inclusionAI/Ring-lite-rl-data huggingface dataset:
A square is inscribed in a unit circle, then a circle is inscribed in this square, followed by inscribing another square in this circle. This process is repeated infinitely. Let \( C \) represent the total area of all circles and \( S \) represent the total area of all squares. Calculate the ratio \( \frac{C}{S} \). Express your answer as a simplified mathematical expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{2}}
\( S \) is the set of all ordered tuples \((a, b, c, d, e, f)\) where \(a, b, c, d, e, f\) are integers and \(a^2 + b^2 + c^2 + d^2 + e^2 = f^2\). Find the largest \( k \) such that \( k \) divides \( a b c d e f \) for all elements in \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A student was determining the interval $D$ where the root $x_0$ of the equation $x^3 = -x + 1$ lies. They set the function $f(x) = x^3 + x - 1$, calculated $f(-1) < 0$, and $f(1) > 0$. In the process, they used the "bisection method" and chose 3 more values, which are $x_1$, $x_2$, $x_3$, to finally determine the interval $D$. What is the interval $D$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0.5, 0.75)}
Let $\mathcal{S}$ be a regular 18-gon. For two vertices in $\mathcal{S}$, define the \textit{distance} between them as the length of the shortest path along the edges of $\mathcal{S}$ between them (e.g., adjacent vertices have distance 1). Find the number of ways to choose three distinct vertices from $\mathcal{S}$ such that no two of them have distances 1, 8, or 9.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Given an ellipse \( C: \frac{x^{2}}{ a^{2}}+\frac{y^{2}}{ b^{2}}=1 (a > b > 0) \) with foci \( F_{1} \) and \( F_{2} \) on the left and right, respectively, and \( A \) as the right vertex. A line \( l \) passing through \( F_{1} \) intersects the ellipse at points \( P \) and \( Q \), such that \(\overrightarrow{A P} \cdot \overrightarrow{A Q}=\frac{1}{2}(a+c)^{2} \). Find the minimum eccentricity of the ellipse \( C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \frac{\sqrt{2}}{2}}
In the plane, there are eight distinct points. We will call a numbering of these points from 1 to 8 "good" if the following condition is met:

There exists a line such that all points lie on one side of this line and at different distances from it, with the distances from the points to this line increasing as the numbering increases. That is, the closest point is numbered 1, the next closest point is numbered 2, and so on.

What is the maximum number of different "good" numberings that can be obtained for the given set of eight points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
6. Given that a regular tetrahedron can accommodate 10 small spheres with a radius of 1, the minimum edge length of the regular tetrahedron is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4+2\sqrt{6}}
Let  $a$  and  $b$  be two real numbers and let $M(a,b)=\max\left \{ 3a^{2}+2b; 3b^{2}+2a\right \}$ . Find the values of  $a$  and  $b$  for which  $M(a,b)$  is minimal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a = -\frac{1}{3}, b = -\frac{1}{3}}
Consider \(13\) marbles that are labeled with positive integers such that the product of all \(13\) integers is \(360\). Moor randomly picks up \(5\) marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
John adopts a dog from a shelter. The dog ends up having health problems and this requires 3 vet appointments. After the first appointment, John paid $100 for pet insurance that covers 80% of the subsequent visits. He paid $660 in total. How much did each vet appointment cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$400}
A moth starts at vertex \(A\) of a certain cube and is trying to get to vertex \(B\), which is opposite \(A\), in five or fewer "steps," where a step consists of traveling along an edge from one vertex to another. The moth will stop as soon as it reaches \(B\). How many ways can the moth achieve its objective?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
已知凸五边形 $A B C D E$ 的对角线 $C E$ 分别与对角线 $B D, A D$ 交于点 $F, G$, 若 $B F: F D=5: 4, A G: G D=1: 1, C F: F G: G E=2: 2: 3$, 求 $\triangle C D F$ 与 $\triangle A B E$ 面积的比。答案应以 \(\frac{m}{n}\) 的形式给出，请计算 $m + n$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
How many ways are there to choose 2010 functions $f_{1}, \ldots, f_{2010}$ from \{0,1\} to \{0,1\} such that $f_{2010} \circ f_{2009} \circ \cdots \circ f_{1}$ is constant? Note: a function $g$ is constant if $g(a)=g(b)$ for all $a, b$ in the domain of $g$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$4^{2010}-2^{2010}$}
Determine whether $0$ belongs to or is not in the empty set $\varnothing$. Fill in the blank with $\in$ or $\notin$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\notin}
What is the real parameter $m$ for which the substitution $y = z^m$ transforms the given differential equation $2x^4yy' + y^4 = 4x^6$ into a first-order homogeneous differential equation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/2}
After 100 new people move into a town, 400 of the original population move out. Every year afterward, the town’s population is halved. After 4 years, the population is 60 people. Before the new people moved in, how many people lived in the town?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1260}
Working alone, Ryan can dig a 10 ft by 10 ft hole in a certain amount of time. Castel can dig the same hole in six hours. It would take them 2.72727272727 hours if they worked together. How many hours does it take for Ryan to dig the hole alone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The quadrilateral formed by the line $l_1$: $x+3y-7=0$, the line $l_2$: $kx+y-2=0$, and the positive semi-axes of the x and y axes has a circumscribed circle. Find the value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
In how many ways can $10$ distinct books be placed onto a $3$-shelf bookcase such that no shelf is empty? Express your answer in the form \( n \cdot 10! \), where \( n \) is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
A regular pentagon \(A B C D E\) is given. On side \(A E\), there is a point \(K\), and on side \(C D\), there is a point \(L\). It is known that \(\angle L A E + \angle K C D = 108^\circ\) and \(A K : K E = 3:7\). Find the ratio \(C L : A B\).

A regular pentagon is a pentagon in which all sides are equal and all angles are equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.7}
In a joint-stock company, there are 2017 shareholders, and any group of 1500 of them has a controlling stake (at least 50% of the shares). What is the maximum share of the company that one shareholder can own?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32.8\%}
In order to complete a large job, $1000$ workers were hired, just enough to complete the job on schedule. All the workers stayed on the job while the first quarter of the work was done, so the first quarter of the work was completed on schedule. Then $100$ workers were laid off, so the second quarter of the work was completed behind schedule. Then an additional $100$ workers were laid off, so the third quarter of the work was completed still further behind schedule. Given that all workers work at the same rate, what is the minimum number of additional workers, beyond the $800$ workers still on the job at the end of the third quarter, that must be hired after three-quarters of the work has been completed so that the entire project can be completed on schedule or before?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{766}
A gardener wants to plant trees in his garden in such a way that the number of trees in each row should be the same. The garden has a rectangular shape and 4 distinct tree types (A, B, C, and D) which must be planted in alternating patterns (ABCDABCD... or DCBADCBA...). The gardener also wants the total number of trees planted to be at least 100. If there are 7 rows or 6 rows or 4 rows, then no tree will be left. Find the least number of trees required that satisfy all these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
We mark point $O_1$ on a unit circle and, using $O_1$ as the center, we mark (by means of a compass) we mark point $O_2$ on the circle (clockwise starting with $O_1$). Using point $O_2$ as a new center, we repeat the procedure in the same direction with the same radius; and so on. After we had marked point $O_{1968}$ a circle is cut through each mark so we get 1968 arcs. How many different arc lengths can we thus procure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the values of \( \alpha \) for which the integral \( \int_0^1 \frac{1-x^\alpha}{1-x} \, \mathrm{d}x \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > -1}
Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci $F_{1}$ and $F_{2}$ respectively, draw a line $l$ through the right focus that intersects the ellipse at points $P$ and $Q$. Find the maximum area of the inscribed circle of triangle $F_{1} PQ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9 \pi}{16}}
Find the least positive integer $n$ such that when $3^n$ is written in base $143^2$ , its two right-most digits in base $143$ are $01$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{195}
A bag of potatoes weighs some amount divided by half of its weight. The bag of potatoes weighs 36 lbs. What is the weight of the bag of potatoes before dividing by half of its weight?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{648}
Define $f(x) = x^2 - 45x + 21$. Find the sum of all positive integers $n$ with the following property: there is exactly one integer $i$ in the set $\{1, 2, \ldots, n\}$ such that $n$ divides $f(i)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2592}
At a certain university, 2/3 of the students are females and 3/5 of the total student population is composed of full-time students. The remaining students are part-time. One-tenth of the male students are foreign students. If the university has 3600 students, how many students are non-foreign male part-time students?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{432}
Point $D$ lies on side $AC$ of isosceles triangle $ABC$ (where $AB = AC$) such that angle $BAC$ is $100^\circ$ and angle $DBC$ is $30^\circ$. Determine the ratio of the area of triangle $ADB$ to the area of triangle $CDB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Problem 7.7. The numbers from 1 to 200 were arranged in a random order on a circle such that the distances between adjacent numbers on the circle are the same.

For any number, the following is true: if you consider 99 numbers standing clockwise from it and 99 numbers standing counterclockwise from it, there will be an equal number of numbers that are smaller than it in both groups. Which number stands opposite the number 113?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{114}
Given a positive integer \( n \geq 2 \) and non-negative real numbers \( a_{i} \) (\(1 \leq i \leq n\)) satisfying \( \sum_{i=1}^{n} a_{i}=1 \), find the maximum value of \( \sum_{1 \leq i<j \leq n}(j - i) a_{i} a_{j} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n-1}{4}}
Given an integrated-circuit production line with a long-term yield of 80% (defective rate of 20%) and a standard deviation of 0.04 in the proportion defective from day to day, assume a Beta prior distribution for the proportion defective ($\theta$). On a specific day, a sample of 100 units is inspected, and 38 defective units are found. Calculate the mean of the posterior distribution of $\theta$ for this day.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2905}
A 3 × 3 grid of blocks is labeled from 1 through 9. Cindy paints each block orange or
lime with equal probability and gives the grid to her friend Sophia.
Sophia then plays with the grid of blocks. She can take the top row of blocks and move
it to the bottom, as shown.
1 2 3
4 5 6
7 8 9
4 5 6
7 8 9
1 2 3
Grid A Grid A0
She can also take the leftmost column of blocks and move it to the right end, as shown.
1 2 3
4 5 6
7 8 9
2 3 1
5 6 4
8 9 7
Grid B Grid B0
Sophia calls the grid of blocks citrus if it is impossible for her to use a sequence of the
moves described above to obtain another grid with the same coloring but a different numbering
scheme. For example, Grid B is citrus, but Grid A is not citrus because moving the
top row of blocks to the bottom results in a grid with a different numbering but the same
coloring as Grid A.
What is the probability that Sophia receives a citrus grid of blocks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{243}{256}}
Determine the values of \( p \) for which the integral \( \int_1^2 \frac{\ln(x-1)}{(4-x^2)^p} \, \mathrm{d}x \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 2)}
In the cube \( A C_{1} \), determine the magnitude of the dihedral angle \( C_{1}-D_{1} B-C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Let $a, b \in \mathbb{R}_+$ with $a+b-ab=0$. If the value of $\ln \left( \frac{m^2}{a+b} \right)$ is always non-positive, then the range of $m$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ [-2, 2] }
Find the quadratic subfield of $\mathbb{Q}(\zeta_7)$ that can be expressed in the form $\mathbb{Q}(\sqrt{D})$, where $D$ is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
Find the integer $n$, $-180 \le n \le 180,$ such that $\sin n^\circ = \cos 682^\circ.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
If the order of the letters in the English word "good" is written incorrectly, there are a total of $\_\_\_\_\_$ possible errors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
The sum of certain positive integers (not necessarily distinct) less than or equal to 10 is $S$. Find all values of $S$ such that, regardless of these integers, they can always be partitioned into two groups, each with a sum less than or equal to 70.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{133}
Given the sequence \(\left(a_{n}\right)_{n=1}^{\infty}\) with \(a_{1}=a_{2}=2\), and the recurrence relation
\[ \frac{2 a_{n-1} a_{n}}{a_{n-1} a_{n+1}-a_{n}^{2}}=n^{3}-n \]
for all \(n=2,3,4, \ldots\), find the integer that is closest to the value of \(\sum_{k=2}^{2011} \frac{a_{k+1}}{a_{k}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3015}
On the sides \( AB \) and \( AC \) of an equilateral triangle \( ABC \) with side length 10, points \( P \) and \( Q \) are taken such that the segment \( PQ \) touches the incircle of the triangle and has a length of 4. Find the area of triangle \( APQ \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{\sqrt{3}}}
Divide the set \( M = \{1, 2, \ldots, 12\} \) of the first 12 positive integers into four subsets each containing three elements, such that in each subset, one number is the sum of the other two. Find the number of different ways to do this.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Tony read 23 books, Dean read 12 books and Breanna read 17 books. Tony and Dean read 3 of the same books and all three had read the same book as well. In total how many different books have Tony, Dean, and Breanna read?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
The Fahrenheit temperature ( $F$ ) is related to the Celsius temperature ( $C$ ) by  $F = \tfrac{9}{5} \cdot C + 32$ . What is the temperature in Fahrenheit degrees that is one-ﬁfth as large if measured in Celsius degrees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
Given that $m$ and $n$ are both positive numbers, the curve $\frac {x^{2}}{m} + \frac {y^{2}}{n}=1$ passes through the fixed point $A(1, \sqrt {2})$. Find the minimum value of $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\sqrt{2} + 1)^2}
The product of two two-digit numbers is 1656. Calculate the sum of these two numbers. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
A list of $11$ positive integers has a mean of $10$, a median of $9$, and a unique mode of $8$. What is the largest possible value of an integer in the list?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$35$}
How many factors are there in the product $1 \cdot 2 \cdot 3 \cdot \ldots \cdot n$ if we know that it ends with 1981 zeros?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7935}
A bag of potatoes weighs 12 lbs divided by half of its weight. How much does the bag of potatoes weigh?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
The hyperbola $C$ has its center at the origin and its foci on the $x$-axis. $C$ intersects the directrix of the parabola $y^{2}=4x$ at points $A$ and $B$, with $AB= \sqrt {3}$. Determine the length of the real axis of $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
### 1.26. Find the center of gravity of a hemisphere

$$
0 \leqslant z \leqslant \sqrt{R^{2}-x^{2}-y^{2}},
$$

where the volume density \( p \) of the material at each point is proportional to the distance from the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{5}R}
Evaluate the integral $$\int _{ c } \frac { \sin(iz) }{ { z }^{ 3 }({ z }^{ 2 }+2i) } dz$$ using residue calculus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In the convex quadrilateral $ABCD$, the sum of $AB+BD+DC$ is at most 2, and the area of the quadrilateral is $1/2$. What can be the length of diagonal $AC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Given the parabola $y^{2}=2px (p\neq 0)$ and the fixed points $A(a,b)$, $B(-a,0)$, where $ab\neq 0$ and $b^{2}\neq 2pa$. Let $M$ be a point on the parabola, and let $M_{1}$ and $M_{2}$ be the other intersection points of lines $AM$ and $BM$ with the parabola, respectively. As $M$ varies, the line $M_{1}M_{2}$ passes through a fixed point. Find the coordinates of this fixed point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(a, \frac{2pa}{b})}
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $b\cos B$ is the arithmetic mean of $a\cos C$ and $c\cos A$, then the angle $B=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
Given the function:
$$
\begin{aligned}
y= & |x-1|+|2x-1|+|3x-1|+|4x-1|+|5x-1|
\end{aligned}
$$
Find the value of x that minimizes the function y. Express your answer as a single real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
If 20 men can build a water fountain 56 metres long in 7 days, and they also need to construct a surrounding stone wall that takes an additional 3 days to build, what total length of a similar water fountain with its surrounding wall can be built by 35 men in 9 days, considering that they will be constructing the wall using a more time-efficient material that reduces wall construction time by 50%?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Timmy plans to ride a skateboard ramp that is 50 feet high, with a 30-degree incline. He knows he needs to go 40 mph at the start to make it all the way to the top. He measures his speed on three trial runs and goes 36, 34, and 38 mph. The wind resistance varies between 3 mph and 5 mph during each trial run. Considering the impact of varying wind resistance on the ramp with different inclines, how much faster does he have to go than his average speed, taking into account the additional factors, to make it up the ramp?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Task 1. Represent in the form of an irreducible fraction

$$
7 \frac{19}{2015} \times 6 \frac{19}{2016}-13 \frac{1996}{2015} \times 2 \frac{1997}{2016}-9 \times \frac{19}{2015}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{19}{96}}
There are two rows of seats with three side-by-side seats in each row. Two little boys, two little girls, and two adults sit in the six seats such that neither little boy sits next to either little girl. Determine the number of different ways these six people can be seated.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{176}
Given $a$, $b$, and $c$ are the sides of $\triangle ABC$, and $a^2-a-2b-2c=0$, $a+2b-2c+3=0$, find the largest angle of this triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
Ed had 5 more marbles than Doug. Doug lost 3 of his marbles at the playground. If Ed had 27 marbles. How many marbles did Doug have initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
A man sells a car to his friend at a 14% loss. The friend sells it for Rs. 54,000 and gains 20%. What was the original cost price of the car?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52,325.58}
Out of the 300 senior students, five percent received a full merit college scholarship, ten percent got a half merit scholarship, three percent received a sports scholarship, and seven percent were granted need-based scholarships. Some students received more than one type of scholarship. Among these students, 1% were given both full merit and sports scholarships, 2% received both half merit and need-based scholarships, and 0.5% obtained a sports scholarship and a need-based scholarship. No students were granted all three types of scholarships. Calculate the number of senior students who did not receive any scholarships.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{236}