inclusionAI/Ring-lite-rl-data huggingface dataset:
A unit cube has a base and a top face, with side edges connecting them. The top face is rotated by an acute angle $\alpha$ around the line connecting the centers of the two faces. If the length of the side edges remains unchanged and the vertices' connections are intact, calculate how much closer the top face gets to the base face.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \sqrt{\cos \alpha}}
对 $n$ 个实数 $a_{1}, a_{2}, \cdots, a_{n}$, 令
$$
b_{k}=\frac{a_{k}+a_{k+1}}{2}, c_{k}=\frac{a_{k-1}+a_{k}+a_{k+1}}{3},
$$
其中， $k=1,2, \cdots, n, a_{0}=a_{n} ; a_{n+1}=a_{1}$. 求最大的正实数 $\lambda$ ，使得对于任意正整数 $n \geqslant 3$ ，以及任意实数 $a_{1}, a_{2}, \cdots, a_{n}$ ，均有
$$
\sum_{k=1}^{n}\left(a_{k}-b_{k}\right)^{2} \geqslant \lambda \sum_{k=1}^{n}\left(a_{k}-c_{k}\right)^{2}
$$原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
An integer $n$ is said to be [i]good[/i] if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the values of the parameter \( \beta \) for which the integral \( \int_0^{+\infty} \frac{\sin x}{x^\beta(1+x^3)}\, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-3, 2)}
Let $S_{n}$ be the sum of the first $n$ terms of the geometric sequence $\{a_{n}\}$, and $S_{4}=5S_{2}$. Determine all possible values of the common ratio $q$ of this sequence. List all possible values of $q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2, -1, 2}
Given a function $f$ for which  
$f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)$  
holds for all real $x,$ what is the largest number of different values that can appear in the list $f(0),f(1),f(2),\cdots,f(999)?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{177}
Several figures can be made by attaching two equilateral triangles to the regular pentagon $ABCDE$ in two of the five positions shown. Find the number of non-congruent figures that can be constructed in this way.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Four congruent semicircles are drawn on the surface of a sphere with radius $2$, as shown, creating a close curve that divides the surface into two congruent regions. The length of the curve is $\pi\sqrt{n}$. What is the value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
一个等腰直角三角形的三个顶点分别在正三棱柱 的三条侧棱上, 已知正三棱柱的底面边长为 2 , 则该三角形的斜边长为 ($\quad$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2\sqrt{3}$}
A jailer is granting partial amnesty by the following procedure: First, he opens all the cells. Next, he locks every second cell (in this prison, all the cells are in a single row). In the third step, he turns the key in every third cell (if a cell is open, it will be locked, and if it is locked, it will be opened). Continuing in this manner, on the $n$-th step, he turns the key in every $n$-th cell. After finishing this process, he releases all prisoners who are in open cells. Indicate the numbers of the cells that contain these lucky prisoners.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 4, 9, 16, 25}
A square steel plate has had a rectangular piece of width 3 cm and another rectangular piece of width 5 cm cut off (as shown in the figure). The total area removed is 81 square centimeters. What was the area of the original square? $\qquad$ square centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
In a quadratic trinomial, the second coefficient and the constant term were switched, and the resulting trinomial was added to the original trinomial. As a result, a third quadratic trinomial was obtained, which has a single root. What can this root be? If there are multiple correct answers, list them separated by semicolons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0; -2}
A stained glass window consists of nine squares of glass arranged in a 3x3 grid. Of these nine squares, \( k \) are red, and the rest are blue. A set of windows is created such that any possible window can be formed in exactly one way by rotating and/or flipping one of the windows in the set. If the total number of red squares in the set exceeds 100, find the value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A circular spinner used in a game has a radius of 15 cm. The probability of winning on one spin of this spinner is $\frac{1}{3}$ for the WIN sector and $\frac{1}{4}$ for the BONUS sector. What is the area, in square centimeters, of both the WIN sector and the BONUS sector? Express your answers in terms of $\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56.25\pi}
What is the maximal dimension of a linear subspace  $ V$  of the vector space of real  $ n \times n$  matrices such that for all  $ A$  in  $ B$  in  $ V$ , we have  $ \text{trace}\left(AB\right) \equal{} 0$  ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n(n-1)}{2}}
A frame for three square photographs has a uniform width everywhere (see the figure). The perimeter of one opening is 60 cm, and the perimeter of the entire frame is 180 cm. What is the width of the frame?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 \text{ cm}}
A hemisphere is placed on a sphere of radius \(100 \text{ cm}\). The second hemisphere is placed on the first one, and the third hemisphere is placed on the second one (as shown below). Find the maximum height of the tower (in cm).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Let \( a, b, c \) be the side lengths of a right triangle, with \( a \leqslant b < c \). Determine the maximum constant \( k \) such that \( a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) \geqslant k a b c \) holds for all right triangles, and identify when equality occurs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + 3\sqrt{2}}
Determine the group to which the smallest group containing the functions \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{x-1}{x} \), under composition, is isomorphic. The functions are defined on the set \( A = \mathbb{R} \setminus \{0,1\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_3}
A $20 \times 20 \times 20$ block is divided into 8000 non-overlapping unit cubes, each assigned a number. In each column of 20 cubes parallel to any edge of the block, the sum of their numbers is 1. One unit cube is assigned the number 10. Three $1 \times 20 \times 20$ slices parallel to the faces of the block contain this unit cube. Calculate the sum of all numbers of the cubes outside these slices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{333}
Consider the sequence \( B_n \) where \( n \in \mathbf{N}^* \). Let the point with coordinates \((1,1)\) be \( M \). \( P_n (x_n, y_n) \) is the circumcenter of triangle \(\triangle A_n B_n M\). Calculate the sum \( x_1 + x_2 + \cdots + x_{100} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2575}
In a school with 1000 students, numbered from 1 to 1000, a group of 500 students is called a 'good group' if there exists a student in the group whose number divides the number of another student in the group. Otherwise, the group is called a 'bad group'. A 'good student' is defined as a student who does not belong to any 'bad group'. Determine the largest number of a 'good student'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
The density function of a random variable \( x \) is given by \( f(x) = ke^{-2x^{2}+10x} \). Determine the upper 5% point of the distribution of the means for a random sample of size 25 drawn from this population.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.66}
Let \( G = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mid a, b, c, d \in \mathbb{Z}, ad - bc = 1 \right\} \) be the group of 2x2 integer matrices with determinant 1. Let \( H = \left\{ \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix} \mid n \in \mathbb{Z} \right\} \) be a subgroup of \( G \). Determine the index \([G:H]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
How do you find the volume of the solid bounded by the surfaces $z=x^2 + y^2$ and $x^2+y^2+z^2=20$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{8}{3} \left(10 \sqrt{5}-19\right) \pi\)}
Given the equation:
\[
[x+0.1]+[x+0.2]+[x+0.3]+[x+0.4]+[x+0.5]+[x+0.6]+[x+0.7]+[x+0.8]+[x+0.9]=104
\]
where \([x]\) denotes the greatest integer less than or equal to \(x\), find the smallest value of \(x\) that satisfies this equation. Express your answer as a decimal value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11.5}
Purple College keeps a careful count of its students as they progress each year from the freshman class to the sophomore class to the junior class and, finally, to the senior class.  Each year at the college one third of the freshman class drops out of school, $40$ students in the sophomore class drop out of school, and one tenth of the junior class drops out of school.  Given that the college only admits new freshman students, and that it wants to begin each school year with $3400$ students enrolled, how many students does it need to admit into the freshman class each year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5727}
Let  $a$  and  $b$  be relatively prime positive integers such that  $a/b$  is the sum of the real solutions to the equation \[\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}.\] Find  $a+b$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A person has 4 colors of light bulbs (enough of each color), and needs to install one light bulb at each of the 6 points $A, B, C, A_{1}, B_{1}, C_{1}$ as shown in the diagram. It is required that the light bulbs at the two ends of the same segment are of different colors. What is the number of ways to install the light bulbs, ensuring that each color is used at least once? (Answer in digits).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216}
For how many ordered triplets $(a, b, c)$ of positive integers less than 10 is the product $a \times b \times c$ divisible by 20?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{102}
Consider a sphere centered at the origin $(0, 0, 0)$ in the $xyz$ coordinate system. Initially, the lower hemisphere (with negative $y$-coordinate) is painted. The sphere is then randomly rotated about any axis passing through the center. Determine the expected number of rotations required to paint the entire sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In the diagram, congruent rectangles $ABCD$ and $DEFG$ share a common vertex $D$. The sides $BC$ and $EF$ intersect at point $H$. Given that $DA = DE = 8$, $AB = EF = 12$, and $BH = 7$, find the area of the region $ABHED$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Find the volume of the solid obtained by rotating the curve \( y = \cosh x \) around the \( y \)-axis from \( y = 0 \) to \( y = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi\left(1 - \frac{1}{e}\right)}
The points \( A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6} \) divide a circle of radius 1 into six equal parts. A ray \( l_{1} \) is drawn from \( A_{1} \) in the direction of \( A_{2} \), a ray \( l_{2} \) is drawn from \( A_{2} \) in the direction of \( A_{3} \), and so on, with a ray \( l_{6} \) drawn from \( A_{6} \) in the direction of \( A_{1} \). From a point \( B_{1} \) on the ray \( l_{1} \), a perpendicular is dropped onto the ray \( l_{6} \), from the foot of this perpendicular another perpendicular is dropped onto \( l_{5} \), and so on. The foot of the sixth perpendicular coincides with \( B_{1} \). Find the length of the segment \( B_{1} A_{1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find an upper bound for the operator norm \( \lvert e^A \rvert \) of a square matrix \( A \) in terms of \( \lvert A \rvert \) and the dimension \( n \) of the matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\lvert A \rvert}}
Find the variance of the stochastic process \(X_{t} = \int_{0}^{t} W^{+}_{s}dW_{s}\), where \(W_{t}\) is a Wiener process and \(a^{+} = \max(a, 0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{t^2}{4}}
Let $\overline{ABCD}$ be a $4$-digit number. What is the smallest possible positive value of $\overline{ABCD}- \overline{DCBA}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
90
}
Under the new AMC 10, 12 scoring method, $6$ points are given for each correct answer, $2.5$ points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between $0$ and $150$ can be obtained in only one way, for example, the only way to obtain a score of $146.5$ is to have 24 correct answers and one unanswered question. Some scores can be obtained in exactly two ways; for example, a score of $104.5$ can be obtained with $17$ correct answers, $1$ unanswered question, and $7$ incorrect, and also with $12$ correct answers and $13$ unanswered questions. There are three scores that can be obtained in exactly three ways. What is their sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
188.5
}
A princess is choosing a groom from one hundred suitors, each better than the last, but they propose in a random order. A groom is considered outstanding if the princess likes him more than all who have proposed before him. The first groom is also outstanding by default. An outstanding groom and all those who propose after him, up to the next outstanding groom, form a sequence. Calculate the expected number of grooms in the first such sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H_{100}}
Find the radius of convergence of the series:
\[ \sum \frac{(K/e)^{3K}}{3K!}(7x)^K. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{7}}
In a microcircuit with 2000 contacts, initially, any two contacts are connected by a separate wire. Hooligans Vasya and Petya take turns cutting the wires, with Vasya (who starts) cutting one wire per turn, and Petya cutting either two or three wires per turn. The hooligan who cuts the last wire connected to any contact loses. Who wins with optimal play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Petya}}
Let $X_1$ and $X_2$ be two independent uniformly distributed random variables on $[0,1]$. Calculate the expectation of $\min(X_1,X_2)$ given that $\max(X_1,X_2) > 0.5$. Provide your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{18}}
A round table has radius \(4\). Six rectangular place mats are placed on the table. Each place mat has width \(1\) and length \(x\) as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length \(x\). Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. Find the value of \(x\). The original answer is in the form \(\frac{k\sqrt{n}-\sqrt{p}}{m}\). Please give the value of \(k+m+n+p\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Provide an example of a continuous function whose integral from 0 to infinity converges, but whose limit as x approaches infinity does not exist.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin(x^2)}
We colour all the sides and diagonals of a regular polygon $P$ with $43$ vertices either
red or blue in such a way that every vertex is an endpoint of $20$ red segments and $22$ blue segments.
A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour.
Suppose that there are $2022$ blue monochromatic triangles. How many red monochromatic triangles
are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{859}
Find the number of elements in the conjugacy class of the permutation $\partial = (1\ 10)(2\ 6\ 3\ 7\ 6\ 8\ 12)(4)(9\ 11)$ in the symmetric group $S_{12}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4989600}
Let $m,n \ge 2$ be such that $m-1$ is a divisor of $n-1$. Let $T$ be a tree with $m$ vertices. Calculate the Ramsey number $R(T,K_{1,n})$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m + n - 1}
Find the limit of the sequence \( \lim_{n\to\infty} \frac{(2n-1)!}{(2n)^n} \) as \( n \) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Calculate the line integral \( \int_C \frac{z\arctan(z)}{\sqrt{1+z^2}}\,dz + (y-z^3)\,dx - (2x+z^3)\,dy \), where the contour \( C \) is defined by the equations \( \sqrt{1-x^2-y^2}=z \) and \( 4x^2+9y^2 = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{2}}
Compute the limit:

\[ \lim_{n \to \infty} \frac{\sum\limits_{i = 0}^{n + 1} \binom{n + 1}{i}^3}{\sum\limits_{i = 0}^n \binom{n}{i}^3} \]

where \( \sum\limits_{i = 0}^n \binom{n}{i}^3 = \text{hypergeom}\left(\left[-n, -n, -n\right], [1, 1], -1\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the pattern in the sequence: $1, 4, 3, 9, 5, 16, 7, \quad \qquad , 36, 11, \quad \qquad$ What are the two numbers that should be filled in the blanks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25, 13}
Given a function \( f: \mathbb{R}^2 \to \mathbb{R} \) such that \( \int_{\mathbb{R}} \left[ \int_{\mathbb{R}} f(x,y) \, dy \right] \, dx = 1 \) and \( \int_{\mathbb{R}} \left[ \int_{\mathbb{R}} f(x,y) \, dx \right] \, dy = -1 \), determine the possible range of values for \( \iint_{\mathbb{R}^2} |f| \, dm \), where \( m \) refers to the Lebesgue measure on \( \mathbb{R}^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let $X$ be a Hilbert space and $\varphi \in X' \setminus \{0\}$. Define the set \( C = \{ x \in X : \varphi(x) = 1 \} \). Find the orthogonal complement \( C^{\perp} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0\}}
Evaluate the limit \( \lim_{N \rightarrow \infty} \frac{g(N)}{N^3} \), where \( g(N) \) is the number of quadruples \((x, y, z, w)\) of positive integers satisfying \( x + y + z + w = N \) and \( x \le y \le z \le w \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{144}}
In the figure, the line $t$ is parallel to the segment $EF$ and tangent to the circle. If $AE = 12$, $AF = 10$, and $FC = 14$, determine the length of $EB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given a right triangle \(ABC\) with the right angle at vertex \(B\). A median \(BD\) is drawn from \(B\). Let \(K\) be the point of tangency of side \(AD\) of triangle \(ABD\) with the incircle of this triangle. Find the acute angles of triangle \(ABC\) if \(K\) bisects \(AD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ, 60^\circ}
For which integers \( n \) from 1 to 6 does the equation

\[ a^n + b^n = c^n + n \]

have a solution in integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 2, 3}
Suppose there are $m$ types of currency with denominations $p_{1}$ units, $p_{2}$ units, $\cdots, p_{m}$ units. What is the number of ways to change $r$ units of currency into these various denominations, denoted as $a_{r}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_{r}}
Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of  $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
A side of a triangle is divided into five equal parts; lines are drawn through the division points parallel to the base.

Find the lengths of the segments of these lines that are enclosed between the other two sides of the triangle, given that the base is 20 units long.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4, 8, 12, 16}
In the convex quadrilateral \( ABCD \), the lines extending the opposite sides intersect at points \( K \) and \( L \), and the diagonals intersect at point \( M \). The diagonal \( AC \) intersects the line \( KL \) at point \( N \). Given that \( AM = 3 \) cm and \( MC = 2 \) cm, find the distance \( CN \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a sequence of positive integers, each sum of 17 consecutive terms is even, and each sum of 18 consecutive terms is odd. Determine the maximum number of terms such a sequence can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Determine the radius of convergence for the power series \(\sum^\infty_{k=1} \frac{(2k-1)^{2k-1}}{2^{2k}(2k)!}x^k\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{e^2}}
Amandine and Brennon play a turn-based game, with Amadine starting.
On their turn, a player must select a positive integer which cannot be represented as a sum of multiples of any of the previously selected numbers.
For example, if $3, 5$ have been selected so far, only $1, 2, 4, 7$ are available to be picked;
if only $3$ has been selected so far, all numbers not divisible by three are eligible.
A player loses immediately if they select the integer $1$.
Call a number $n$ feminist if $\gcd(n, 6) = 1$ and if Amandine wins if she starts with $n$. Compute the sum of the feminist numbers less than $40$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{192}
Calculate the integral \( \int_\gamma \frac{1}{\sin(z)} \, dz \) where \( \gamma(t) = 4e^{2 i \pi t} \) for \( t \in [0,1] \). Use the residue theorem to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi i}
Fifteen people are each randomly assigned two numbers from the set \( \{1, 2, \ldots, 25\} \). A person is considered happy if the sum of their two numbers is 25. Calculate the expected number of happy people.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{5}}