inclusionAI/Ring-lite-rl-data huggingface dataset:
Given skew lines l₁ and l₂ with direction vectors $\overrightarrow {a}$ = (-1, 1, 0) and $\overrightarrow {b}$ = (1, 0, -1) respectively, find the size of the angle formed by skew lines l₁ and l₂.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{π}{3}}
Given that there are 25 cities in the County of Maplewood, and the average population per city lies between $6,200$ and $6,800$, estimate the total population of all the cities in the County of Maplewood.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{162,500}
Carlson has three boxes, each containing 10 candies. One box is labeled with the number 4, another with the number 7, and the third with the number 10. In one operation, Carlson performs the following two steps sequentially:

- Takes from any box the number of candies equal to the number written on it.
- Eats 3 candies from those taken, and puts the remaining candies into any other box.

What is the maximum number of candies Carlson can eat as a result of several such operations?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
When some percentage of 7000 is subtracted from 1/10 of 7000, the difference is 700. What is the percentage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0\%}
Given that 2 defective items and 3 good items are mixed together, they need to be distinguished through inspection. Each time, one product is randomly inspected and not returned, until 2 defective items or 3 good items are inspected, at which point the inspection ends. The probability that the inspection stops exactly after four inspections is _______. (Answer with a number.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{5}}
How many two-digit factors does $555555$ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\circ}$ counterclockwise from $A_0$ on $C_1$. Circle $C_2$ has radius $r^2$ and is internally tangent to $C_1$ at point $A_1$. In this way a sequence of circles $C_1,C_2,C_3,\cdots$ and a sequence of points on the circles $A_1,A_2,A_3,\cdots$ are constructed, where circle $C_n$ has radius $r^n$ and is internally tangent to circle $C_{n-1}$ at point $A_{n-1}$, and point $A_n$ lies on $C_n$ $90^{\circ}$ counterclockwise from point $A_{n-1}$, as shown in the figure below. There is one point $B$ inside all of these circles. When $r = \frac{11}{60}$, the distance from the center $C_0$ to $B$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
in a school with 604 students , the average age of the boys is 12 years and that of the girls is 11 years . if the average age of the school is 11 years 9 months , then the number of girls in the school is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{151}
How many rectangles can be formed where each of the four vertices are points on a 4x4 grid with points spaced evenly along the grid lines?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
One standard balloon can lift a basket with contents weighing not more than 80 kg. Two standard balloons can lift the same basket with contents weighing not more than 180 kg. What is the weight, in kg, of the basket?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
设数列 \{x_n\} 满足 x_1=2001，x_{n+1}=x_n+y_n，其中 y_n 等于 x_n 的个位数，则 x_{2024}=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12108}
An inverted frustum with a bottom diameter of 12 and height of 18, filled with water, is emptied into another cylindrical container with a bottom diameter of 24. Assuming the cylindrical container is sufficiently tall, what will be the height of the water level in the cylindrical container?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.5}
Sergeev I.N.

At the base $A_{1} A_{2} \ldots A_{n}$ of the pyramid $S A_{1} A_{2} \ldots A_{n}$ lies a point $O$, such that $S A_{1}=S A_{2}=\ldots=S A_{n}$ and $\angle S A_{1} O=\angle S A_{2} O=$ $\ldots=\angle S A_{n} O$.

For what least value of $n$ does it follow from this that $S O$ is the height of the pyramid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Isabela bought twice as many cucumbers as pencils, with both initially costing $20 each. She received a 20% discount on the pencils and a 30% discount on a third item, a notebook, which also cost $20 initially. However, there is an additional 5% tax on the pencils and a 10% tax on the cucumbers. If she bought 100 cucumbers and 25 notebooks, calculate the total amount she spent to buy all these items.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$3390}
Let $P$ be a point on the right branch of the hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{16} = 1$, and let $M$, $N$ be points on the circles $(x+5)^{2} + y^{2} = 4$ and $(x-5)^{2} + y^{2} = 1$ respectively. Find the maximum value of $|PM| - |PN|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Claudia has a total of 12 coins consisting of 5-cent and 10-cent coins, with which she can combine to make exactly 17 different values using some or all of these coins. Calculate the number of 10-cent coins Claudia has. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
设 $F$ 是椭圆 $E: \frac{x^{2}}{3}+y^{2}=1$ 的左焦点, 过点 $F$ 斜率为正的直线 $l$ 与 $E$ 相交于 $A, B$ 两点, 过点 $A, B$ 分别作直线 $A M$ 和 $B N$ 满足 $A M \perp l, B N \perp l$, 且直线 $A M, B N$ 分别与 $x$ 轴相交于 $M$ 和 $N$. 试求 $|M N|$ 的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt{6}$}
Find the minimum value of the discriminant of a quadratic trinomial whose graph does not intersect the regions below the x-axis and above the graph of the function \( y = \frac{1}{\sqrt{1-x^2}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
设a，b是正整数，使得79$\vert ( a + 7 7 b )$，$7 7 \vert ( a + 7 9 b )$，求和$a + b$可能存在的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{193}
How many ways are there to color the ten balls of a triangular array that is free to rotate using 2 colors? The triangular array is arranged such that a single ball is in the apex [first row]; the next row is composed of two balls; the third row is three balls, and the fourth row is four balls.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{352}
In the tetrahedron \( A B C D \), \( A B = A C = A D = B C = 1 \), \( B D = \sqrt{3} \), \( C D = \sqrt{2} \). Find the angle between the edges \( A D \) and \( B C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
The Evil League of Evil is plotting to poison the city's water supply. They plan to set out from their headquarters at  $(5, 1)$  and put poison in two pipes, one along the line  $y=x$  and one along the line  $x=7$ . However, they need to get the job done quickly before Captain Hammer catches them. What's the shortest distance they can travel to visit both pipes and then return to their headquarters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{5}}
A square board is divided by seven lines parallel to one side of the board and seven lines parallel to the other side of the board, creating 64 rectangular cells that are colored in a checkerboard pattern (alternating black and white). The distances between neighboring lines are not necessarily equal, so the cells may have different sizes. It is known that the ratio of the area of any white cell to the area of any black cell is at most 2. Find the maximum possible ratio of the total area of the white cells to the total area of the black cells.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{4}}
A farmer bought two sheep in 1996. That year, he didn't have any offspring. The first sheep gave birth to one lamb every three years, and the second sheep gave birth to one lamb every two years. All the sheep born each year also gave birth to one lamb annually. How many sheep will the farmer have in the year 2000?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find an integrating factor for the following differential equation, where the integrating factor is a function of $x^2-y^2$: $$ (x^2 - y^2 + x) dy + ydx = 0. $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{x^2-y^2}}
Marie is painting a $4 \times 4$ grid of identical square windows. Initially, they are all orange but she wants to paint $4$ of them black. How many ways can she do this up to rotation and reflection?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
Let \( S = \left\{ A = \left(a_1, \cdots, a_8\right) \mid a_i = 0 \text{ or } 1, \, i = 1, 2, \cdots, 8 \right\} \). For two elements \( A = \left(a_1, \cdots, a_8\right) \) and \( B = \left(b_1, \cdots, b_8\right) \) in \( S \), define
\[ d(A, B) = \sum_{i=1}^{8} \left| a_i - b_i \right|, \]
which is called the distance between \( A \) and \( B \). 

What is the maximum number of elements that can be taken from \( S \) such that the distance between any two of them is \( \geqslant 5 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The numbers $1, 2, \ldots, 49, 50$ are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers $a, b, c$ from the board, replaces them by their sum $a + b + c$, and writes $(a + b)(b + c)(c + a)$ in her notebook. Ann performs such operations until only two numbers remain on the board (in total 24 operations). Then she calculates the sum of all 24 numbers written in the notebook. Let $A$ and $B$ be the maximum and minimum possible sums that Ann can obtain. Find the value of $\frac{A}{B}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the number of Hamiltonian paths in a complete bipartite graph K(m,n) where m and n are distinct positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m! * n!}
A \(4 \times 4\) Sudoku grid is filled with digits so that each column, each row, and each of the four \(2 \times 2\) sub-grids that compose the grid contains all of the digits from 1 to 4. 

Find the total number of possible \(4 \times 4\) Sudoku grids.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
Let$n = 2 ^ { 3 1 } 3 ^ { 1 9 } .$ How many positive integer divisors of $n ^ { 2 }$ are less than n but do not divide n?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{589}
Find the maximum value of the parameter \( m \) for which the inequality 
\[ m \sqrt{m}\left(x^{2}-6 x+9\right)+\frac{\sqrt{m}}{\left(x^{2}-6 x+9\right)} \leq \sqrt[4]{m^{3}} \cdot\left|\cos \frac{\pi x}{5}\right| \]
has at least one solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0625}
In the triangle $ABC$, $\angle ABC = \angle ACB = 78^\circ$. On the sides $AB$ and $AC$, respectively, the points $D$ and $E$ are chosen such that $\angle BCD = 24^\circ$, $\angle CBE = 51^\circ$. Find the measure of angle $\angle BED$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
50 students with blond hair, brown hair, and red hair are sitting around a round table. It is known that in any group of students sitting consecutively, there is at least one student with brown hair between any two students with blond hair, and at least one student with red hair between any two students with brown hair. What is the minimum number of students with red hair that can be sitting at this table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Given the equation \( x e^{-2 x} + k = 0 \) has exactly two real roots in the interval \((-2, 2)\), find the range of values for \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(-\frac{1}{2 e}, -\frac{2}{e^{4}}\right)}
From a deck of bridge cards, two players have taken the clubs, hearts, and spades numbered 1 to 6 cards, and play the following game with these 18 cards. The player first draws two cards from the deck held by the banker. If the sum of the numbers on the drawn cards is 7, the player wins; if the sum is greater than 7, the player loses; and if the sum is less than 7, the player draws a third card, and wins only if the new number complements the sum of the first two cards to make 7. What is the probability that the player wins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{33}{136}}
In the isosceles triangle \( ABC \) (\( AB = BC \)), medians \( AD \) and \( EC \) intersect at point \( O \). The ratio of the radius of the circle inscribed in triangle \( AOC \) to the radius of the circle inscribed in quadrilateral \( ODBE \) is \(\frac{2}{3}\). Find the ratio \(\frac{AC}{BC}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20/17}
If today is Monday, starting from tomorrow, the first day after $2^{50}$ days is Sunday \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Saturday}}
Given vectors a = {3, -2m} and b = {8, 3m}, find the value of the real number m such that vectors a and b are perpendicular. Express your answer as a real number, including the ± symbol if applicable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±2}
In triangle \( ABC \), the sides \( AC = 14 \) and \( AB = 6 \) are known. A circle with center \( O \), constructed on side \( AC \) as the diameter, intersects side \( BC \) at point \( K \). It turns out that \( \angle BAK = \angle ACB \). Find the area of triangle \( BOC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
The rightmost nonzero digit in the decimal expansion of 101! is the same as the rightmost nonzero digit of \( n! \), where \( n \) is an integer greater than 101. Find the smallest possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{103}
Fluffball and Shaggy the squirrels ate a basket of berries and a pack of seeds containing between 50 and 65 seeds, starting and finishing at the same time. Initially, Fluffball ate berries while Shaggy ate seeds. Later, they swapped tasks. Shaggy ate berries six times faster than Fluffball, and seeds three times faster. How many seeds did Shaggy eat if Shaggy ate twice as many berries as Fluffball?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Find the integral of $$\int\frac{dx}{(x^{2}-36)^{3/2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{x}{36\sqrt{x^{2}-36}}+C}
已知 $f(x)=|1-2 x|, x \in[0,1]$, 则方程 $f(f(f(x)))=\frac{1}{2} x$ 的解的个数是
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircles?

$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6:\pi}
Over a span of 7 years, a small college reduced its faculty twice. First, they reduced the faculty by 15% over the first 3 years, and then by an additional 25% over the next 4 years, leaving them with a faculty of 180 professors. What was the original number of faculty members?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{282}
A certain store sells a batch of thermal shirts, with an average daily sales of 20 pieces and a profit of $40 per piece. In order to increase sales and profits, the store has taken appropriate price reduction measures. After investigation, it was found that within a certain range, for every $1 decrease in the unit price of the thermal shirts, the store can sell an additional 2 pieces per day on average. If the store aims to make a daily profit of $1200 by selling this batch of thermal shirts and minimizing inventory, the unit price of the thermal shirts should be reduced by ______ dollars.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
设 $k$ 是正整数, 且对任何正整数 $n \geqslant 2$, 如下不等式都成立: $\left[\frac{n}{\sqrt{3}}\right]+1>\frac{n^{2}}{\sqrt{3 n^{2}-k}}$ ，其中 $[x]$ 表示不超过 $x$ 的最大整数。  
求 $k$ 的最大值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given that sequence {a_n} is an equal product sequence, with a_1=1, a_2=2, and a common product of 8, calculate the sum of the first 41 terms of the sequence {a_n}.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{94}
What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16.\overline{6}\%}
There are 120 girls, 96 boys, and 72 teenagers from various age groups who want to participate in 6th-grade intramurals. The girls consist of 40 who prefer basketball, 50 who prefer volleyball, and 30 who prefer soccer. The boys consist of 48 who prefer basketball and 48 who prefer soccer. The teenagers consist of 24 who prefer volleyball and 48 who prefer mixed sports. Each team must have the same number of girls, boys, and teenagers, as well as balance the sports preferences within each team. What is the greatest number of teams that can participate in intramurals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Two knights placed on distinct squares of an $8 \times 8$ chessboard, where each square is a unit square, are said to attack each other if the distance between the centers of the squares on which the knights lie is $\sqrt{5}$. In how many ways can two identical knights be placed on distinct squares of an $8 \times 8$ chessboard such that they do NOT attack each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1848}
Justin needs to read at least 100 pages or more of his book in one week to pass his class. He has read 10 pages already on the first day. Assuming he reads twice the amount of pages as the first day each day in the remaining 6 days, how many pages will he have read after the week is over?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{130}
A number squared results in the first perfect square that begins with two 2s. The sum of the digits of the next perfect square to begin with two 2s is 13. What is the number that, when squared, gives the first perfect square that begins with two 2s?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
Two right cones each have base radius 4 and height 3, such that the apex of each cone is the center of the base of the other cone. Find the surface area of the union of the cones.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62\pi}
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 + \sqrt{5}}{2}}
6.1. In February of a non-leap year, Kirill and Vova decided to eat ice cream according to certain rules. If the date was even and the day of the week was Wednesday or Thursday, they each ate seven portions of ice cream. If the day of the week was Monday or Tuesday and the date was odd, they each ate three portions of ice cream. If the day of the week was Friday, the number of portions each ate was equal to the date on the calendar. Ice cream was forbidden on all other days and under other conditions. What is the maximum number of portions of ice cream that Vova could have eaten in February under these conditions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
Let $a$ and $b$ be the numbers obtained by rolling a pair of dice twice. The probability that the equation $x^{2}-ax+2b=0$ has two distinct real roots is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Xiao Ming places some chess pieces into a $3 \times 3$ grid. Each small square within the grid can have zero, one, or more chess pieces. After counting the number of chess pieces in each row and each column, he obtains 6 different sums. What is the minimum number of chess pieces needed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The Fermat point is a point located inside a triangle such that the sum of its distances to the three vertices of the triangle is minimized. When all three interior angles of a triangle are less than $\frac{2\pi}{3}$, the angles formed by connecting the Fermat point with the three vertices of the triangle are all $\frac{2\pi}{3}$. Given that point $P$ is the Fermat point of triangle $ABC$, with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, if $\cos A = 2\sin \left(C-\frac{\pi}{6}\right)\cos B$ and $b^{2} = \left(a-c\right)^{2}+6$, then the value of $PA \cdot PB + PB \cdot PC + PA \cdot PC$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
With about six hours left on the van ride home from vacation, Wendy looks for something to do.  She starts working on a project for the math team.

There are sixteen students, including Wendy, who are about to be sophomores on the math team.  Elected as a math team officer, one of Wendy's jobs is to schedule groups of the sophomores to tutor geometry students after school on Tuesdays.  The way things have been done in the past, the same number of sophomores tutor every week, but the same group of students never works together.  Wendy notices that there are even numbers of groups she could select whether she chooses  $4$  or  $5$  students at a time to tutor geometry each week:
\begin{align*}\dbinom{16}4&=1820,\dbinom{16}5&=4368.\end{align*} 
Playing around a bit more, Wendy realizes that unless she chooses all or none of the students on the math team to tutor each week that the number of possible combinations of the sophomore math teamers is always even.  This gives her an idea for a problem for the  $2008$  Jupiter Falls High School Math Meet team test:

\[\text{How many of the 2009 numbers on Row 2008 of Pascal's Triangle are even?}\]

Wendy works the solution out correctly.  What is her answer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1881}
We consider positive integers $n$ having at least six positive divisors. Let the positive divisors of $n$ be arranged in a sequence $(d_i)_{1\le i\le k}$ with 
\[ 1 = d_1 < d_2 < \dots < d_k = n \quad (k \ge 6). \]
Find all positive integers $n$ such that 
\[ n = d_5^2 + d_6^2. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
What is the 18 th digit after the decimal point of $\frac{10000}{9899}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}