inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the number of terms needed in the series expansion to approximate π to within $10^{-3}$ using the formula:

$$\frac{\pi}{4} = 4\arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right)$$

and the Maclaurin series for the arctangent function:

$$\arctan(x)=\sum_{i=1}^n\frac{(-1)^{i+1}x^{2i-1}}{2i-1}$$

Find the smallest integer $n$ such that the approximation error is less than $10^{-3}$. Use the inequality:

$$\frac{4}{2n+1} \left(\frac{4}{5^{2n+1}}-\frac{1}{239^{2n+1}}\right)<10^{-3}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
There are 10 cups on a table, 5 of them with the opening facing up and 5 with the opening facing down. Each move involves flipping 3 cups simultaneously. What is the minimum number of moves required to make all cup openings face the same direction?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Three semicircles of radius $1$ are constructed on diameter $\overline{AB}$ of a semicircle of radius $2$. The centers of the small semicircles divide $\overline{AB}$ into four line segments of equal length, as shown. Find the area of the shaded region that lies within the large semicircle but outside the smaller semicircles. The original answer is in the form \(\frac{k}{m}\pi - \frac{\sqrt{n}}{p}\). Please calculate the value of k + m + n + p.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
20. (MON 4) Find the least natural number $n$ such that if the set $\{1,2, \ldots, n\}$ is arbitrarily divided into two nonintersecting subsets, then one of the subsets contains three distinct numbers such that the product of two of them equals the third.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Find the number of distinct subgroups that can be generated by two elements in the group \( \mathbb{Z}_q \oplus \mathbb{Z}_q \oplus \mathbb{Z}_q \), where \( q \) is a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2q^2 + 2q + 3}
Evaluate the integral \( \int_0^\infty \ln(x)\,x^2e^{-x}\,\mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 - 2\gamma}
设二位数AB与三位数ABC的和仍为三位数BAD，其中A、B、C、D 为相异的数字；设BAD最大可能之值为x，则$1 0 0 0 - x = \underline { }$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
If 183 is divisible by 11, find the value of the smallest natural number n?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the number of roots of the polynomial \( Q(z) = 3z^5 - 2z^3 + z^2 - z + 1 \) in the region where \( \operatorname{Re}(z) > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the quadratic variation of the process \( Y_t = B_{2t} \), where \( (B_t)_{t \geq 0} \) is a standard Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2t}
Evaluate the limit: \[ \lim_{x \to 0} \left(\frac{2f(x)-x}{x^2}\right) \] where \( f(x) = \lim_{n \to \infty} \frac{x}{n} \left(\frac{1}{1+e^{-x/n}}+\frac{1}{1+e^{-2x/n}}+\dots+\frac{1}{1+e^{-x}} \right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Find all sets of three distinct integers \(a, b, c\) such that \(W(a) = W(b) = W(c)\), where \(W(x) = x^4 - 3x^3 + 5x^2 - 9x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
In the Laurent series for \(f(z) = \frac{1}{z-4}\) centered at \(z=1\), find the coefficient of \((z-1)^{-2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
An ancient civilization has a tribe of 12 members organized hierarchically. The tribe has one main chief, two supporting chiefs (Senior and Junior), and each supporting chief has three inferior officers. If the tribe has 12 members in total, in how many ways can the leadership structure of the tribe be formed under these restrictions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2217600}
A ball is floating on the surface of a lake. After the lake freezes and the ball is removed, a hole with a diameter of 24cm and a depth of 8cm is left on the ice surface. Find the surface area of the ball. Express your answer in terms of π.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{676\pi}
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Let \(a_{1}, a_{2}, \cdots, a_{n}\) be all the irreducible proper fractions with a denominator of 60. Calculate the value of \(\sum_{i=1}^{n}\left(\cos \frac{a_{i} \pi}{2}\right)^{2}\). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $s$ be the set of all rational numbers $r$ that satisfy the following conditions: $0<r<1$, and $r$ can be represented as a repeating decimal of the form $0.abcabcabc\cdots = 0.\dot{a}b\dot{c}$, where the digits $a, b, c$ do not necessarily have to be distinct. How many different numerators are there when the elements of $s$ are written as reduced fractions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{660}
Let $S$ be the smallest subset of the integers with the property that $0 \in S$ and for any $x \in S$, we have $3x \in S$ and $3x + 1 \in S$. Determine the number of non-negative integers in $S$ less than $2008$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
To guard a military installation around the clock, a day shift and a night shift are required. A sentry guard may take either a day, or a night shift, or work around the clock. In these cases the guard is given a leave of absence of not less than $1, 1.5$ or 2.5 full days, respectively. What is the least number of guards necessary to ensure the security of the installation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A ball is passed around a circular table with 10 seats, each occupied by a person. In the first run, the ball moves to the $1^{1} = 1$st position. In the second run, it moves an additional $2^{2} = 4$ positions, ending up in the 5th position. In the third run, it moves another $3^{3}$ positions, ending up in the 2nd position. Determine the position of the ball after the 2017th run.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the order of accuracy for numerically integrating a function \( f(x,y) \) over the square domain \([0,1]\times[0,1]\) using a two-dimensional Cartesian grid with equal spacing \( \Delta x = \Delta y \) and evaluating the function at the grid cell centers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the value of \( k \) if the function \( f(x) = \begin{cases} ax(x-1)+b & ; x < 1 \\ x+2 & ; 1 \le x \le 3 \\ px^2+qx+2 & ; x > 3 \end{cases} \) is continuous for all \( x \in \mathbb{R} \) except \( x=1 \), but \( |f(x)| \) is differentiable everywhere, and \( f'(x) \) is continuous at \( x=3 \). Additionally, \( |a+p+b+q| = k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The front tires of a car wear out after $25,000$ kilometers, and the rear tires after $15,000$ kilometers. When you need to swap tires so that the car can travel the longest possible distance with the same tires?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9375}
Let $A$, $B$, $C$, $D$ be four points in the plane, with $C$ and $D$ on the same side of the line $AB$, such that $AC \cdot BD = AD \cdot BC$ and $\angle ADB = 90^{\circ}+\angle ACB$. Find the ratio
\[\frac{AB \cdot CD}{AC \cdot BD} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Paul fills in a \( 7 \times  7 \) grid with the numbers 1 through 49 in a random arrangement. He then erases his work and does the same
 thing again, to obtain two different random arrangements of the numbers in the grid. What is the expected number of pairs of
 numbers that occur in either the same row as each other or the same column as each other in both of the two arrangements?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{147}{2}}
Given a $2\times2$ matrix $A$ with real entries that is invertible and has a determinant $\text{Det}(A) = k$. If $\text{Det}(A + k\text{adj}(A)) = 0$, where $\text{adj}(A)$ denotes the adjugate of $A$, find the value of $\text{Det}(A - k\text{adj}(A))$. \(\text{adj}(A)\) is the adjugate of \(A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $f_n$ be the Fibonacci numbers, defined by $f_0 = 1$, $f_1 = 1$, and $f_n = f_{n-1}+f_{n-2}$. For each $i$, $1 \le i \le 200$, we calculate the greatest common divisor $g_i$ of $f_i$ and $f_{2007}$. What is the sum of the distinct values of $g_i$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
25
}
A $7 \times 7$ board is filled with $49$ distinct numbers, one in each cell. You can ask questions where each question reveals the set of numbers in any $3$ chosen cells. What is the minimal number of questions required to determine the number in each cell of the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Determine the value of \(\alpha\) for which the series \(\sum \frac{\sqrt{a_{n}}}{n^{\alpha}}\) converges, given that \(\sum a_{n}\) converges and \(a_{n}\) are positive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > \dfrac{1}{2}}
Consider sequences that consist entirely of $A$'s and $B$'s and that have the property that every run of consecutive $A$'s has even length, and every run of consecutive $B$'s has odd length. Examples of such sequences are $AA$, $B$, and $AABAAA$, while $BBAB$ is not such a sequence. How many such sequences have length 14?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{172
}
Given that $X$ and $Y$ are independent central chi-square random variables with degrees of freedom $m \geq 3$ and $n \geq 3$, respectively, and that $E \left(\frac{X}{Y}\right) = 3$ with $m + n = 14$, find the expected value $E \left(\frac{Y}{X}\right)$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{7}}
Evaluate the integral \( \int_0^y w^{a-1}(1-w)^{b-1} \, dw \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{B(y; a, b)}
A convex polygon has at least one side with length 1. If all diagonals of the polygon have integer lengths, what is the maximum number of sides the polygon can have? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A cat is going up a stairwell with ten stairs. The cat can jump either two or three stairs at each step, or walk the last step if necessary. How many different ways can the cat go from the bottom to the top?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let \( f(x) \) be a real-valued function defined on the interval \((0,\infty)\). Determine the smallest possible value of \( \alpha \) such that \(|f'(x)| < |f(x)|\) for all \( x \in (\alpha,\infty)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the permutations \(\alpha = (1\ 2\ 3\ 7\ 11\ 10\ 9\ 5)\), \(\beta = (2\ 3\ 4\ 8\ 12\ 11\ 10\ 6)\), \(\gamma = (5\ 6\ 7\ 11\ 15\ 14\ 13\ 9)\), and \(\delta = (6\ 7\ 8\ 12\ 16\ 13\ 14\ 10)\), determine the order of the permutation group generated by these cycles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20922789888000}
Jackson buys a computer game for $66 which had a discount of 15% off the original price. He also buys three movie tickets, each originally $12, with a 10% entertainment tax added to the ticket price. Additionally, he spends $7 on snacks and $5 on transportation for each movie trip. What is the total amount he spent on entertainment, additional expenses, after incorporating the discount and tax?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$117.60}
Find all continuous functions \( g: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the functional equation \( g(x+y) = g(x) + g(y) \) for all real numbers \( x \) and \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g(x) = kx}
In an equilateral triangle $\triangle PRS$, if $QS=QT$ and $\angle QTS=40^\circ$, what is the value of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
There are 7 numbers from 9 to 79 that are exactly divisible by a certain divisor. What is the divisor?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given that $A(z_1)$, $B(z_2)$, and $C(z_3)$ are the vertices of an equilateral triangle $ABC$, determine the value of $\arg\left(\frac{z_2+z_3-2z_1}{z_3-z_2}\right)$. Provide your answer in radians.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{2}}
Given the differential equation \( \frac{dx}{dy} \cdot \frac{d^2y}{dx^2} = k \frac{dy}{dx} \), where \( k \) is a constant, find the value of the gradient \( \frac{dy}{dx} \) at the point where \( y = 2 \). It is known that the gradient \( \frac{dy}{dx} = 1 \) at the point \( (0,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^k}
Chris gave his 10 friends 12 pieces of candy each. Then, he gave another 7 friends 1.5 times the amount of candy he gave the first 10 friends. Lastly, he gave the remaining 5 friends an equal share of 20% more candy than the total amount given to the first 17 friends combined. How many pieces of candy did Chris give overall?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{541}
On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was \( 6 : {57} \) AM,
 and the sunset as \( 8 : {15} \) PM. The length of daylight and sunrise were correct, but the sunset was wrong. When did the sun real
 set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5:21 PM
}
Consider the initial value problem \(\frac{dy}{dx} = 3y^{\frac{2}{3}}\) with the initial condition \(y(0) = 0\). Determine the number of distinct solutions for this IVP, considering the possible definitions of \(y^{\frac{2}{3}}\) for \(y < 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the set of all real numbers $x$ such that $\sin x > \ln x$. Express your answer in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2.219)}
In 2019, a team, including professor Andrew Sutherland of MIT, found three cubes of integers which sum to 42: $42=\left(-8053873881207597 \_\right)^{3}+(80435758145817515)^{3}+(12602123297335631)^{3}$. One of the digits, labeled by an underscore, is missing. What is that digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the value of the expression \( \sum_{r = 0}^{15}(2-3r)^{15}C_r \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-671744}
Determine the disk of convergence for the power series \( \sum_{k=0}^\infty b_k\cdot z^k,\) where \(b_k = \begin{cases} 1 &, k = 2^n\\ 0 &, \bigl(\forall n\bigr)(k \neq 2^n).\end{cases}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{ z \in \mathbb{C} \mid |z| < 1 \}}
In a school, there are $1200$ students. Each student is part of exactly $k$ clubs. For any group of $23$ students, they are part of a common club. Furthermore, there is no club to which all students belong. Determine the smallest possible value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Find the angle between the asymptotes of the hyperbola given by the equation \(5x^2 - 2\sqrt{7}xy - y^2 - 2x + 1 = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
The sum \( A = \left\lfloor \frac{8}{9} \right\rfloor + \left\lfloor \frac{8^2}{9} \right\rfloor + \cdots + \left\lfloor \frac{8^{2014}}{9} \right\rfloor \) when divided by 63 leaves a remainder of \(\qquad\). (\(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to the real number \( x \)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Determine the proportion of nonplanar graphs with $v$ vertices that satisfy either $e > 3(v - 2)$ or $e > 2(v - 2)$ (if triangle-free) to the total number of nonplanar graphs with $v$ vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the largest positive integer  $n$  ( $n \ge 3$ ), so that there is a convex  $n$ -gon, the tangent of each interior angle is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n_{\text{max}} = 8 }
Stephen made 10 round trips up and down a 40,000-foot tall mountain. On each ascent, he reached 3/4 of the mountain's height, but on each descent, he only went down 2/3 of the height he had climbed due to an automated transportation system at certain heights. Throughout his trips, he stopped to rest at two different resting points on the way up, carrying an additional 50 lbs on every even trip. Furthermore, during two of his trips, he faced harsh weather conditions that forced him to reroute and cover an extra 10% of the ascent distance each time. Calculate the total distance he covered on his trips, taking into consideration the resting points, additional weight, and weather conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{506,000}
Find the greatest common divisor (GCD) of all the numbers in the set \(\{(n+2014)^{n+2014}+n^n \mid n \in \mathbb{N}, n > 2014^{2014}\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In approximate calculations, the number of significant figures can be used to measure the degree of approximation of an approximate number, simplifying the calculation. However, if not careful, it can also bring us trouble. For example, $2^{100} = 1267650600228229401496703205367$, in a certain calculation, $(2^{100} + 10^{10}) - 2^{100} = 0$, please point out the maximum number of significant figures taken in this approximate calculation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Determine the Lebesgue measure of the set \( A = \left \{x \in [0,1] \cap (\mathbb{R} - \mathbb{Q}) \mid \text{in the decimal expansion of } x, \text{ the first 1 (if it exists) is followed by 2} \right \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{10}}
Determine the normalizer of the subgroup \(\langle(1,2,...,p)\rangle\) in the symmetric group \(S_p\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{AGL(1, p)}
A point \( P \) is chosen randomly inside a triangle with sides 13, 20, and 21. Find the probability that the circle centered at \( P \) with radius 1 will intersect at least one of the sides of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{75}{196}}
Find the sum of the series: $$\sum_{k=0}^\infty\frac{(k+2)^3-3(k+2)^2+2(k+2)}{(k+4)!}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30 - 11e}
Evaluate the limit: $$\lim_{x\rightarrow 0^{+}}\frac{\int_{1}^{+\infty}\frac{e^{-xy}-1}{y^3}dy}{\ln(1+x)}$$ as x approaches 0 from the right.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Find a sufficient condition on \(\alpha\) that ensures the integral is finite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > -1}
What is the area of the quadrilateral formed by the lines $y=8$, $y=x+3$, $y=-x+3$, and $x=5$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Determine all natural numbers $n$ such that $2^n$ divides $3^n - 1$. Provide your answer as a set of numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{1, 2, 4\}}
What is the probability that Sisyphus must labor forever, given that each rock he places on Mount Olympus is either vaporized with probability 10%, rolled back into the valley with probability 50%, or split into two rocks that roll back into the valley with probability 40%? Assume he starts with one rock in the valley and none on Mount Olympus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
In how many ways can a commander issue a series of 8 orders to a row of 2018 soldiers, where each order instructs either all soldiers in even positions or all soldiers in odd positions to leave the formation, such that exactly 7 soldiers remain in formation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Out of 410 students of a school, some play football, 175 play cricket and 50 neither play football nor cricket. There are 140 students who play both football and cricket. How many students play football?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{325}
Compute the flux of the vector field \( \nabla \ln \sqrt{x^2 + y^2 + z^2} \) across the surface of an icosahedron centered at the origin. Let \( S \) be the surface of the icosahedron, and \( f(x,y,z) = \ln \sqrt{x^2+y^2+z^2} \). Calculate the flux \( \iint_S (\nabla f \cdot n) \, d\sigma \), where \( n \) is the outward unit normal vector on \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
The height of a right-angled triangle, dropped to the hypotenuse, divides this triangle into two triangles. The distance between the centers of the inscribed circles of these triangles is 1. Find the radius of the inscribed circle of the original triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{2}}
Calculate the flux of the vector field \( v = (\cos{2 \pi x}, y^3, z^3) \) through the unit sphere centered at the origin. The flux is given by the integral \( \iint_S(v \cdot n) \, ds \), where \( n \) is the outward unit normal vector to the surface \( S \), and \( ds \) is the area element on \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8\pi}{5}}
Tokarev S.I.

Find the smallest natural number that can be represented as the sum of 2002 natural addends with the same sum of digits and as the sum of 2003 natural addends with the same sum of digits.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10010}
Determine the number of group homomorphisms from $\Bbb Z/2\Bbb Z \times \Bbb Z/2\Bbb Z$ to $S_3$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
What is the least positive integer \( m \) such that \( m \cdot  2! \cdot  3! \cdot  4! \cdot  5!\cdots {16}! \) is a perfect square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
70
}