inclusionAI/Ring-lite-rl-data huggingface dataset:
On the plane  $S$  in a space, given are unit circle  $C$  with radius 1 and the line  $L$ . Find the volume of the solid bounded by the curved surface formed by the point  $P$  satifying the following condition  $(a),\ (b)$ . $(a)$  The point of intersection  $Q$  of the line passing through  $P$  and perpendicular to  $S$  are on the perimeter or the inside of  $C$ . $(b)$  If  $A,\ B$  are the points of intersection of the line passing through  $Q$  and pararell to  $L$ , then  $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Sue can either borrow $10,\!000$ dollars for $5$ years with a simple interest of $7\%$ annually or an interest which compounds annually for $6\%$. How much more money, rounded to the nearest dollar, would she have to pay back for the more expensive interest than the less expensive interest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{118 \text{ dollars}}
Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a $\spadesuit$, and the third card is a 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{11050}}
In a TV factory, an average of 63 TVs are produced per day for the first 25 days of a 30-day month. A varying number of workers fall ill in the next 3 days such that on the first day, 2 workers fall ill, on the second day, 4 workers fall ill, and on the third day, 5 workers fall ill, reducing the daily average TV production for the month to 58 sets/day. Find the average production per day for the last 5 days of the month, considering that some workers recover after the third day and join back on the fourth day in the last 5 days.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Let \( R \) be the rectangle in the Cartesian plane with vertices at \((0,0)\), \((2,0)\), \((2,1)\), and \((0,1)\). \( R \) can be divided into two unit squares. Pro selects a point \( P \) uniformly at random in the interior of \( R \). Find the probability that the line through \( P \) with slope \(\frac{1}{2}\) will pass through both unit squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/4}
Given that every high school in the town of Pythagoras sent a team of 3 students to a math contest, and Andrea's score was the median among all students, and hers was the highest score on her team, and Andrea's teammates Beth and Carla placed 40th and 75th, respectively, calculate the number of schools in the town.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
两个半径都是 $1$ 的球 $O_{1}$ 和球 $O_{2}$ 相切，且均与直二面角 $\alpha-l-\beta$ 的两个半平面都相切，另有一个半径为 $r(r<1)$ 的小球 $O$与这个二面角的两个半平面也都相切, 同时,与球 $O_{1}$ 和球 $O_{2}$ 都外切。若原始答案的形式为 $m - \sqrt{n}$，请给出 m + n 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Kayla rolls four fair 6-sided dice. Find the probability that at least one of the numbers Kayla rolls is greater than 4 and at least two of the numbers she rolls are greater than 2. Express your answer as a fraction in simplest form, \(\frac{k}{m}\). Please provide the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{142}
Given $\log_{14} \Big(\log_4 (\log_3 y) \Big) = 1$, solve for $y^{-1/3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{-4^{14}/3}}
Determine the time complexity of the recurrence relation \( T(n) = 2T(\sqrt{n}) + \sqrt{\frac{2}{n}} \) for \( n > 3 \), given that \( T(n) \) is constant for \( n \leq 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(\log n)}
Let $X_n$ and $Y_m$ be independent Poisson random variables with means $n$ and $m$, respectively. Determine the limiting distribution of \( \frac{X_n - Y_m - (n-m)}{\sqrt{X_n + Y_m}} \) as $n, m \to \infty$. Provide your answer in terms of a standard distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N(0,1)}
To make a rectangular box with a lid, whose volume is $72cm^{3}$, and the ratio of the lengths of the two adjacent sides of the base is $1:2$, then when its height is \_\_\_\_\_\_, the surface area can be minimized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Six people of different heights are getting in line to buy donuts. Compute the number of ways they can arrange themselves in line such that no three consecutive people are in increasing order of height, from front to back.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{349}
$ABCDEF GH$ is a regular octagon with a side length of $10$ units. Consider a circle with center $A$ and radius $AC$, which intersects another circle with center $D$ and radius $CD$ at a point $I$, different from $C$. What is the length of the segment $IF$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A train of a certain length running at the speed of 72 km/hr takes 13.598912087033037 seconds to cross a bridge 132 m in length. How long is the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140}
Given a regular 2007-gon. Find the minimal number  $k$  such that: Among every  $k$  vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1506}
A permutation $a_1,a_2,\cdots ,a_6$ of numbers $1,2,\cdots ,6$ can be transformed to $1,2,\cdots,6$ by transposing two numbers exactly four times. Find the number of such permutations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
Determine the pointwise convergence of the series \( \sum_{n=1}^{+\infty}(-1)^n\frac{nx^3}{n^2+x^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
A tank has a leak in the bottom that can empty the tank in 6 hours. There are two inlet pipes that fill the tank: Pipe A fills the tank at a rate of 4 liters/minute, and Pipe B fills the tank at a rate of 6 liters/minute. When both inlet pipes are opened and the leak is present, the tank empties in 10 hours. What is the capacity of the tank in liters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2250}
What is the Dirichlet density of the set of all prime ideals in a number field $K$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given an odd function $f$ with domain $\mathbb{R}$ satisfying $f(x+4)=f(x)$, and for $x \in (0,2)$, $f(x)=\ln(x^2+a)$ where $a>0$. If the function $f(x)$ has 9 zeros in the interval $[-4,4]$, find the range of values for the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,1)}
$ABCDEF GH$  is a regular octagon with  $10$  units side . The circle with center  $A$  and radius  $AC$  intersects  the circle  with center  $D$  and radius  $CD$  at point  $ I$ , different from  $C$ . What is the length of the segment  $IF$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Calculate the limit of the expression \( \lim_{n\rightarrow \infty }\sum_{k=n}^{5n}\binom{k-1}{n-1}\left( \frac{1}{5} \right )^n\left( \frac{4}{5} \right )^{k-n} \) as \( n \) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
A polynomial $P$ of degree $2015$ satisfies the equation $P(n)=\frac{1}{n^2}$ for $n=1, 2, \dots, 2016$. Find $\lfloor 2017P(2017)\rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-9}
Determine the values of the variable \( x \) such that the four expressions \( 2x - 6 \), \( x^2 - 4x + 5 \), \( 4x - 8 \), and \( 3x^2 - 12x + 11 \) differ from each other by the same constant amount, regardless of the order of the expressions. Find all possible integer values of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $h_n$ and $k_n$ be the unique relatively prime positive integers such that \[\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\frac{h_n}{k_n}.\] Let $L_n$ denote the least common multiple of the numbers $1, 2, 3, \ldots, n$. For how many integers with $1\le{n}\le{22}$ is $k_n<L_n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$8$}
A cafe has 3 tables and 5 individual counter seats. People enter in groups of size between 1 and 4, and groups never share a table. A group of more than 1 will always try to sit at a table, but will sit in counter seats if no tables are available. Conversely, a group of 1 will always try to sit at the counter first. One morning, \( M \) groups consisting of a total of \( N \) people enter and sit down. Then, a single person walks in and realizes that all the tables and counter seats are occupied by some person or group. What is the minimum possible value of \( M + N \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A plane $P$ is parmeterized by
\[\mathbf{v} = \begin{pmatrix} 1 \\ 6 \\ 7 \end{pmatrix} + t \begin{pmatrix} 2 \\ -1 \\ -1 \end{pmatrix} + s \begin{pmatrix} 2 \\ -3 \\ -5 \end{pmatrix},\]and line $L$ is parameterized by
\[\mathbf{w} = \begin{pmatrix} 7 \\ 4 \\ 1 \end{pmatrix} + u \begin{pmatrix} 3 \\ 0 \\ -1 \end{pmatrix}.\]Find the intersection of plane $P$ and line $L.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\begin{pmatrix} 1 \\ 4 \\ 3 \end{pmatrix}}
If p and q are positive integers each greater than 1, and a certain number times (p + 1) = 28(q + 1), the least possible value of p + q is 43. What is the certain number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{392}
In the concave quadrilateral ABCD shown, ∠BCD = 90°, AB = 12, BC = 4, CD = 3, and AD = 13. Find the area S of the concave quadrilateral ABCD. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
there are certain number of hats and gloves in a box . they are of 41 red , 23 green , 11 orange . power gone but a woman can differentiate between hats and gloves . how many draws are required to obtain a pair of each color ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78}
What is the maximum number of checkers that can be placed on an $8 \times 8$ board so that each one is being attacked?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. Find the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone. The original answer is in \(\frac{k+\sqrt{m}}{n}\) format, please give the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given a rhombus \(ABCD\) with diagonals equal to 3 cm and 4 cm. From the vertex of the obtuse angle \(B\), draw the altitudes \(BE\) and \(BF\). Calculate the area of the quadrilateral \(BFDE\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.32}
Find the smallest three-digit number with the property that if you append to it a number that is greater by 1 on the right, the result (a six-digit number) will be a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{183}
How many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring of length at least two? (A [i]substring[/i] is a continuous block of letters that is part of the string. A string is [i]palindromic[/i] if it is the same when read backwards.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
The probability of an event occurring in each of 900 independent trials is 0.5. Find the probability that the relative frequency of the event will deviate from its probability by no more than 0.02.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.7698}
Given a line \( l \) tangent to the unit circle \( S \) at point \( P \), point \( A \) lies on the same side of \( l \) as circle \( S \), and the distance from \( A \) to \( l \) is \( h \) where \( h > 2 \). From point \( A \), two tangents to \( S \) are drawn, intersecting \( l \) at points \( B \) and \( C \). Find the product of the lengths of line segments \( PB \) and \( PC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{h}{h-2}}
A reader mentioned that his friend's house in location $A$, where he was invited for lunch at 1 PM, is located 1 km from his own house in location $B$. At 12 PM, he left $B$ in his wheelchair heading towards location $C$ for a stroll. His friend, intending to join him and help him reach on time for lunch, left $A$ at 12:15 PM heading towards $C$ at a speed of 5 km/h. They met and then proceeded to $A$ together at a speed of 4 km/h, arriving exactly at 1 PM.

How much distance did our reader cover in the direction of $C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/3}
Ten balls of equal radius are stacked in the shape of a triangular pyramid such that each ball touches at least three others. Find the radius of the sphere into which the pyramid of balls is inscribed, given that the radius of the ball inscribed at the center of the pyramid of balls, that touches six identical balls, is $\sqrt{6}-1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5(\sqrt{2} + 1)}
In a movie screening, there were initially 80 guests. Of these guests, 35% were women, 30 were men, and the rest were children. During the first half of the movie, 25% of the men and 10% of the children left, while 5 new guests (2 men and 3 children) arrived. In the second half of the movie, an additional 20% of the remaining men, 15% of the remaining children, and 2 women decided to leave, whereas 3 more children joined. Calculate the total number of people left at the end of the movie.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{69}
Let $N$ be the greatest integer multiple of $8$, no two of whose digits are the same. What is the remainder when $N$ is divided by $1000$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
To what power must the root \( x_{0} \) of the equation \( x^{11} + x^{7} + x^{3} = 1 \) be raised to obtain the number \( x_{0}^{4} + x_{0}^{3} - 1 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
It is said that the price is 17.25 won per 1 gram (g) of a certain mineral. When one 1000 gram (g) weight and one 10 gram (g) weight were placed on a balance, and a lump of the mineral was placed on the other side, it was level. How much is a mineral worth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17422.5}
$\sin 10^{\circ}\cos 40^{\circ}-\cos 50^{\circ}\cos 10^{\circ}=\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Let  $x,y,z$  be complex numbers such that $\hspace{ 2cm} \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=9$  $\hspace{ 2cm} \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=64$  $\hspace{ 2cm} \frac{x^3}{y+z}+\frac{y^3}{z+x}+\frac{z^3}{x+y}=488$ If  $\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}=\frac{m}{n}$  where  $m,n$  are positive integers with  $GCD(m,n)=1$ , find  $m+n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
The bisector of angle \(ABC\) forms with its sides an angle that is three times smaller than the adjacent angle to \(ABC\). Find the measure of angle \(ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72^\circ}
There are four entrances into Hades. Hermes brings you through one of them and drops you off at the shore of the river Acheron where you wait in a group with five other souls, each of whom had already come through one of the entrances as well, to get a ride across. In how many ways could the other five souls have come through the entrances such that exactly two of them came through the same entrance as you did? The order in which the souls came through the entrances does not matter, and the entrance you went through is fixed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{270}
The number of students in Jakob's graduating class is more than 100 and fewer than 200 and is 2 less than a multiple of 4, 3 less than a multiple of 5, and 4 less than a multiple of 6. How many students are in Jakob's graduating class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{122}
Suppose $P, Q$ are points on the circle $({x}^{2}+{{\left( y-1 \right)}^{2}}=3)$ and the ellipse $(\dfrac{{{x}^{2}}}{4}+{{y}^{2}}=1)$, respectively. The maximum distance between $P$ and $Q$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7\sqrt{3}}{3}}
Cat and Claire are having a conversation about Cat’s favorite number. Cat says, "My favorite number is a two-digit perfect square!"

Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I’d know for certain what it is?"

Cat says, "Yes! Moreover, if I told you a number and identified it as the sum of the digits of my favorite number, or if I told you a number and identified it as the positive difference of the digits of my favorite number, you wouldn’t know my favorite number."

Claire says, "Now I know your favorite number!" What is Cat’s favorite number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
For what value of \(a\) does the inequality
\[
\log _{\frac{1}{a}}\left(\sqrt{x^{2}+a x+5}+1\right) \cdot \log _{5}\left(x^{2}+a x+6\right) + \log _{a} 3 \geq 0
\]
have exactly one solution?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In Figure 4, AD, BC, and CD are tangents to the circle with its center at O and diameter AB = 12. If AD = 4, find the value of BC.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the sum $\sum_{r=0}^{n}\frac{1}{(n+r+1)(n+r+2)(n+r+3)}$ as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( ABC \) be a triangle with \( AB = 5 \), \( BC = 6 \), and \( CA = 7 \). Let \( D \) be a point on ray \( AB \) beyond \( B \) such that \( BD = 7 \), \( E \) be a point on ray \( BC \) beyond \( C \) such that \( CE = 5 \), and \( F \) be a point on ray \( CA \) beyond \( A \) such that \( AF = 6 \). Compute the area of the circumcircle of \( DEF \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{251}{3}\pi}
A necklace consists of 50 blue beads and some quantity of red beads. It is known that on any segment of the necklace containing 8 blue beads, there are at least 4 red beads. What is the minimum number of red beads that can be in this necklace? (The beads in the necklace are arranged cyclically, meaning the last bead is adjacent to the first one.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Two uniform solid spheres of equal radii are so placed that one is directly above the other. The bottom sphere is fixed, and the top sphere, initially at rest, rolls off. At what point will contact between the two spheres be "lost"? Assume the coefficient of friction is such that no slipping occurs. The original answer is in the format \theta=\arccos (m / n), please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, *special* if $a$ and $b$ are positive integers whose sum is 15. Find how many distinct integers can be written as the sum of two, not necessarily different, special fractions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given positive integers \(a\) and \(b\) such that \((a, b)=1\), and there exist a prime number \(p\) and an integer \(c\) such that the congruence equation \(x^{a} + y^{b} \equiv c \pmod{p}\) has no integer solutions, find the smallest possible value of \(a + b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $A$ and $B$ be two square matrices of order $2\times 2$, where $\det(A)=1$ and $\det(B)=2$. Find the value of $$\det(A+\alpha B)-\det(\alpha A+B)$$ where $\alpha \in \mathbb{R}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha^2-1}
A line of soldiers 1 mile long is jogging. The drill sergeant, in a car, moving at twice their speed, repeatedly drives from the back of the line to the front of the line and back again. When each soldier has marched 15 miles, how much mileage has been added to the car, to the nearest mile?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Hexagon $ABCDEF$ has an inscribed circle $\Omega$ that is tangent to each of its sides. If $AB = 12$, $\angle FAB = 120^o$, and $\angle ABC = 150^o$, and if the radius of $\Omega$  can be written as $m +\sqrt{n}$ for positive integers $m, n$, find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Given the function $f(x)=\sin (\omega x+\varphi)$ $(\omega > 0,|\varphi|\leqslant \frac {\pi}{2})$, $x=- \frac {\pi}{4}$ is a zero of $f(x)$, $x= \frac {\pi}{4}$ is the axis of symmetry for the graph of $y=f(x)$, and $f(x)$ is monotonic in $( \frac {\pi}{18}, \frac {5\pi}{36})$, the maximum value of $\omega$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
5. Simplify $C_{n}^{0} C_{n}^{1}+C_{n}^{1} C_{n}^{2}+\cdots+C_{n}^{n-1} C_{n}^{n}=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{C_{2n}^{n-1}}
Freda wants to write a number in each of the nine cells of a figure so that the sum of the three numbers on each diameter is 13 and the sum of the eight numbers on the circumference is 40. What number must be written in the central cell? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
设 $A, B, C, D$ 是空间四个不共面的点, 以 \frac{1}{2} 的概率在每对点之前连一条边, 任意两对点之间是否连边是相互独立的, 则 $A, B$ 可用空间折线（一条或若干条边组成的）连接的概率为。原始答案是\frac{m}{n}格式，请给出m + n的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Megan is delivering meals on wheels. Out of her 80 clients, 15 need vegan meals, 18 need kosher meals, 12 need gluten-free meals, 10 need halal meals, 8 need dairy-free meals, and 4 need nut-free meals. Among these clients: 5 need meals that are both vegan and kosher, 6 need meals that are both vegan and gluten-free, 3 need meals that are both kosher and gluten-free, 4 need meals that are both halal and dairy-free, and 2 need meals that are both gluten-free and nut-free. Two clients need meals that are vegan, halal, and gluten-free, and one client needs a meal that is kosher, dairy-free, and nut-free. How many meals does Megan deliver that do not fall into any of these dietary restriction categories?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}