inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the range of values for $m$ such that the equation $|x^2-4|+x^2+mx=0$ has exactly 2 distinct roots in the interval $(1,3)$. Provide your answer as an interval for $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-4, -2)}
A number is additive if it has three digits, all of them are different and the
 sum of two of the digits is equal to the remaining one. (For example,
 \( {123}\left( {1 + 2 = 3}\right) ,{945}\left( {4 + 5 = 9}\right) ) \) . Find the sum of all additive number
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
48840}
Consider the vector spaces in \(\mathbb{R}^2\) with the basis \(B=\{ (2,1), (5,3)\}\) and \(C=\{ (1,1), (2,3)\}\). Let the linear transformation \(f: \mathbb{R}^2 \to \mathbb{R}^2\) be defined by \(f(x,y)=(x+y,-2x+y)\). The matrix representation of \(f\) with respect to the bases \(B\) and \(C\) is \(Cf_B=\begin{pmatrix} \alpha_{11} & \alpha_{12}\\ \alpha_{21} & \alpha_{22} \end{pmatrix}\). Compute the value of \(\alpha_{12} + 2\alpha_{22}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A system consists of 7 components with independent, identically distributed lifetimes. The system will operate until any one of the components fails. If the lifetime \( X \) of each component has the probability density function:
\[
f(x) =
\begin{cases}
\frac{3}{x^4},  & \text{for } x > 1 \\
0, & \text{otherwise}
\end{cases}
\]
what is the expected lifetime until failure of the system?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{21}{20}}
Determine the asymptotic complexity of the recurrence relation \(B(n) = 5\cdot B(n/3) + cn^2\) with the initial condition \(B(1) = c\). Express your answer in the form \(O(n^d)\) for some value of \(d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^2)}
Estimate the upper bound of the number of solutions \(N_{\sqrt{2}}(1)\) for the inequality:
\[ \left \lvert \sqrt{2} - \frac{p}{q} \right \rvert < \frac{1}{q^{3}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the matrix \(A = \begin{bmatrix} 1 & 3 & 4\\ 3 & 6 & 9\\ 1 & 6 & 4 \end{bmatrix},\) where \(B\) is an invertible \(3\times 3\) matrix that satisfies \(A \cdot A^{T} \cdot A +3B^{-1} =0,\) determine the value of \(\det( \operatorname{adj} (A^{-1}(B^{-1}){2B^{T}})).\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{64}{81}}
Determine all values of \(N\) such that for any arrangement of \(N\) cubes (using an infinite supply of white, blue, and red cubes), the final remaining cube's color is independent of the robot's starting position. We call such \(N\) lucky. Find all lucky \(N\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the values of $n$ such that $$(1+2i)^{4n}=(1-2i)^{4n}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
$ABCD$ is a regular tetrahedron with side length $1$. Find the area of the cross section of $ABCD$ cut by the plane that passes through the midpoints of $AB$, $AC$, and $CD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$ \frac{1}{4}$
}
Find a closed formula for the determinant of an \( n \times n \) matrix where the diagonal elements are 0, the last row is \([0, 0, \ldots, 1]\), the first row is \([1, 0, \ldots, 0]\), and all other elements are 0. For example, when \( n=2 \), the matrix is \( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \), and when \( n=3 \), the matrix is \( \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^{\frac{n(n-1)}{2}}}
Determine the maximum degree of a Lagrange polynomial required to approximate the function \( f(x) = \cos(x) \) on the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\) with an error of less than or equal to 0.01 using the Lagrange error bound formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the disc of convergence for the series \( \sum_{n=0}^\infty z^{n^3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{ z \in \mathbb{C} \mid |z| < 1 \}}
Solve the differential equation \( y'=\frac{\sin x + e^x (\sin x + \cos x) + x\cos x - y(1 + \cos x + e^x)}{e^x + \sin x + x - 2y} \) and find the curve that passes through the point (0,0).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = \sin x}
Determine the number of ring homomorphisms from \(\mathbb{Z}_{12}\) to \(\mathbb{Z}_{28}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the longest geometric progression that can be formed using prime numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of different ring homomorphisms from \(\mathbb{Z}[i]\) to \(\mathbb{Z}/(85)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the dimension of the algebraic set $X = V(y-x^2, y^2-y+z^2)$ over an algebraically closed field $F$. The dimension of an algebraic set is defined as the transcendence degree of its rational function field over $F$ if the set is irreducible, or the maximum dimension of its irreducible components if the set is reducible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given two sequences encoded as follows: a=1, b=2, c=4, d=8, ..., y=2^24, z=2^25, and a '|' symbol indicating the end of a message, determine the time at which the sequences synchronize. For example, the sequence "*ea|babab" has an offset of 17 and a length of 25, while "*d|abaca" has an offset of 8 and a length of 17. Calculate the synchronization time for these sequences.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Minimize the value of the function \( P = a^2 + b^2 + c^2 + 2\sqrt{3abc} \) given that \( a, b, c \) are positive real numbers such that \( a + b + c = 1 \). Find the minimum value of \( P \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Let $z$ and $w$ be differentiable functions of $x$ and $y$ satisfying the following equations:

$$ xw^3+yz^2+z^3=-1$$

$$ zw^3-xz^3+y^2w=1$$

Find $\frac{\partial z}{\partial x}$ and evaluate it at the point $(x,y,z,w)=(1,-1,-1,1)$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{5}{4}}
Determine the dimension of the subspace $W$ of $\mathcal{L}(\mathbb{R}[x])$, where $W$ is generated by linear operators on $\mathbb{R}[x]$ that have no adjoint. The vector space $\mathbb{R}[x]$ consists of all polynomials with real coefficients, and the inner product is defined by $\langle f,g\rangle=\int_{0}^{1}f(t)g(t)dt$. What is the dimension of $W$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the contour integral \( \oint_{|z| = 1} \! dz \log\left(\frac{z - a}{z - b}\right) \) given \(|a| < 1\) and \(|b| < 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i (b - a)}
Compute the limit \( \lim_{a\to \infty}\frac{1}{a}\int_1^a a^{\frac{1}{x}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( \omega \) be a circle with center O and radius 10, and let H be a point such that \( OH = 6 \). A point P is called snug if, for all triangles ABC with circumcircle \( \omega \) and orthocenter \( H \), P lies on \( \triangle ABC \) or in the interior of \( \triangle ABC \). Find the area of the region consisting of all snug points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\pi}
Evaluate the derivative \( \frac{d}{dx}\int_{0}^{x} \sin(1/t) \, dt \) at \( x = 0 \). Determine if the derivative exists at this point, and if so, find its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the coefficient of $x^{60}$ in the expansion of the product $\prod_{m=1}^{11} (x^m - m)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that $a$ and $b$ are roots of the equation $x^2 - 3cx - 8d = 0$, and $c$ and $d$ are roots of the equation $x^2 - 3ax - 8b = 0$, find the value of $a + b + c + d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Find the value of $p + q$ if the following two lines intersect at an angle of $60^\circ$, given that $a = -(p/q)$, where $p$ and $q$ are coprime positive integers:

$$ \frac{x+2}{a} = \frac{y-1}{a-2} = \frac{z+3}{a+2} \\
 x = 0 \; \& \; y = z. $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given the ideals $I = (2, x)$ and $J = (3, x)$, determine whether the product $I \cdot J$ is equal to the ideal $(6, x)$ or if $(6, x)$ is contained in $I \cdot J$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I \cdot J = (6, x)}
Let $F$ be a field of characteristic $p > 0$ and let $f(t) = t^p - t - a \in F[t]$. Suppose also that $a \neq b^p - b$ for any $b \in F$. Determine the Galois group $Gal(K/F)$, where $K$ is a splitting field of $f$. \( \text{Provide your answer in terms of group structure.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/p\mathbb{Z}}
Determine the value of $k \in \mathbb{R}$ such that the stochastic process \( Y_t = (W_t + t)e^{-W_t + kt}, \quad t \geq 0 \) has zero drift, where \( W_t \) is a Wiener process.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Let $A$ be an abelian group of order $p^5$, where $p$ is a prime number. Given that there exists a subgroup $N$ such that $A/N \simeq \mathbb{Z}/p^2\mathbb{Z}$, determine the number of possible groups $A$ that satisfy this condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A stick of unit length is cut into three pieces of lengths $X$, $Y$, and $Z$, with $X \le Y \le Z$. First, the stick is cut into two pieces at a randomly chosen point uniformly distributed along its length. Then, the smaller of these two pieces is cut again at a randomly chosen point uniformly distributed along its length. These two cuts are conducted independently. Find the covariance $\text{Cov}(X,Y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{2304}}
How many ways are there to place $31$ knights in the cells of an $8 \times  8$ unit grid so that no two attack one another?
(A knight attacks another knight if the distance between the centers of their cells is exactly $\sqrt5$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
68
}
Express the function $f(x) = g(x) \prod_{n=1}^x{n^k}$ in an explicit form without using the product operator, such that the resulting function $h(x) = \frac{f(x)}{g(x)}$ satisfies the conditions $h(0) = 0$, $h(1) = 1$, and for all natural numbers $n$, $h(n) = n^k \cdot h(n-1)$. What is the explicit form of $h(x)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x!)^k}
Find a 1-dimensional sufficient statistic for \( \theta \) given that \( X_1, X_2, \ldots, X_n \) is a sample from a normal distribution with probability density function:
\[ f(x\mid\theta)=\frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{1}{2}(x-\theta)^2\right)} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sum_{i=1}^n X_i}
Given a signal $s(t)$ sampled at $1000$ observations, where $h(k) = s(k\Delta t + t_0)$ for $k = 0, 1, \dots, 999$, with $\Delta t = 1/200$ and $t_0 = -2$ seconds, the discrete Fourier transform $|\hat{h}(m)|$ has its largest values at $m = 193$ and $m = 807$. What is the corresponding positive frequency in hertz for the signal $s$, rounded up to the nearest integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
Compute the partial derivative \( u_{xy} \) of the function \( u(x,y) = \frac{xy(x^2-y^2)}{x^2+y^2} \) at the point \((0,0)\), given that \( u(0,0) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let $A$ be the set of nonnegative sequences $(a_n)$ such that $\sum_{n=1}^{\infty}a_n=1.$ Determine the range of the map $P:A\to \mathbb{R}$ defined by \[P((a_n))= \prod_{n=1}^\infty(1+a_n).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2, e)}
Find the norm of the linear functional $L: \mathcal{H} \rightarrow \mathbb{F}$ defined by $L(\{\alpha_n\}) = \sum_{n=0}^\infty \alpha_n\lambda^n$, where $|\lambda|>1$ and $\mathcal{H} = l^2(\mathbb{N}\cup \{0\})$. \( \mathbb{F} \) denotes the field of scalars, which can be either the real numbers \( \mathbb{R} \) or the complex numbers \( \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Calculate the volume of the region in \( \mathbb{R}^n \) defined by the inequalities \( x_1 + \cdots + x_n \geq a \) and \( x_i > 0 \) for all \( i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given a $5 \times 5$ matrix $B$ over the field $\mathbb{R}$, compute the dimension of the vector space of $5 \times 5$ matrices that commute with $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find all non-zero polynomials \( P(x) \) such that \( P\left(x^{2}-2 x\right) \equiv [P(x-2)]^{2} \) for all \( x \in \mathbf{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x + 1)^n}
Let $b_1, b_2, b_3, \ldots$ be the sequence of positive integers $b$ such that $b$ and $a$ are relatively prime, and when $a$ is divided by $b$, the remainder is $4$ and the quotient is $7$. Find $b_{2014}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4031}
Evaluate the integral \( \int_0^\pi \sin(\sin(x))\sin(x)\, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi J_1(1)}
Find the residue of the function \( f(z) = \frac{1}{e^{e^{\frac{1}{z}}}} \) at the point \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{e}}
The minimum value of the function \( y = |\cos x| + |\cos 2x| \) (for \( x \in \mathbf{R} \)) is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\sqrt{2}}{2}"]}
Determine the range of values for \(a\) such that the function \(f(x) = (a+2)x^3 - 3ax^2 + 9ax - 1\) is decreasing for all real values of \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -3]}
Arnaldo and Bernaldo train for a marathon along a circular track, which has in its center a mast with a flag raised. Arnaldo runs faster than Bernaldo, so that every $30$ minutes of running, while Arnaldo gives $15$ laps on the track, Bernaldo can only give $10$ complete laps. Arnaldo and Bernaldo left at the same moment of the line and ran with constant velocities, both in the same direction. Between minute $1$ and minute $61$ of the race, how many times did Arnaldo, Bernaldo and the mast become collinear?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
10
}
Evaluate the expression \( \displaystyle \underset{(i \ne j \ne k)}{\sum_{i=0}^\infty \sum_{j=0}^\infty \sum_{k=0}^\infty \frac{1}{3^i 3^j 3^k}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{81}{208}}
Calculate the expected first-passage time \( \mathbf{E}[T_{a,b}] \) for a standard Brownian motion \((B_t)_t\) with \(B_0=0\), where \(T_{a,b} = \inf\{t>0 : B_t \not\in(a,b)\}\) for \(a<0<b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-ab}
Find the first derivative of the function \( f(X) = \log \det(XX^TA^{-1}) \) with respect to the matrix \( X \), where \( X \in \mathbb{R}^{n \times r} \) and \( A \in \mathbb{R}^{n \times n} \) with \( A^{-1} \) being the inverse of \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(XX^T)^{-1}X}
Determine the number of Polish group topologies that can be placed on \( \text{PSL}_2(\mathbb{C}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given a linear transformation $T$ on a finite vector space $V$ such that $Ker(T) = Im(T)$, determine the minimal polynomial of $T$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2}
Given 11 sets \( M_{1}, M_{2}, \cdots, M_{11} \), each with exactly 5 elements, and for all \( i, j \) such that \( 1 \leq i < j \leq 11 \), \( M_{i} \cap M_{j} \neq \varnothing \), determine the minimum possible value of the maximum number of sets that have a non-empty intersection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Consider the initial value problem given by the differential equation \( x' = x + \mu(t + x^2) \) with the initial condition \( x(0) = 1 \). Find the sensitivity of the solution \( x(t, \mu) \) with respect to the parameter \( \mu \) by calculating \( \frac{\partial x}{\partial \mu} \bigg|_{\mu=0} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{2t} - t - 1}
Evaluate the limit of the sequence given by:
\[ \lim_{n \rightarrow \infty} \left( \frac{n \pi}{4} - \left( \frac{n^2}{n^2+1^2} + \frac{n^2}{n^2+2^2} + \cdots + \frac{n^2}{n^2+n^2} \right) \right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Determine strictly positive real numbers \( {a}_{1},{a}_{2},\ldots ,{a}_{n} \) if for any \( n \in  {N}^{ * } \) takes
 place equality: \( {a}_{1}^{2} + {a}_{2}^{2} + \ldots  + {a}_{n}^{2} = {a}_{1} + {a}_{2} + \ldots  + {a}_{n} + \frac{n\left( {{n}^{2} + {6n} + {11}}\right) }{3} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_n = n + 2}
Segment $BD$ and $AE$ intersect at $C$, as shown, $AB=BC=CD=CE$, and $\angle A = \frac 52 \angle B$. What is the degree measure of $\angle D$?
[asy] unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4;  pair C=(0,0), Ep=dir(35), D=dir(-35), B=dir(145); pair A=intersectionpoints(Circle(B,1),C--(-1*Ep))[0]; pair[] ds={A,B,C,D,Ep};  dot(ds); draw(A--Ep--D--B--cycle);  label("$A$",A,SW); label("$B$",B,NW); label("$C$",C,N); label("$E$",Ep,E); label("$D$",D,E); [/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( 52.5^\circ \).}
Determine the number of distinct Boolean functions \( f(x_1, x_2, \dots, x_n) \) that can be constructed using only AND and OR operators, without NOT, where the input is from \( \{0,1\}^n \) and the output is in \( \{0,1\} \). The function must output \(1\) for all inputs with all bits set to \(1\) and \(0\) for all inputs with all bits set to \(0\). Constant functions \(f \equiv 1\) and \(f \equiv 0\) are not valid. What is the number of such functions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{D(n) - 2}
A puzzle club meets in a schoolroom every Friday after school. Last Friday, a member said, "I’ve hidden a list of numbers in this envelope that add up to the number of this room." A girl asked, "If you told us the number of numbers in the envelope and their product, would that be enough to determine the room number?" He replied, "No." She then said, "Well, at least I’ve worked out their product." What is the number of the school room they meet in?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \(a\) and \(b\) be elements of a group such that \(|a| = 2\), \(b \neq e\), and \(aba = b^2\). Determine the order of \(b\), denoted as \(|b|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The area of a convex quadrilateral $ABCD$ is $18$. If $|AB|+|BD|+|DC|=12$, then what is $|AC|$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$6\sqrt{2}$}
Evaluate the integral \( \int_0^{2\pi}\cos(\sin(e^{i \theta}))d\theta \) using complex analysis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Given the vector field \(\vec{F} (x, y, z) = xy\hat{i} + (4x - yz)\hat{j} + (xy - z^{1/2}) \hat{k}\) and a circle \(C\) of radius \(R\) lying in the plane \(x + y + z = 5\), oriented counterclockwise when viewed from above, find the radius \(R\) if the line integral \(\int_C \vec{F} \cdot d\vec{r}\) equals \(\pi \sqrt{3}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{3}}{2}}
Vasya, Petya, and Kolya live in the same house. Vasya and Petya go fishing at a lake, while Kolya stays home. Vasya starts walking home, and Kolya sets out on his bicycle to meet him. When Kolya meets Vasya, he gives him a ride home. Petya starts walking home at the same speed as Vasya when Kolya meets Vasya. Kolya then goes back to meet Petya and gives him a ride home. Petya's travel time from the lake to home is $\frac{4}{3}$ of Vasya's travel time. Calculate how many times slower Vasya's journey would be if he walked the entire way home.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $\lbrace N(t)\rbrace_{t\geq 0}$ be a Poisson process with intensity $\lambda = 3$. Compute the probability $P\left[N(6) = 2 \,|\, N(8) = 4, N(3) = 1\right].$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{36}{125}}
Let $U$ be the vector subspace of $\mathbb{R}^5$ generated by $\{(1,3,-3,-1,-4),(1,4,-1,-2,-2),(2,9,0,-5,-2)\}$, and let $V$ be the vector subspace of $\mathbb{R}^5$ generated by $\{(1,6,2,-2,3),(2,8,-1,-6,-5),(1,3,-1,-5,-6)\}$. Determine the dimension of the vector space $U \cap V$. \( \text{Provide your answer as a single integer.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of ordered pairs \((x, y)\) that satisfy the following system of equations: \((\cos^{-1} x)^2 + \sin^{-1}(y) = 1\) and \(\cos^{-1}(x) + (\sin^{-1}y)^2 = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the eigenvalues of the linear map $g: V_3 \rightarrow V_3$ defined by $p(X) \mapsto p'(X)$, where $V_3$ is the vector space of polynomials with degree at most 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
What is the smallest positive odd integer \( n \) such that the product of
 \( {2}^{\frac{1}{7}},{2}^{\frac{3}{7}},\ldots ,{2}^{\frac{2 - n + 1}{7}} \)
 is greater than 1000 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Compute the integral \( \int_{|z|=2} z^n(1 - z)^m\ dz \) for negative integers \( n \) and \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the order of the distribution $S$ defined by $S(\psi)=\psi^{(n)}(y_0)$ for $\psi \in C_c^\infty (\mathbb R)$ and $n\in \mathbb N$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Evaluate the integral \( \int_{0}^{2\pi } \frac{\sin^{2} (x) }{5+4\cos(x)}\,\mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Given the function $f(x) = x^2 - 2ax + a(a+1)$ with the domain $f:[a, \infty) \to [a, \infty)$, one solution to the equation $f(x) = f^{-1}(x)$ is $5049$. Determine the other possible solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5050}