 10, and 15 minutes, while Train B stops for 7, 12, and 20 minutes. Considering these factors, determine how many minutes both trains A and B stop per hour on average.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
(Mid-term training Valbonne 2019 group B)

Let $n \geqslant 1$ be an integer. We have $n$ cubes with sides $1, 2, \ldots, n$. We want to stack them in a certain order, such that a cube of side $k$ can only be placed on a cube of side $l$ with $l \geqslant k-2$. How many such different stackings exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{n-2}\times2}
There are two people each wearing blue and yellow clothes, and one person wearing red clothes. Now, these five people are to be arranged in a row, with the requirement that people wearing the same color clothes cannot be adjacent. The total number of different arrangements is     .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
In triangle $ABC$, $AC=24$ inches, $BC=10$ inches, $AB=26$ inches. The radius of the inscribed circle is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$4\text{ in}$}
Two players, A and B, take turns rolling a fair six-sided die, with A rolling first. To win, a player must roll a 4 immediately after the other player rolls a 1. What is the probability that player A wins the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{36}{73}}