inclusionAI/Ring-lite-rl-data huggingface dataset:
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. If $a_5=3$ and $S_{10}=40$, then the minimum value of $nS_n$ is .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-32}
Let \( p(x) \) be a polynomial of degree less than 1992 such that \( p(0) \), \( p(1) \), and \( p(-1) \) are all non-zero. The 1992nd derivative of \( \frac{p(x)}{x^3 - x} \) is expressed as \( \frac{f(x)}{g(x)} \) for polynomials \( f(x) \) and \( g(x) \). Determine the smallest possible degree of \( f(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3984}
Find a matrix $T$ that transforms vectors in a 2D vector space with a weighted inner product defined by a positive-definite matrix $Q$ such that the distance between vectors in the transformed space matches the distance in the Euclidean space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Q^{1/2}}
In the parallelogram \(ABCD\), points \(K, L,\) and \(M\) are the midpoints of sides \(AB, BC,\) and \(CD\) respectively. Given that quadrilaterals \(KBLM\) and \(BCDK\) are inscribed, determine the ratio \(AC : AD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the last two digits of the sum of the eighth powers of 100 consecutive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Let $A$ be a $7 \times 7$ matrix satisfying $2A^2 - A^4 = I$. If $A$ has two distinct eigenvalues and each eigenvalue has geometric multiplicity 3, determine the number of non-zero entries in the Jordan Canonical Form of $A$. \( \text{(Express your answer as a single integer.)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The left and right foci of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$ are $F_1$ and $F_2$, respectively. The chord $AB$ passes through point $F_1$. If the circumference of the incircle of $\triangle ABF_2$ is $\pi$, and the coordinates of points $A$ and $B$ are $(x_1,y_1)$ and $(x_2,y_2)$, respectively, then find the value of $|y_1-y_2|=$ __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{3}}
Find the largest positive integer $N$ for which there exists a harmonious coloring for a regular $N$-gon, where each side and each diagonal is colored in one of 2018 different colors, and no two sides or diagonals are all the same color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
In an equilateral triangular prism \( S-ABC \), the lateral edges are equal in length to the edges of the base. If \( E \) and \( F \) are the midpoints of \( SC \) and \( AB \) respectively, find the angle between the skew lines \( EF \) and \( SA \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Determine the minimum value of $d - n$ such that the probability that the images of $n$ $2n$-dimensional linearly independent vectors over $\mathbb{F}_2$ remain linearly independent after projection onto a random $d$-dimensional subspace is at least $1 - \frac{1}{\mathrm{poly}(n)}$. Provide your answer in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log n}
Person A, Person B, and Person C compare the number of candies they have.

Person A says: "I have 13 candies, 3 less than Person B, and 1 more than Person C."
Person B says: "I do not have the least, Person C and I differ by 4 candies, Person A has 11 candies."
Person C says: "I have fewer candies than Person A, Person A has 10 candies, Person B has 2 more candies than Person A."

If each person has one false statement out of the three they made, how many candies does the person with the fewest candies have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
The Fibonacci numbers are defined by $F_{0}=0, F_{1}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 2$. There exist unique positive integers $n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}$ such that $\sum_{i_{1}=0}^{100} \sum_{i_{2}=0}^{100} \sum_{i_{3}=0}^{100} \sum_{i_{4}=0}^{100} \sum_{i_{5}=0}^{100} F_{i_{1}+i_{2}+i_{3}+i_{4}+i_{5}}=F_{n_{1}}-5 F_{n_{2}}+10 F_{n_{3}}-10 F_{n_{4}}+5 F_{n_{5}}-F_{n_{6}}$. Find $n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1545}
Calculate the expected number of coin tosses needed to obtain the sequence THH (T: tail, H: head) when flipping a fair coin indefinitely.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $x, y, z$ be positive real numbers such that $(1 + x + y + z)(1 + \frac{1}{x} + \frac{1}{y} + \frac{1}{z}) = 25$. Find the value of $x + y + z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
If a sequence $\{a_n\}$ satisfies $$\frac {1}{a_{n+1}}- \frac {1}{a_{n}}=d$$ (where $n\in\mathbb{N}^*$, $d$ is a constant), then the sequence $\{a_n\}$ is called a "harmonic sequence". Given that the sequence $\{\frac {1}{x_{n}}\}$ is a "harmonic sequence", and $x_1+x_2+\ldots+x_{20}=200$, the maximum value of $x_3x_{18}$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given the function \( f(x) = ax^2 + bx + c \), where \( a, b, c \) are real numbers and \( a > 100 \), determine the maximum number of integers \( x \) such that \( |f(x)| \leq 50 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the projective dimension of the ring $A = \mathbb{Z}_l[T,T^{-1}]$, where $l$ is a prime number and $\mathbb{Z}_l$ is the ring of $l$-adic integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the torque around the x-axis of the triangle with vertices (0, 0), (1, 4), and (1, 0), assuming the density is 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{3}}
In a life test, 100 items are tested simultaneously, each with a lifetime that is an independent exponential random variable with a mean of 200 hours. The test ends after 5 failures. Let T be the time at which the test concludes. Calculate the expected value of T, denoted as E[T].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10.206}
Let $S$ be the set of all ordered pairs $(x,y)$ of integer solutions to the equation $$6x^2+y^2+6x=3xy+6y+x^2y.$$  \$S$ contains a unique ordered pair $(a,b)$ with a maximal value of $b$. Compute $a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
34
}
Put the numbers $0-9$ in some order so that every $2$-digit substring creates a number which is either a multiple of $7$, or a power of $2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7028491635}
Calculate the integral of the pullback of the form \( \omega = \frac{xdy-ydx}{x^2+y^2} \) over the interval \([0, 2\pi]\), where \( g(t) = (e^t \sin(17t), e^{t^2} \cos(17t)) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34\pi}
Determine the cardinality of a sigma algebra generated by countably many sets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Two equilateral triangles are inscribed in a circle of radius \( R \) such that when they intersect, each side is divided into three equal segments. Find the area of the intersection of these triangles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3} R^2}{2}}
Let $ ABCD$ be a convex quadrilateral. We have that $ \angle BAC=3\angle CAD$, $ AB=CD$, $ \angle ACD=\angle CBD$. Find angle $ \angle ACD$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( 30^\circ \)

}
Find the simple closed curve that minimizes the line integral \( \int_C F \cdot d\mathbf{x} \), where \( F:\mathbb{R}^2 \to \mathbb{R}^2 \) is defined by \( F(x,y) =\begin{bmatrix} x^2 +y^2 +4y \\ x^3 +xy^2 +2xy\end{bmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3x^2 + y^2 = 4}
Integers $a$ and $b$ are randomly chosen without replacement from the set of integers with absolute value not exceeding $10$. What is the probability that the polynomial $x^3 + ax^2 + bx + 6$ has $3$ distinct integer roots?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{105}}
Let $x, y, z > 0$ such that $x + y + z = 1$. Find the minimum value of the expression \[ T = \sqrt[5]{\frac{x}{1 - x}} + \sqrt[5]{\frac{y}{1 - y}} + \sqrt[5]{\frac{z}{1 - z}}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the Galois group of the polynomial \((x^3 - 2)(x^2 + 3)\) over \(\mathbb{Q}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_3}
Evaluate the line integral \( \int_\gamma y\,dx + z\,dy + x\,dz \) using Stokes' theorem, where \( \gamma = \{(x,y,z) : x^2 + y^2 + z^2 = 9, x + y + z = 0\} \) with positive orientation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-9\sqrt{3}\pi}
Find the limit points of the set \( B = \left\{ \frac{m}{n} - \frac{n}{m} \mid m, n \in \mathbb{N} \right\} \subseteq \mathbb{R} \) where \( \mathbb{R} \) has the standard metric.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
已知 $x_{1}, x_{2}, x_{3}, x_{4}$ 是两两不等的实数， $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 是 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意一个排列，若记 $X=\max \left\{\min \left\{x_{i 1}, x_{i 2}\right\}, \min \left(x_{i 3}, x_{i 4}\right\}\right\}, Y=$ $\min \left\{\max \left\{x_{i 1}, x_{i 2}\right\}, \max \left\{x_{i 3}, x_{i 4}\right\}\right\}$ ，则 $X>Y$ 的概率是$\qquad$.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{4}
Determine the value of Arg(f(z)) after a complete counter-clockwise rotation of the point z about the origin, starting at z = 2 with the angle there being 0, where \( f(z) = \sqrt{z^2+2z-3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
The sides of a rectangular park are in the ratio 3 : 2 and its area is 3750 sq m. The cost of fencing it at a certain price per meter is $100. What is the cost of fencing per meter in ps?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Given non-negative real numbers \( a, b, c, x, y, z \) such that \( a+b+c = 1 \) and \( x+y+z = 1 \), determine the maximum value of the expression \( \left(a-x^{2}\right)\left(b-y^{2}\right)\left(c-z^{2}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{729}}
Given the function $y=\log_{a}(x+3) (a > 0,a\neq 1)$, its graph passes through point $A$. If point $A$ also lies on the graph of the function $f(x)=3^{x}+b$, then $f(\log_{3}2)=$ _____ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{17}{9}}
Determine the values of \(x\) for which the infinite series \(\sum_{n=1}^{\infty} -\sin(nx)\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x = k\pi}
Let $X_1, X_2, \ldots, X_n, \ldots$ be independent and identically distributed random variables with $P(X_n=0)=P(X_n=1)=P(X_n=2)=P(X_n=3)=\frac{1}{4}$. Define the sequence $\{Y_n\}$ by $Y_0=0$ and for all $n \in \mathbb{N}$, \[ Y_n=\begin{cases} 3 & \text{if } X_n=3, \\ \min\{Y_{n-1},X_n\} & \text{if } X_n<3. \end{cases} \] Compute $\lim_{n \to +\infty} E[Y_n Y_{n-1}]$. \(\text{Express your answer as a single number.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{19}{12}}
Find the coefficient of \(t^{20}\) in the expansion of \((1+t+t^2)^{15}\) using the concept of derivatives.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{531531}
Solve for $c$ in the equation $4 - \cos(-1) = c + 5\ln(c-1)$. You may use the transformation $e^{4 - \cos(1) - c} = (c-1)^5$ to assist in finding the solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.269}
$N$ is a $50-$digit number (in the decimal scale). All digits except the $26^{\text{th}}$ digit (from the left) are $1$. If $N$ is divisible by $13$, what is the $26^{\text{th}}$ digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3
}
平面上有 101 条不同的射线，任两条不共线、没有相同的端点，任三条不共点. 求无序射线对 (a,b) 个数的最大值，使得 a, b 相交，且若交点为 C，则 C 到 a, b 端点的距离相等.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2550}
In parallelogram \(ABCD\), point \(M\) is located on side \(BC\). Given that \(\angle A = 45^{\circ}\), \(\angle AMD = 90^{\circ}\), and the ratio \(BM : MC = 2 : 3\), determine the ratio of the lengths of the adjacent sides of the parallelogram.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\sqrt{2}}{5}}
Determine the cardinality of the set $S$ of all analytic functions $h: D \rightarrow D$, where $D = \{z \in \mathbb{C}: |z| < 1\}$, such that $h(z)^2 = \overline{h(z)}$ for all $z \in D$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the dimension of the vector space \(\mathbb{C}\) over the field \(\mathbb{Q}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Determine the number of field structures that can be defined on the set \(\mathbb{R} \times \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the limit \( \lim_{c\to\infty} \frac{E\left[ X e^{-\frac{(X-c)^2}{2}} \right]}{E\left[e^{-\frac{(X-c)^2}{2}}\right]} \) where \(X\) is a bounded random variable with \(|X| \le a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a}
Find the value of $A + B + C$ such that \[ S = \sum_{n=1}^{\infty} \frac{1}{n^2 \binom{2n}{n}} = \frac{A}{B} \zeta(C). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find a nonzero solution \((x, y, z) \in \mathbb{F}_7^3\) to the equation:

\[2x^2 + y^2 + 2yz + 2z^2 = 0\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 1, 1)}
Peter chooses a positive integer \( a \) such that the sum of its digits is 2012. Paul needs to determine this number. On each move, Paul selects a positive integer \( x \), and Peter tells him the sum of the digits of \(|x - a|\). What is the minimal number of moves required for Paul to determine Peter's number with certainty?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the least number of sets in a partition of all \(n!\) permutations of \(1, 2, 3, \ldots, n\), such that any two permutations in the same set can be transformed into each other by a sequence of the following moves:
- If \(a\) is the median of \(a, b, c\), replace \(a, b, c\) with \(b, c, a\).
- If \(c\) is the median of \(a, b, c\), replace \(a, b, c\) with \(c, a, b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}}
Evaluate the integral \( \oint\limits_{|z|=3}\frac{1}{(z^5-1)(z-2)} dz \) using the method of residues, including the residue at infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( \{X_n\} \) be a sequence of i.i.d. exponentially distributed random variables with mean 1. Define \( M_n = \max(X_1, X_2, \ldots, X_n) \). Find a sequence \( \{a_n\} \) such that \( P[M_n - a_n \leq x] \) converges to a non-degenerate distribution function \( G(x) \) as \( n \to \infty \), for every \( x \in \mathbb{R} \) where \( G(x) \) is continuous.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln n}
For $a > 0$, consider the function \( f(x) = \frac{2}{a}\sqrt{1-x}\left(a^2+\frac{1}{1-2x}\right) \) defined for \( 0 \leq x < \frac{1}{2} \). Determine the value of \( a \) for which \( f(x) \) attains its minimum value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( f(n) \) be the sum of the first \( n \) good numbers, where a natural number \( k > 1 \) is called good if there exist natural numbers \( a_1 < a_2 < \cdots < a_k \) such that \( \frac{1}{\sqrt{a_1}} + \frac{1}{\sqrt{a_2}} + \cdots + \frac{1}{\sqrt{a_k}} = 1 \). Find the sum of all values of \( n \) for which \( \frac{f(n+5)}{f(n)} \) is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Find the angle between the tangent line to the logarithmic spiral \(\alpha(t) = e^{-t}(\cos(t),\sin(t))\) at the point where \(t = \phi + 2\pi k\) (for some integer \(k\)) and the ray from the origin at angle \(\phi\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Evaluate the expression $E[E[X|Y]|X]$. What is the result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X}
Calculate the integral part of the number \( T = \frac{2}{1} \cdot \frac{4}{3} \cdot \frac{6}{5} \cdot \frac{8}{7} \cdots \frac{2016}{2015} \). Given that \( T = 2017\int_{0}^{\frac{\pi}{2}} \sin^{2017}(x)dx \), and the identity \( \int_0^{\frac{\pi}{2}} \sin^{2n+1}(x) dx = \frac{2}{3} \cdot \frac{4}{5} \cdot \frac{6}{7} \cdots \frac{2n}{2n+1} \), find the integer part of \( 2017\int_{0}^{\frac{\pi}{2}} \sin^{2017}(x)dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Determine the range of values for \( b \) such that the function \( g(x) = \int_{0}^{x} \left|\log_3\left(\log_4\left(\log_2\left(\sin t + b\right)\right)\right)\right| dt \) is decreasing for all \( x \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Given the context-free grammar with variables $V_n = \{expr, term, factor, number\}$, terminals $V_t = \{(,), -, *, 0...9\}$, productions $P$ as follows:
\[\begin{align} expr &\to expr - expr \mid term \\
term &\to term * factor \mid factor \\
factor &\to number \mid (expr) \\
number &\to 0|1|2|3|4|5|6|7|8|9 \end{align}\]
and start symbol $S = expr$, determine the number of distinct derived trees for the expression $1-2-(3-4)*5*6$. \(\text{Provide your answer as a single number.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a function \( F: \mathbb{R}^2 \rightarrow \mathbb{R} \) that is \( C^2 \) with \( F_x(1,1)=F_{yy}(1,1)=1 \), \( F_y(1,1)=F_{xx}(1,1)=0 \), and a function \( g: \mathbb{R}^2 \rightarrow \mathbb{R} \) defined by \( g(r,\theta)=F(r\cos\theta,r\sin\theta) \), find the value of \( g_{r\theta}(\sqrt{2},\pi/4) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the smallest positive integer \( k \) such that \( 2^{w(n)} \leq k \sqrt[4]{n} \) holds for all positive integers \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the gradient of the expression \( \mathrm{trace}(A'(X\circ X-1)) \) with respect to the matrix \( X \), where \( \circ \) denotes the Hadamard product, \( A \) is a matrix of positive real values, and \( 1 \) is a matrix of all ones.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2A \circ X}
Given that \( \Im[f'(z)]=6x(2y-1) \) and \( f(0)=3-2i \), \( f(1)=6-5i \), find the value of \( f(1+i) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 + 3i}
For how many consecutive integers \(x\) can the quadratic polynomial \(ax^2 + bx + c\) be a perfect square, given that \(a, b, c \in \mathbb{Z}\), \(a \neq 0\), and \(b^2 - 4ac \neq 0\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
On a $2\times2$ chessboard, a king starts from the upper left corner and moves with equal probability to any adjacent square. Calculate the approximate probability that the king will be in the bottom right corner after exactly 1000 moves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Determine the multiplicity of the root \(x = a\) for the polynomial \(Q(x) = \frac{1}{2}(x-a)(p'(x)+p'(a)) - p(x) + p(a)\), where \(p(x)\) is a polynomial of degree \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the angle between the line \((6, 8, 5) + t (3, 4, 5)\), where \(t \in \mathbb{R}\), and the sphere \(x^2 + y^2 + z^2 = 25\) at their points of intersection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Determine the minimum number of vertices $m$ such that the complete graph $K_n$ is a vertex minor of the path graph $P_m$. What is the value of $m$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 1}
Calculate the flux of the vector field \( \mathbf{F}(x,y,z) = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \) through the cylindrical surface \( S = \{ (x,y,z) \mid x^2 + y^2 \le 9, 0 \le z \le 2 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54\pi}
Let the positive numbers \( x \) and \( y \) satisfy \( x^{3} + y^{3} = x - y \). Find the maximum value of the real number \( \lambda \) such that \( x^{2} + \lambda y^{2} \leq 1 \) always holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + 2\sqrt{2}}
Find the function $f(x)$ that satisfies the integro-differential equation: $$\int_0^x f(u)du - f'(x) = x.$$ Note that the initial condition $f(0)$ is not provided.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{C(e^x + e^{-x}) + 1}
How many 7-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7 such that among the four odd digits, any three are not all adjacent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2736}
When two circular rings of diameter $12$ on the Olympic Games Logo intersect, they meet at two points, creating a $60^o$ arc on each circle. If four such intersections exist on the logo, and no region is in $3$ circles, the area of the regions of the logo that exist in exactly two circles is $a\pi - b\sqrt{c}$ where $a$, $b$, $c$ are positive integers and $\sqrt{c}$ is fully simplified find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{123}
A coin with diameter \(d\) and thickness \(t\) is flipped such that it rotates uniformly around a horizontal axis parallel to one of its faces. Determine the ratio \(\frac{d}{t}\) such that the probability of the coin landing on its edge is \(\frac{1}{3}\), assuming that any orientation of the coin is equally likely.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
If the decimal representation of $n$ is 99…9 (with 20 digits of 9), then the decimal representation of $n^3$ contains     number of digit 9.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
In triangle \(ABC\), angle \(C\) is a right angle, and \(AC: AB = 4: 5\). A circle with its center on leg \(AC\) is tangent to the hypotenuse \(AB\) and intersects leg \(BC\) at point \(P\), such that \(BP: PC = 2: 3\). Determine the ratio of the radius of the circle to leg \(BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{20}}
Given that \( \sin A + \sin^2 A = 1 \) and \( a \cos^{12} A + b \cos^8 A + c \cos^6 A - 1 = 0 \), find the value of \( b + \frac{c}{a} + b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}