inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the number of distinct cyclic subgroups of order 10 in the group \( \mathbb{Z}_{30} \oplus \mathbb{Z}_{120} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given $n$ complete graphs, each with $m$ nodes, determine the minimum number of edges required to connect these graphs such that there exists a path of length 2 between any two nodes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m(n - 1)}
Find the centroid of the part of a spherical shell centered at the origin with radius 30 that lies in the first octant. The shell is defined by the equation \( z = \sqrt{30^2 - x^2 - y^2} \). Assume symmetry in the first octant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(15, 15, 15)}
For a real 2x2 matrix $B$, determine the conditions on the eigenvalues of $B$ that ensure the convergence of the geometric series used in the expression for $A^{-1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|\lambda| < 1}
In a graph \( G \) with \( m \) vertices of a certain degree, if \( G \) is isomorphic to its complement, determine the number of vertices of the complementary degree in \( G \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m}
Determine the values of \( \alpha \in \mathbb{R} \) for which the integral \( \int_{\mathbb{R}^2} \frac{1}{(1+x^2+xy+y^2)^\alpha} \, dx \, dy \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > 1}
Determine the conditions on the fixed point (a, b) such that the mapping from the group of all polynomials with real coefficients under addition to itself, which assigns to each polynomial its antiderivative passing through (a, b), is a homomorphism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(a, 0)}
Find the smallest positive value of \( x \) such that \( \cos(2 \pi x) \), \( \cos(4 \pi x) \), \( \cos(8 \pi x) \), \( \cos(16 \pi x) \), and \( \cos(32 \pi x) \) are all nonpositive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{21}{64}}
Three lines from the family of lines given by \( C: x \cos t + (y + 1) \sin t = 2 \) enclose an equilateral triangle \( D \). What is the area of the region \( D \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\sqrt{3}}
Determine the extension degree of the constructible subfield $\Omega^{\text{constr}}$ of the splitting field $\Omega^f_{\mathbb{Q}}$ for the polynomial $f=x^9-2x^7+3x^2-6$, given that it is a subfield of the largest real subfield of $\Omega^f_{\mathbb{Q}}$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that with a probability of \( \frac{1}{3} \), exactly one of eight identical-looking envelopes contains a bill, and with a probability of \( \frac{2}{3} \), all eight envelopes are empty, you open one envelope. Let \( A \) be the event that there is a bill in this envelope, and \( B \) be the event that the other seven envelopes have already been opened and turned out to be empty. Calculate \( P(A \mid B) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{17}}
Find the smallest possible value of the expression

$$
\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor,
$$

where \(a, b, c\) and \(d\) vary over the set of positive integers.

(Here \(\lfloor x\rfloor\) denotes the greatest integer less than or equal to \(x\).)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine the number of solutions in natural numbers \((x, y)\) for the system of equations: \[\begin{cases} \text{GCD}(x, y) = 20! \\ \text{LCM}(x, y) = 30! \end{cases}\] where \(n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
How many six-digit natural numbers exist in which the digits are arranged in ascending order?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
An ant starts at the point \((1,0)\). Each minute, it walks from its current position to one of the four adjacent lattice points until it reaches a point \((x, y)\) with \(|x|+|y| \geq 2\). What is the probability that the ant ends at the point \((1,1)\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{24}}
Let \(B = (B_t)_{t \ge 0}\) be a standard Brownian motion starting at zero. Consider the non-negative stochastic process \(X_t\) that solves the stochastic differential equation:

$$dX_t = \frac{1}{X_t} dt + dB_t$$

Calculate \(E[\sigma]\), where \(\sigma = \inf \{ t \ge 0 : X_t = 1 \}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Evaluate the infinite series: $$\sum_{k=1}^\infty\frac{k(k+1)(k+2)}{3!}\left(\frac{1}{3}\right)^k$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{16}}
Given a series of $n$ geomagnetic readings modeled as independent observations from a random variable uniformly distributed on the interval $(0, \theta)$, where $\theta > 1$, and $m$ out of $n$ readings are less than $1$, find the maximum likelihood estimate (MLE) of $\theta$. Assume the remaining $n-m$ readings are greater than $1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n}{m}}
Use Green's Theorem to compute the area enclosed by the parametric curve given by \( x = 2\cos(t) \) and \( y = \sin(2t) \) for \( 0 \le t \le 2\pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16}{3}}
Evaluate the limit: $$\lim_{n \to \infty} \cos^2(\pi \sqrt[\leftroot{-2} \uproot{1} 3]{n^3+n^2+2n})$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Determine the number of solutions \((x, y, z)\) to the system of equations \(\begin{cases}x+2y+4z=9\\ 4yz+2xz+xy=13\\ xyz=3\end{cases}\) such that at least two of \(x, y, z\) are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A bag contains 100 ropes. In each step, two rope ends are randomly selected, tied together, and returned to the bag. This process continues until no free ends remain. What is the expected number of loops formed at the end of this process?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sum_{k=1}^{100} \frac{1}{2k - 1}}
Using each of the nine digits exactly once, form prime numbers (numbers that are divisible only by 1 and themselves) such that their sum is minimized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{207}
In a regular 2019-gon, numbers are placed at the vertices such that the sum of the numbers in any nine consecutive vertices is 300. It is known that the number at the 19th vertex is 19, and the number at the 20th vertex is 20. What number is at the 2019th vertex?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
Among the standard products of a certain factory, on average 15% are of the second grade. What is the probability that the percentage of second-grade products among 1000 standard products of this factory differs from 15% by less than 2% in absolute value?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9232}
Evaluate the integral \( \oint_{|z|=2} \frac{\sin z}{z^3} \, dz \) using contour integration methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( x \neq y \), and suppose the two sequences \( x, a_{1}, a_{2}, a_{3}, y \) and \( b_{1}, x, b_{2}, b_{3}, y, b_{1} \) are both arithmetic sequences. Determine the value of \( \frac{b_{4}-b_{3}}{a_{2}-a_{1}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8/3}
In quadrilateral \(ABCD\), the side lengths \(AB\) and \(BC\) are equal to 1, \(\angle B = 100^\circ\), and \(\angle D = 130^\circ\). Find \(BD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the complex integral \( \int_{\gamma} {\frac{z}{\overline z}}dz \) where \( \gamma \) is the edge of the region \( \{1 < |z| < 2 \text{ and } \Im (z) > 0\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Approximate the integral \( \int_0^1 e^{x^2} \, dx \) using the Maclaurin series expansion of \( e^{x^2} \). Ensure that the approximation error is less than 0.0005 from the actual value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.4625}
A regular triangular prism \( ABC A_1 B_1 C_1 \) is inscribed in a sphere. The base of the prism is \(ABC \) and the lateral edges are \(AA_1, BB_1, CC_1\). Segment \(CD\) is the diameter of this sphere, and points \(K\) and \(L\) are the midpoints of edge \(AA_1\) and \(AB\) respectively. Find the volume of the prism if \(DL = \sqrt{2}\) and \(DK = \sqrt{3}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given $x \gt 0$, $y \gt 0$, and $\ln \left(xy\right)^{y}=e^{x}$, find the minimum value of $x^{2}y-\ln x-x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the value of \((\boldsymbol{x'}\cdot\nabla)^2\boldsymbol{x}\) where \(\boldsymbol{x'},\boldsymbol{x}\in \mathbb{R}^3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\boldsymbol{0}}
What is the shortest distance from the origin to the circle defined by \( x^2 - 30x + y^2 - 8y + 325 = 0 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{241} - 2\sqrt{21}}
There is a cone circumscribed around a sphere with a radius of 1, and the height of the cone is twice the diameter of the sphere. Find the ratio of the total surface area of the cone to the surface area of the sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( S \) be the smallest uncountable ordinal, and assume both the Continuum Hypothesis and the Axiom of Choice. Determine the cardinality of the set of all functions from \( S \) to the continuum that map all but a countable number of elements of \( S \) to zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_1}
Let $ABCD$ be a convex quadrilateral with $AC = 20$, $BC = 12$, and $BD = 17$. If $\angle CAB = 80^\circ$ and $\angle DBA = 70^\circ$, then find the area of $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
Calculate \( \frac{\partial^2 g}{\partial v \, \partial u}(2,2) \) for \( g(u,v)=vf(v^2, 2uv) \) given that \( f(4,8)=2 \), \( \nabla f(4,8)=(5,1) \), \( \frac{\partial^2f}{\partial x^2}(4,8)= \frac{\partial^2f}{\partial y^2}(4,8)=2 \) and \( \frac{\partial^2f}{\partial x \, \partial y}(4,8)=-2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given an ellipse $$\frac {x^{2}}{a^{2}}$$ + $$\frac {y^{2}}{b^{2}}$$ = 1 with its right focus F, a line passing through the origin O intersects the ellipse C at points A and B. If |AF| = 2, |BF| = 4, and the eccentricity of the ellipse C is $$\frac {\sqrt {7}}{3}$$, calculate the area of △AFB.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}}
In the sequence $\{a_n\}$, $a_1 = 1$, $a_2 = 2$, $a_{n+2}$ is equal to the remainder of $a_n + a_{n+1}$ divided by 3. Find the sum of the first 89 terms of $\{a_n\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Evaluate the triple integral \( \int_{-2}^{1} \int_{-2}^{1} \int_{-2}^{1} \frac{x^2}{x^2+y^2+z^2} \,dx \,dy \,dz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
What is the maximum number of even entries possible in a $3 \times 3$ matrix $A$ with integer entries, given that $\det(A) = 1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
For the function \( f(x,y) = \begin{cases} (x^3+y^5-5xy)(x^2+y^2)^{-\beta}, &(x,y) \neq (0,0)\\ 0, & (x,y) = (0,0). \end{cases} \) with \( \beta > 0 \), determine the values of \( \beta \) for which the function is continuous at the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\beta < 1}
Carolyn wants to embroider her new jeans. When embroidering flowers, she can sew 4 stitches/minute, but when embroidering unicorns, she can sew 5 stitches/minute, and when embroidering Godzilla, she can sew 3 stitches/minute. A flower takes 60 stitches to embroider, a unicorn takes 180 stitches, and Godzilla takes 800 stitches. If Carolyn wants to embroider Godzilla crushing 3 unicorns and 50 flowers, and she takes a 5-minute break after every 30 minutes of embroidering, how many total minutes does she need to spend (including breaks) to complete her embroidering project?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1310}
Given a skew-symmetric matrix $M_{ij}$ constructed from $n$ random points in three dimensions $(x_i, y_i, z_i)$, determine the number of purely complex eigenvalues of the matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the value of the principal square root of the sum of a complex number and its square, given that the square of the complex number is equal to its principal square root but not equal to the number itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i}
Determine the range of a uniform distribution from which N random numbers are drawn such that, on average, half of the unique numbers drawn are smaller than M/2. Assume that only unique numbers smaller than M/2 are counted.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, M]}
Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer is $S$ is [i]also[/i] removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises of $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36.8}
Calculate the integral \( \int_{S(a,\frac{|a|}{2})} \frac{1}{|x|} \, dx \), where \( S(x,r) = \{ x \mid x_1^2 + x_2^2 + x_3^2 = r^2 \} \) and \( 0 \neq a \in \mathbb{R}^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi |a|}
Determine the first-order differential equation for which the solution is given by \( y = \left(\frac{y'}{2}\right)^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(y')^2 = 4y}
Given two 3x3 matrices A and B, where det(A) = 0 and every element of B is 1, and det(A + B) = 1, find the determinant of the matrix A + 2014B.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
直接从下列 $z$ 变换看出它们所对应的序列。$X(z)=-2 z^{-2}+2 z+1 \quad(0<|z|<\infty)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\delta(n)+2 \delta(n+1)-2 \delta(n-2)$}
Calculate the degree of the map \( S: \mathbb{R}^{m+n} \rightarrow \mathbb{R}^{m+n} \) defined by \( S(x, y) = (y, x) \) using the oriented bases of the tangent spaces.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^{mn}}
5. The unit square in the first quadrant of the coordinate plane ( $0 \leq x, y \leq 1$ ) is divided into squares with side length $2 \cdot 10^{-4}$. How many nodes of this partition (inside the unit square) lie on the parabola $y=x^{2}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Given a triangle \(ABC\) with a perimeter of 1. A circle is inscribed in angle \(BAC\), lying outside the triangle \(ABC\) and touching side \(BC\) (and the extensions of sides \(AB\) and \(AC\)). Find the length of segment \(AM\), where \(M\) is the point where the circle touches line \(AC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
By what common fraction does $0.\overline{06}$ exceed $0.06$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{1650}}
Determine the number of distinguishable ways to paint a cube using four distinct colors, such that each color appears on at least one face of the cube. Two cubes are considered distinguishable if they cannot be rotated to look identical.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68}
Find a $C^∞$ map $f: X \to Y$ where $X \subseteq \mathbb{R}^m$ and $Y \subseteq \mathbb{R}^n$, such that $f$ is one-to-one and onto, its inverse is continuous, but $f$ is not a diffeomorphism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = x^3}
A crude approximation is that the Earth travels in a circular orbit about the Sun at constant speed, at a distance of $150,000,000\text{ km}$ from the Sun. What is the closest value for the acceleration of the Earth in this orbit? Provide your answer as a number without including the unit, and multiply the original answer by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the dimension of the moduli space of complex structures on \( \mathbb{R}^{4} \) with the standard complex structure at infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
已知椭圆$\Gamma ：\frac{{{x}^{2}}}{{{a2}}}+\frac{{{y}^{2}}}{{{b2}}}=1$与双曲线$\Omega ：\frac{{{x}^{2}}}{{{m2}}}-\frac{{{y}^{2}}}{{{n2}}}=1$共焦点，F$_{1}$、F$_{2}$分别为左、右焦点，曲线$\Gamma $与$\Omega $在第一象限交点为$P$，且离心率之积为1.若$\sin \angle {{F1}}P{{F}_{2}}=2\sin \angle P{{F1}}{{F}_{2}}$，则该双曲线的离心率为____________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\sqrt{5}+1}{2}$}
Find the lowest common multiple of 10, 14, and a number. The lowest common multiple of these numbers is 140. What is the third number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In the figure, \( AB \) is the diameter of a circle, \( CD \) is a chord parallel to \( AB \), and \( AC \) intersects \( BD \) at \( E \). Given that \(\angle AED = \alpha\), calculate the ratio of the areas of \( \triangle CDE \) and \( \triangle ABE \) in terms of \(\alpha\). Express your answer as a function of \(\alpha\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cos^2 \alpha}
Evaluate the limit \( \lim_{x \to 0^-} \left(1-(1-\exp(tx^{-\frac{1}{v}}))^v\right)^{1/x} \), where \( t \in (-\infty, 0) \) and \( v \in (0, 1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the order of the smallest non-commutative ring without unity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the coefficient of the term $x^3y^4z^5$ in the expansion of $(x + y + z)^8(x + y + z + 1)^8$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1940400}
Evaluate the integral \( \int_{\mathcal{C}(0,2)} \frac{\sin(1/z)}{z+1}\,dz \) using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A circle is tangent to side \(BC\) of triangle \(ABC\) at its midpoint \(M\), passes through point \(A\), and intersects segments \(AB\) and \(AC\) at points \(D\) and \(E\) respectively. Find angle \(A\), given that \(BC=12\), \(AD=3.5\), and \(EC=\frac{9}{\sqrt{5}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Determine the number of distinct cyclic subgroups of order 10 in the group \( \mathbb{Z}_{30} \oplus \mathbb{Z}_{120} \). Identify the generators of these subgroups by locating elements with order 10, ensuring to account for duplicates that generate the same subgroup.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given the function \( f(x) = a \sin x + b \sqrt[3]{x} + c \ln \left(x + \sqrt{x^{2} + 1}\right) + 1003 \), where \( a \), \( b \), and \( c \) are real numbers, and the condition \( f\left(\lg^{2} 10\right) = 1 \), find the value of \( f(\lg \lg 3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2005}
Determine the limit in the sense of distributions of the sequence of functions \( f_n = n^2\chi_{[-1/n,0)} - n^2\chi_{(0,1/n]} \), where \( \chi \) denotes the characteristic function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta'}
Distinct points \( P,Q,R,S \) lie on the circle \( {x}^{2} + {y}^{2} = {25} \) and have integer coordinates. The distances \( {PQ} \) and \( {RS} \) are
 irrational numbers. What is the greatest possible value of the ratio \( \frac{PQ}{RS} \) ?
 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
7
}
In a series of three training matches, each lasting 90 minutes, coach F of country F evaluates seven players: $A_{1}, A_{2}, \ldots, A_{7}$. At any given moment, exactly one player is on the field. The total playing time for each of $A_{1}, A_{2}, A_{3}, A_{4}$ must be divisible by 7, and the total playing time for each of $A_{3}, A_{5}, A_{6}, A_{7}$ must be divisible by 13. How many different configurations of total playing times are possible for the players?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32398}
Let  $n{}$  be a natural number. The playing field for a "Master Sudoku" is composed of the  $n(n+1)/2$  cells located on or below the main diagonal of an  $n\times n$  square. A teacher secretly selects  $n{}$  cells of the playing field and tells his student


- the number of selected cells on each row, and
- that there is one selected cell on each column.
The teacher's selected cells form a Master Sudoku if his student can determine them with the given information. How many Master Sudokus are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}}
Find the last three digits of the product of all odd numbers from 1 to 1997.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{375}
One hundred musicians are organizing a festival with several concerts. During each concert, some musicians perform on stage while the others are in the audience. What is the minimum number of concerts required so that each musician has the opportunity to listen to every other musician perform on stage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let \( t \to y(t) \) be a real-valued smooth function on any open interval containing \([0,1]\). Suppose that \( y(t) \) satisfies the differential equation:
\[ y''(t) + w(t)y(t) = \lambda y(t) \]
where \( \lambda \) is a real constant and \( w(t) \) is a given function. Assume that \( y(t) > 0 \) for all \( t \in [0,1] \) and that \( y'(0) = y'(1) = 0 \). Given that \( \int_{0}^{1}\left(\frac{y'}{y}\right)^{2}\,dt = 10 \) and \( \int_{0}^{1}w(t)\,dt = 20 \), find the value of \( \lambda \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}