inclusionAI/Ring-lite-rl-data huggingface dataset:
We have a calculator with two buttons that displays and integer $x$. Pressing the first button replaces $x$ by $\lfloor \frac{x}{2} \rfloor$, and pressing the second button replaces $x$ by $4x+1$. Initially, the calculator displays $0$. How many integers less than or equal to $2014$ can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, $\lfloor y \rfloor$ denotes the greatest integer less than or equal to the real number $y$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
14. Find the area of the plane region $S$ $=\left\{(x, y) \mid x, y \in\left[0, \frac{\pi}{2}\right], \sin ^{2} x-\sin x \sin y+\sin ^{2} y \leqslant \frac{3}{4}\right\}$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi^{2}}{6}}
Let $T$ be a totally ordered family of countable sets under set inclusion. If the cofinality of $T$ is uncountable, determine the cardinality of $\bigcup_{E \in T} E$. \(\text{Provide your answer as a cardinality.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_1}
In an $8 \times 8$ chessboard, how many ways are there to select 56 squares so that all the black squares are selected, and each row and each column has exactly seven squares selected?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{576}
Determine the spectrum of the operator $S: l^q \to l^q$, where $1 \leq q \leq \infty$, defined by $$ S(x_1, x_2, x_3, x_4, \ldots )=(x_2, -x_1, x_4, -x_3, \ldots). $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{i, -i\}}
Simplify the expression: $$\frac{d}{dx}\left(\int_{-(x^2)}^{x^3}\left(\frac{\sin(t)}{1+\cos(t)}+1\right)dt+\int_{x^3}^{x^2}\left(\frac{\sin(t)}{1+\cos(t)}-1\right)dt\right)$$ for \(-1 \le x \le 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6x^2}
Find the value of $a$ such that the derivative $f'(x) = 2xe^{-x} - x^2 e^x$ has a tangent line that passes through the origin, and the function $f(x)$ also passes through the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that the sequence $\{a_{n}\}$ is a geometric sequence with a common ratio $q\neq 1$, $a_{1}=3$, $3a_{1}$, $2a_{2}$, $a_{3}$ form an arithmetic sequence, and the terms of the sequence $\{a_{n}\}$ are arranged in a certain order as $a_{1}$, $a_{1}$, $a_{2}$, $a_{1}$, $a_{2}$, $a_{3}$, $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $\ldots$, determine the value of the sum $S_{23}$ of the first 23 terms of the new sequence $\{b_{n}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1641}
Compute the dimension of the Dolbeault cohomology group \(H^{(0,1)}\left(B(2)-B(1)\right)\), where \(B(r)\) is the ball of radius \(r\) in \(\mathbb{C}^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a random graph with $n$ nodes, where each node is connected to $k$ randomly chosen nodes, determine the probability that the graph is connected if $k = \frac{\log n + c}{n}$, where $c$ is a constant that tends to infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruits is the banana!" The last fruit responds "No, one of the \emph{last} forty fruits is the banana!" The fruit in the middle yells "I'm the banana!" In how many positions could the banana be?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Let \( f(n) \) be the integer nearest to \( \sqrt{n} \). Find the value of 
\[ 
\sum_{n=1}^{\infty} \frac{\left(\frac{3}{2}\right)^{f(n)} + \left(\frac{3}{2}\right)^{-f(n)}}{\left(\frac{3}{2}\right)^{n}}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
4. Let $n$ be a positive integer. Find the smallest positive integer $k$ with the property that for any colouring of the squares of a $2 n \times k$ chessboard with $n$ colours, there are 2 columns and 2 rows such that the 4 squares in their intersections have the same colour.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n^{2}-n+1}
Consider the non-linear differential equation \( y'(t) = y^2 + t \). Determine the maximum number of times the solution \( y(t) \) is differentiable for \( t \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the number of ways to consider \(\Bbb Q\) as an \(\Bbb R\)-module.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The vertices $M$, $N$, $K$ of rectangle $KLMN$ lie on the sides $AB$, $BC$, $CA$ respectively of a regular triangle $ABC$ in such a way that $AM = 2$, $KC = 1$. The vertex $L$ lies outside the triangle. Find the value of $\angle KMN$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
A number, when divided by 44, gives 432 as quotient and a certain remainder. When dividing the same number by 34, the remainder is 2. What is the original remainder when dividing by 44?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Team A and team B play a series. The first team to win three games wins the series. Before each game, each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If it turns out that team B won the second game and team A won the series, what is the conditional probability that team B won the first game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{5}$}
Find the formula for the order of the wreath product of two groups, G and H, where H is a permutation group on n elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|G|^n \cdot |H|}
During a secret meeting, 20 interns elect their favorite animator. Each votes for two animators. It is known that if you take any two interns, there is always at least one animator for whom they have both voted. What is the minimum number of votes received by the animator who wins the election?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
The manufacturing cost of a shoe is Rs. 210, and the transportation cost is Rs. 500 for 100 shoes. The selling price is Rs. 258. What is the percentage gain on the selling price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16.67\%}
There are $1000$ rooms in a row along a long corridor. Initially, the first room contains $1000$ people, and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different rooms will have people in them?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
Given the parabola with a focus at \((-1, -1)\) and a tangent line at the point \((7, 13)\) described by the equation \(y = 3x - 8\), find the length of the latus rectum of the parabola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4\sqrt{65}}{13}}
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2037\) and \(y = |x-a| + |x-b| + |x-c|\) has exactly one solution. Find the minimum possible value of \(c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1019}
After simple interest for two months at $5$% per annum was credited, a Boy Scout Troop had a total of $\textdollar 255.31$ in the Council Treasury. The interest credited was a number of dollars plus the following number of cents
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$11$}
An $80 \times 80$ grid is colored orange and black. A square is black if and only if either the square below it or the square to the left of it is black, but not both. If there is no such square, consider it as if it were orange. The only exception is the bottom left square, which is black. Consider the diagonal from the upper left to the lower right. How many black squares does this diagonal have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Point $A$ has two tangents drawn to a circle, touching the circle at points $M$ and $N$, respectively. A secant from the same point $A$ intersects the circle at points $B$ and $C$, and intersects the chord $MN$ at point $P$. Given that $AB:BC = 2:3$, find the ratio $AP:PC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4:3}
$100$ chicks are sitting in a circle. The first chick in the circle says the number $1$, then the chick $2$ seats away from the first chick says the number $2$, then the chick $3$ seats away from the chicken that said the number $2$ says the number $3$, and so on. The process will always go clockwise. Some of the chicks in the circle will say more than one number while others might not even say a number at all. The process stops when the $1001$th number is said. How many numbers would the chick that said $1001$ have said by that point (including $1001$)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21
}
In a regular triangular prism \(P-ABC\) where all edge lengths are \(1\), points \(L\), \(M\), and \(N\) are the midpoints of edges \(PA\), \(PB\), and \(PC\) respectively. Determine the area of the cross-section of the circumscribed sphere intersected by the plane \(LMN\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
Given 2414 cards, each with a unique natural number from 1 to 2414. We need to choose two cards such that the sum of the numbers on them is divisible by 100. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29112}
The doll maker crafts one doll in 1 hour and 45 minutes. After making every three dolls, the maker takes a 30-minute break. Ten dolls need to be made for gifts. At what time the following day (specify hours and minutes) will the order be completed if the maker started crafting dolls at 10 AM and also worked throughout the night?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5:00}
In a movie theater, five friends took seats from 1 to 5 (the leftmost seat is number 1). During the movie, Anya left to get popcorn. When she came back, she found that Varya had moved one seat to the right, Galya had moved three seats to the left, and Diana and Ellya had swapped seats, leaving the end seat for Anya. On which seat was Anya sitting before she left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A sequence $a_1, a_2, \ldots$ satisfies $a_1 = \dfrac 52$ and $a_{n + 1} = {a_n}^2 - 2$ for all $n \ge 1.$ Let $M$ be the integer which is closest to $a_{2023}.$ Find the last digit of $M$. Express your answer as a single digit (0-9).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In the diagram, \(ABCD\) is a trapezoid with \(BC\) parallel to \(AD\) and \(AB = CD\). Point \(E\) is on \(AD\) such that \(BE\) is perpendicular to \(AD\). Point \(F\) is the point of intersection between \(AC\) and \(BE\). If \(AF = FB\) and \(\angle AFE = 50^\circ\), what is the measure of \(\angle ADC\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65^\circ}
Consider a dice game with an n-sided fair die. On the first roll, rolling an n results in a win, while rolling a 1 results in a loss. For any other result, the player rolls again. On subsequent rolls, the player wins by rolling an n and loses by rolling the current or lower values. What is the probability of winning as the number of sides on the die approaches infinity (n tends to infinity)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Santana has 10 siblings in total. 4 of them have birthdays in March, 1 has a birthday in May, 1 has a birthday in June, 1 has a birthday in October, 1 has a birthday in November, and 2 of them were born in December. Apart from buying each of her siblings a birthday present and a Christmas present, she also buys them a present each for a family holiday celebrated in August. How many more presents does she have to buy in the second half of the year than the first half of the year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Alice chooses three primes \( p, q, r \) independently and uniformly at random from the set of primes of at most 30. She then calculates the roots of \( p x^{2} + q x + r \). What is the probability that at least one of her roots is an integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{200}}
A fair coin is flipped eight times in a row. Let $p$ be the probability that there is exactly one pair of consecutive flips that are both heads and exactly one pair of consecutive flips that are both tails. If $p=\frac{a}{b}$, where $a, b$ are relatively prime positive integers, compute $100a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1028}
Calculate the arc length of the curve defined by the equation \(6xy - y^4 = 3\) from the point \(\left(\frac{19}{12}, 2\right)\) to the point \(\left(\frac{14}{3}, 3\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{4}}
The number \(5^{867}\) is between \(2^{2013}\) and \(2^{2014}\). Find how many pairs of integers \((m,n)\) there are such that \(1\leq m\leq 2012\) and  
\[5^n<2^m<2^{m+2}<5^{n+1}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{279}
3.17. A circle of radius $R$ is circumscribed around an isosceles trapezoid with an acute angle $\alpha$ at the base. Find the perimeter of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8R}{\sin\alpha}}
Given the set $M=\{x\in\mathbb{N}|8-x\in\mathbb{N}\}$, count the number of elements in set $M$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
集合 $U=\{1,2,3, \cdots, 10\}$ ，则 $U$ 的元素两两互素的三元子集个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Given a group \( G \) with order \( |G| = 231 \), determine the maximum number of subgroups of order 3 that \( G \) can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the inverse Mellin transform of the function \((ix)^{-s}\) for \(x > 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Asli will distribute $100$ candies among her brother and $18$ friends of his. Asli splits the friends of her brother into several groups and distributes all the candies into these groups. In each group, the candies are shared in a fair way such that each child in a group takes the same number of candies, and this number is the largest possible. Then, Asli's brother takes the remaining candies of each group. At most, how many candies can Asli's brother have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In a triangle, let the angle bisectors of angles \(\alpha\) and \(\beta\) intersect the circumcircle at points \(D\) and \(E\) respectively. Determine the angles formed by the segment \(DE\) with the angle bisector of angle \(\gamma\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Find the ratio of the volumes obtained by the successive rotation of a parallelogram around each of its two adjacent sides.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a}{b}}
The steamboat "Rarity" travels for three hours at a constant speed after leaving the city, then drifts with the current for an hour, then travels for three hours at the same speed, and so on. If the steamboat starts its journey in city A and goes to city B, it takes it 10 hours. If it starts in city B and goes to city A, it takes 15 hours. How long would it take to travel from city A to city B on a raft?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
I4.4 A boy walks from home to school at a speed of $p \mathrm{~km} / \mathrm{h}$ and returns home along the same route at a speed of $3 \mathrm{~km} / \mathrm{h}$. If the average speed for the double journey is $\frac{24}{q} \mathrm{~km} / \mathrm{h}$, find $q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given that the lines $l_{1}$: $ax+y+3=0$ and $l_{2}$: $2x+\left(a-1\right)y+a+1=0$ are parallel, find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
There were 1238 apples and 374 more pears than apples in the fruit shop. Of these, a few apples were sold, and 276 pears were bought, so the total of apples and pears was 2527. How many apples have been sold?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{599}
What is the maximum number of sides of a convex polygon that can be divided into right triangles with acute angles measuring 30 and 60 degrees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given the complex-number sequence $\left\{z_{n}\right\}$ satisfying $z_{1}=1$ and $z_{n+1}=\overline{z_{n}}+1+n \mathrm{i}$ for $n=1,2,3 \cdots$, where $\mathrm{i}$ is the imaginary unit and $\overline{z_{n}}$ represents the complex conjugate of $z_{n}$, find the value of $z_{2015}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015 + 1007\mathrm{i}}
Evaluate the limit \( \lim_{{k \to \infty}} \int_{0}^{\infty} k^{3/2} f(x) e^{-kx} \, dx \), where \( f(x) = \int_{0}^{x} \frac{\sin(t)}{t^{3/2}} \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\pi}}
Let $A = \{ z \in \mathbb{C} : |z-1| \leq |z-i| \text{ and } |z-(2+2i)| \leq 1 \}$. Find the modulus of $z \in A$ such that the argument of $z$ is minimized. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{7}}
Given the equation \(x^2 + xy + y^3 = 1\), find the value of \(y'''\) at the point where \(x = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
A total amount of Rs. 10,000 is to be distributed amongst p, q, r, and s such that "p" gets twice as that of "q" and "s" gets 4 times as that of "r." Additionally, "q" and "r" are to receive equal amounts, and the amount received by "s" is half of that of "p." What is the difference between the amounts received by s and p?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2,000}
Calculate the integral of the 1-form \( \alpha = dx_1 + dx_2 - dx_3 \) over the image of the map \( G: S^1 \rightarrow S^1 \times S^1 \times S^1 \) given by \( G(q) = (q, q^2, q^3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( {\left\{  {a}_{n}\right\}  }_{n = 0}^{\infty },{\left\{  {b}_{n}\right\}  }_{n = 0}^{\infty } \) ,and \( {\left\{  {c}_{n}\right\}  }_{n = 0}^{\infty } \) be sequences of real numbers such that for all \( k \geq  1 \) ,
\( {a}_{k} = \left\lfloor  {\sqrt{2} + \frac{k - 1}{2024}}\right\rfloor   + {a}_{k - 1} \)
\( {b}_{k} + {c}_{k} = 1 \)
 \( {a}_{k - 1}{b}_{k} = {a}_{k}{c}_{k}. \)
Suppose that \( {a}_{0} = 1,{b}_{0} = 2 \) ,and \( {c}_{0} = 3 \) . Given that \( \sqrt{2} \approx  {1.4142} \) ,compute
$\sum_{k=1}^{2024}(a_kb_k-a_{k-1}c_k).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2862}
Compare the smallest positive roots of the equations \( \cos(\sin x) = x \) and \( \sin(\cos x) = x \) within the interval \([0, \frac{\pi}{2}]\). Let \(x_1\) be the root of \(\sin(\cos x) = x\) and \(x_2\) be the root of \(\cos(\sin x) = x\). Determine which is greater: \(x_1\) or \(x_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x_2}
On a $4 \times 4$ board, the numbers from $1$ to $16$ are written, one in each box. Andres and Pablo choose four numbers each. Andres chooses the biggest number from each row, and Pablo chooses the biggest number from each column. The same number can be chosen by both. Then, all chosen numbers are removed from the board. What is the greatest possible sum of the numbers left on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78}
Let $\triangle ABC$ have side lengths $AB = 12$, $AC = 16$, and $BC = 20$. Inside $\angle BAC$, two circles are positioned, each tangent to rays $\overline{AB}$ and $\overline{AC}$, and the segment $\overline{BC}$. Compute the distance between the centers of these two circles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\sqrt{2}}
给定空间中 10 个点, 其中任意四点不在一个平面上, 将某些点之间用线段相连, 若得到的图形中没有三角形也没有空间四边形, 试确定所连线段数目的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$15$}
Dana creates a "rectangular spiral" by drawing line segments of lengths, in cm, of \(1,1,2,2,3,3,4,4, \ldots\). Dana's pen runs out of ink after the total of all the lengths he has drawn is \(3000 \, \text{cm}\). Calculate the length of the longest line segment that Dana draws. Express your answer as a whole number in cm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Let $[x]$ denote the greatest integer not exceeding $x$, e.g. $[5.6]=5$ and $[3]=3$. How many different integers are there in the sequence $\left[\frac{1^{2}}{2006}\right], \left[\frac{2^{2}}{2006}\right], \left[\frac{3^{2}}{2006}\right], \ldots, \left[\frac{2006^{2}}{2006}\right]$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1505}
Find the number of ordered 2012-tuples of integers $\left(x_{1}, x_{2}, \ldots, x_{2012}\right)$, with each integer between 0 and 2011 inclusive, such that the sum $x_{1}+2 x_{2}+3 x_{3}+\cdots+2012 x_{2012}$ is divisible by 2012.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2012^{2011}}
Let $f(n)$ be a function defined as $f(n)= 1! +2! +3!+... +n!$. If there exist polynomials $P(x)$ and $Q(x)$ such that $f(n+2)=P(n)f(n+1)+Q(n)f(n)$ for all $n \geq 1$, determine the degrees of the polynomials $P(x)$ and $Q(x)$. Provide your answer as a pair of integers representing the degrees of $P(x)$ and $Q(x)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 1)}
Find the number of 7-tuples \( \left( {{n}_{1},\ldots ,{n}_{7}}\right) \) of integers such that
 \( \mathop{\sum }\limits_{{i = 1}}^{7}{n}_{i}^{6} = {96957}. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2688}
Determine the function $f(x)$ that satisfies the equation $f(x^2 + yf(x)) = f(x^2)$ for all real numbers $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = c}