inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the positive integers expressible in the form  $\frac{x^2+y}{xy+1}$ , for at least  $2$  pairs  $(x,y)$  of positive integers
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A ladder  $10\text{ m}$  long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of  $1\text{ m/s}$ , how fast is the top of the ladder sliding down the wall when the bottom of the ladder is  $6\text{ m}$  from the wall?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4} \text{ m/s}}
Three numbers, $a_1$, $a_2$, $a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}$. Three other numbers, $b_1$, $b_2$, $b_3$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3$, with the sides of the brick parallel to the sides of the box. If $p$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In the initial draft of the floor plan for the CCTV Headquarters Building shown in the figure, $A B C D E F$ is a regular hexagon with an area of 60. Points $G, H, I, J$ are the midpoints of the sides $A B, C D, D E, F A$, respectively. Calculate the area of the shaded region.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Let \( M \) be a set composed of a finite number of positive integers,
\[ 
M = \bigcup_{i=1}^{20} A_i = \bigcup_{i=1}^{20} B_i, \text{ where}
\]
\[ 
A_i \neq \varnothing, B_i \neq \varnothing \ (i=1,2, \cdots, 20)
\]
satisfying the following conditions:
1. For any \( 1 \leqslant i < j \leqslant 20 \),
\[ 
A_i \cap A_j = \varnothing, \ B_i \cap B_j = \varnothing;
\]
2. For any \( 1 \leqslant i \leqslant 20, \ 1 \leqslant j \leqslant 20 \), if \( A_i \cap B_j = \varnothing \), then \( \left|A_i \cup B_j\right| \geqslant 18 \).

Find the minimum number of elements in the set \( M \) (denoted as \( |X| \) representing the number of elements in set \( X \)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Tom enters a digital maze composed of 20 rooms. Each room contains a card marked with a number as shown in the figure. Tom starts in the first room and collects card 1. Subsequently, he can enter an adjacent room, but each room can only be entered once. If the number on the card in the current room is greater than any number on the cards he has already collected, he collects that card. What is the maximum number of cards Tom can collect by navigating through the maze?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
To make a very modern painting for his school, Roberto divides a square canvas into eight parts with four stripes of the same width and a diagonal, as shown in the figure. He paints the picture in blue and green, so that two adjacent parts always have different colors. In the end, he notices that he used more green than blue. What fraction of the picture was painted blue?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{8}}
Given the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, let the points $A$ and $B$ be the two endpoints of its real axis. Let $P$ be a moving point on this hyperbola, and let the slopes of lines $AP$ and $BP$ be $k_1$ and $k_2$, respectively. Find the eccentricity of the hyperbola when the expression $k_1k_2-2\left( \ln |k_1|+\ln |k_2| \right)$ reaches its minimum value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
A wooden rectangular parallelepiped with all sides measured in whole centimeters was painted red. After that, it was cut parallel to its faces into 1 cm³ cubes. It turned out that one-third of the resulting cubes had at least one face painted red, while the remaining two-thirds had all faces unpainted. Find the length of the parallelepiped if it is 2 cm longer than its width and 4 cm longer than its height.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18 \text{ cm}}
We consider the following game for one person. The aim of the player is to reach a fixed capital  $C>2$ . The player begins with capital  $0<x_0<C$ . In each turn let  $x$  be the player’s current capital. Define  $s(x)$  as follows:  $$ s(x):=\begin{cases}x&\text{if }x<1C-x&\text{if }C-x<11&\text{otherwise.}\end{cases} $$  Then a fair coin is tossed and the player’s capital either increases or decreases by  $s(x)$ , each with probability  $\frac12$ . Find the probability that in a finite number of turns the player wins by reaching the capital  $C$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{X_0}{C}}
Find the value of the following infinite sum, where $n$ and $z$ are fixed natural numbers with $n, z > 1$:
$$\sum_{k=1}^{\infty}\frac{(k-1)(k-2)(k-3)\cdot\ldots\cdot(k-n)}{n!z^k}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{(z-1)^{n+1}}}
Task 2. (10 points) Task 2. (10 points) A circle touches the extensions of two sides $A B$ and $A D$ of square $A B C D$, and the points of tangency cut off segments of length 4 cm from vertex $A$. Two tangents are drawn from point $C$ to this circle. Find the side of the square if the angle between the tangents is $60^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4(\sqrt{2}-1)}
In the right triangular prism $ABC-A_{1}B_{1}C_{1}$ with a volume of $3\sqrt{7}$, where $AB=2$ and $AC=3$, the minimum value of the surface area of the circumscribed sphere of this prism is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18\pi}
If (10 ^ 4 * 3.456789) ^ some exponent is written as a single term, there would be 28 digits to the right of the decimal place. What is the exponent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Points $A, B, C$ in the plane satisfy $\overline{A B}=2002, \overline{A C}=9999$. The circles with diameters $A B$ and $A C$ intersect at $A$ and $D$. If $\overline{A D}=37$, what is the shortest distance from point $A$ to line $B C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Given that 70% of the light bulbs are produced by Factory A with a pass rate of 95%, and 30% are produced by Factory B with a pass rate of 80%, calculate the probability of buying a qualified light bulb produced by Factory A from the market.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{133}{200}}
Define the polynomials $P_0, P_1, P_2, \ldots$ by:
\[ P_0(x) = x^3 + 213x^2 - 67x - 2000 \]
\[ P_n(x) = P_{n-1}(x-n), \quad n \in \mathbb{N} \]
Find the coefficient of $x$ in $P_{21}(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61610}
In a $9 × 9$ square table, 9 cells are marked: those at the intersection of the 2-nd, 5-th and 8-th rows with the 2-nd, 5-th and 8-th column. In how many ways can one get from the lower left cell to the upper right one moving only upwards and to the right without entering marked cells?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{678}
In the AU tribe's language, there are two letters - "a" and "u". Certain sequences of these letters form words, where each word contains no fewer than one and no more than 13 letters. It is known that if any two words are written consecutively, the resulting sequence will not be a word. Find the maximum possible number of words in such a language.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16256}
In how many ways can each of the integers $1$ through $11$ be assigned one of the letters $L$, $M$, and $T$ such that consecutive multiples of $n$, for any positive integer $n$, are not assigned the same letter?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A $9 \times 9$ grid square is made of matches, with each side of the cells being one matchstick. Petya and Vasya take turns removing one matchstick at a time, starting with Petya. The winner is the player after whose move there are no complete $1 \times 1$ squares left. Which player can ensure their victory, regardless of how their opponent plays?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Vasya}
When Jack traveled around the world, he had to go through customs and quarantine in multiple countries, as well as deal with time zone changes and layovers. In Canada, he spent 20 hours waiting in customs and 14 days in quarantine. After that, he traveled to Australia with a layover of 12 hours in a timezone 8 hours behind. In Australia, he spent 15 hours in customs and 10 days in quarantine. Finally, Jack visited Japan, passing through a layover of 5 hours in a timezone 2 hours ahead, and ended up spending 10 hours in customs and 7 days in quarantine there. Calculate the total time in hours Jack spent on his journey considering customs, quarantines, layovers, and time zone adjustments.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{806}
Evaluate the improper integral: $$I=\int_{0}^\infty \frac{1-e^{-\frac{1}{x^2}}}{\sqrt{x}}{\rm d}x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\pi}}
如果非负整数 m 及其各位数字之和均为 6 的倍数，则称 m 为"六合数". 求小于 2012 的非负整数中六合数的个数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
Determine the shape of the section of a cube by a plane passing through its center perpendicular to its diagonal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Regular Hexagon}}
A parallelogram $ABCD$ is inscribed in the ellipse $\frac{x^{2}}{4}+y^{2}=1$, where the slope of the line $AB$ is $k_{1}=1$. Determine the slope of the line $AD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{4}}
Let $A=20072009$, and $n$ is an 800-digit number formed by writing 100 $A$s side by side. Calculate the remainder when $n$ is divided by 11. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A person named Jia and their four colleagues each own a car with license plates ending in 9, 0, 2, 1, and 5, respectively. To comply with the local traffic restriction rules from the 5th to the 9th day of a certain month (allowing cars with odd-ending numbers on odd days and even-ending numbers on even days), they agreed to carpool. Each day they can pick any car that meets the restriction, but Jia’s car can be used for one day at most. The number of different carpooling arrangements is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Bernie has 2020 marbles and 2020 bags labeled \( B_{1}, \ldots, B_{2020} \) in which he randomly distributes the marbles (each marble is placed in a random bag independently). If \( E \) is the expected number of integers \( 1 \leq i \leq 2020 \) such that \( B_{i} \) has at least \( i \) marbles, compute the closest integer to \( 1000 E \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
In triangle \(ABC\), angle \(A\) is the largest angle. Points \(M\) and \(N\) are symmetric to vertex \(A\) with respect to the angle bisectors of angles \(B\) and \(C\) respectively. If \(\angle MAN = 50^\circ\), find the measure of \(\angle A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
In a division operation with a remainder of 0, a student mistakenly used the divisor as 102 instead of 201 and obtained a quotient of 753. In addition, the quotient was accidentally divided by 3 due to a calculation error. What is the correct quotient of the original division operation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1146}
Determine the values of \( p \in [1, \infty) \) for which the sequences \( x_k \) and \( y_k \) converge with respect to the \( l^p \) norm, where \( x_k \) has the first \( k \) entries as 1 followed by 0's, and \( y_k \) has the first \( k \) entries as \( k^{-\frac{1}{3}} \) followed by 0's.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3, \infty)}
Determine for which values of the parameter \( \beta \) the integral \( \int_{10}^{+\infty}\frac{1}{(x-5)^{\beta}\sqrt{(x-5)^2+9}}\,dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\beta > 0}
An athletic facility has $5$ tennis courts. Pairs of players arrive at the courts and use a court for an exponentially distributed time with mean $40$ minutes. Suppose a pair of players arrives and finds all courts busy and $k$ other pairs waiting in queue. What is the expected waiting time to get a court?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8(k + 1)}
Let \( {ABC} \) be a triangle with \( {BC} = {20} \) and \( {CA} = {16} \) ,and let \( I \) be its incenter. If the altitude from \( A \) to \( {BC} \) ,the perpendicular
 bisector of \( {AC} \) , and the line through \( I \) perpendicular to \( {AB} \) intersect at a common point, then the length \( {AB} \) can be written as
 \( m + \sqrt{n} \) for positive integers \( m \) and \( n \) . What is \( {100m} + n \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{460
}
Mr. Stevens, a businessman, owned and operated a small retail shop in Boone Mall from 1995 to 2015. During his first year in business, he sold 1,200 items. Over the next 10 years, his sales increased by 150 items each year. In his final year of business, he decided to have a closing sale and managed to sell 2,000 items. How many items in total did Mr. Stevens sell from 1995 to 2015?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47750}
Three rods (with negligible thickness) are fixed perpendicularly to each other at a common point at one end. The lengths of the rods are 1, 2, and 3. This construction is placed on a table in such a way that the free ends of the rods lie on the plane of the tabletop. Determine the exact height of the fixed point above the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{6}{7}}
Given an arithmetic-geometric sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_3 - 4a_2 + 4a_1 = 0$, find the value of $\frac{S_8}{S_4}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
a sum fetched total simple interest of 4016.25 at the rate of 11 p . c . p . a . in 5 years . what is the sum ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7302.27}
A factory implements a time-based wage system, where each worker is paid 6 yuan for every hour worked, for a total of 8 hours per day. However, the clock used for timing is inaccurate: it takes 69 minutes for the minute hand to coincide with the hour hand once. Calculate the amount of wages underpaid to each worker per day.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.60}
Find the greatest value of \( \arg z \) achieved on the circle \( |z-6i|=3 \) in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi}{3}}
A high school student has two types of postcards, A and B, celebrating the school's anniversary: 2 type A postcards and 3 type B postcards. He wants to pick 4 postcards to send one to each of his 4 middle school teachers. Calculate the total number of different ways he can choose the postcards. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the values of $p$ for which the $p$-norm $||x||_p = \left(\sum_{k=1}^n |x_k|^p\right)^{\frac{1}{p}}$ satisfies the Triangle Inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, \infty)}
Find the range of values for \( \phi \) that satisfy the inequality:
\[ -1.96 \leq \frac{6.4-10\phi}{\sqrt{1-\phi^2}} \leq 1.96 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0.467, 0.766]}
Determine the minimum number of coin flips required to achieve a probability of at least 90% of obtaining three consecutive heads using a fair coin. Consider the problem as a Markov Chain with states representing the number of consecutive heads achieved.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
A square region $ABCD$ is externally tangent to the circle with equation $x^2+y^2=1$ at the point $(0,1)$ on the side $CD$. Vertices $A$ and $B$ are on the circle with equation $x^2+y^2=4$. What is the side length of this square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{2\sqrt{19}-4}{5}$}
A certain set of integers is assigned to the letters of the alphabet such that $H=10$. The value of a word is the sum of its assigned letter values. Given that $THIS=50$, $HIT=35$ and $SIT=40$, find the value of $I$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Arwen, Elrond, Galadriel and Legolas went out to pick some flowers. They agreed to only pick a maximum of 50 flowers per person. Arwen was able to get 20 tulips, 18 roses, and 6 sunflowers. Elrond was able to get twice as many tulips as Arwen did and thrice the number of roses Arwen picked, without exceeding the 50-flower limit. Galadriel picked three times the amount of tulips that Elrond gathered and twice the amount of roses Arwen has, also following the flower limit. Legolas picked as many sunflowers as Arwen and Galadriel combined and an equal number of roses and tulips. How many flowers in total did they collect?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{194}
2. The eight-digit number $\square 2 \square 0 \square 2 \square 2$ (the digits in $\square$ can repeat) is a multiple of 72. There are $\qquad$ such eight-digit numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
If $f(x)$ is a polynomial such that $x^2-x+a$ divides $f(x)$, and $f(0)=90$ and $f(1)=92$, find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, it is given that $A B = B C = 4$ and $A A_{1} = 2$. Point $P$ lies on the plane $A_{1} B C$, and it holds that $\overrightarrow{D P} \cdot \overrightarrow{P B} = 0$. Find the area of the plane region formed by all such points $P$ satisfying the given condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{36\pi}{5}}
Given the function $f(x) = x^4 + ax^3 + 2x^2 + b$, where $a, b \in \mathbb{R}$. If the function $f(x)$ only has an extremum at $x=0$, find the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[-\frac{8}{3}, \frac{8}{3}\right]}
Given six balls numbered 1, 2, 3, 4, 5, 6 and boxes A, B, C, D, each to be filled with one ball, with the conditions that ball 2 cannot be placed in box B and ball 4 cannot be placed in box D, determine the number of different ways to place the balls into the boxes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
Find the limit of the following expression as x approaches 0:
$$\lim_{x \to 0} \left( \frac{\tan (\sin (x))-\sin (\tan (x))}{x^7} \right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/30}
In a $2017 \times 2017$ grid, each cell contains a light, which can be either on or off. A light is termed "faulty" if and only if it has an even number of neighboring lights that are on. Find the minimum possible number of faulty lights in this $2017 \times 2017$ grid.

Note: Two lights are considered neighbors if their cells share a common edge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In the interval \(1 \leqslant n \leqslant 10^{6}\), how many integers \(n\) satisfy the equation \(n = x^y\) for non-negative integer solutions \(x, y\) where \(x \neq n\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1111}
Find \(\cos \frac{\alpha - \beta}{2}\), given \(\sin \alpha + \sin \beta = -\frac{27}{65}\), \(\tan \frac{\alpha + \beta}{2} = \frac{7}{9}\), \(\frac{5}{2} \pi < \alpha < 3 \pi\) and \(-\frac{\pi}{2} < \beta < 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{27}{7 \sqrt{130}}}
In the Cartesian coordinate system, there are 5 points on the positive x-axis and 3 points on the positive y-axis. These points are connected to form 15 line segments. Find the maximum number of intersection points these 15 line segments can have in the first quadrant. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
The side length of square \(ABCD\) is 4. Point \(E\) is the midpoint of \(AB\), and point \(F\) is a moving point on side \(BC\). Triangles \(\triangle ADE\) and \(\triangle DCF\) are folded up along \(DE\) and \(DF\) respectively, making points \(A\) and \(C\) coincide at point \(A'\). Find the maximum distance from point \(A'\) to plane \(DEF\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\sqrt{5}}{5}}
## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{1} \frac{(4 \sqrt{1-x}-\sqrt{x+1}) d x}{(\sqrt{x+1}+4 \sqrt{1-x})(x+1)^{2}}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{8}\ln5}
Calculate the value of $\sin^2 36^\circ + \tan 62^\circ \tan 45^\circ \tan 28^\circ + \sin^2 54^\circ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Two identical rulers are placed together. Each ruler is exactly 10 cm long and is marked in centimeters from 0 to 10. The 3 cm mark on each ruler is aligned with the 4 cm mark on the other. The overall length is \( L \) cm. What is the value of \( L \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
In a neighborhood of 200 people who own pets, the following information is given:

- 30 people own only dogs
- 25 people own only cats
- 10 people own only birds
- 7 people own only snakes
- 12 people own only fish
- 15 people own both cats and dogs
- 12 people own both birds and dogs
- 8 people own both birds and cats
- 3 people own both snakes and dogs
- 4 people own both snakes and cats
- 2 people own both snakes and birds
- 9 people own both fish and dogs
- 6 people own both fish and cats
- 14 people own both fish and birds
- 11 people own both fish and snakes
- 5 people own cats, dogs, and snakes
- 4 people own cats, dogs, and birds
- 6 people own cats, birds, and snakes
- 9 people own dogs, birds, and snakes
- 7 people own cats, fish, and dogs
- 5 people own birds, fish, and dogs
- 3 people own birds, fish, and cats
- 8 people own snakes, fish, and dogs
- 4 people own snakes, fish, and cats
- 6 people own snakes, fish, and birds
- 10 people own all five pets

How many total snakes are owned by the people in this neighborhood?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}