inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the number of 4-digit numbers such that if any single digit is removed, the resulting 3-digit number divides the original 4-digit number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In class 12th grade (12), there are 6 students, with 2 wearing red clothes, 2 in blue, and the other two each wearing black and yellow, respectively. They need to stand in a line for a photo, with the requirement that students wearing the same color clothes do not stand next to each other, and the students wearing black and yellow must stand next to each other. There are a total of ___ different ways to arrange this.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Let $\pi_1$, $\pi_2$, $\pi_3$ be three planes in a projective space such that $\pi_1 \cap \pi_2 \cap \pi_3 = \emptyset$, $\dim(\pi_1 \cap \pi_2) = 1$, and $\dim(\pi_1 \cap \pi_3) = \dim(\pi_2 \cap \pi_3) = 0$. Determine the dimension of $\pi_1 \lor \pi_2 \lor \pi_3$. \(\lor\) denotes the span of the planes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the range of values for the parameter \(\lambda\) such that the transformation \(T\) defined by \((Tf)(x) = x + \lambda \int_0^x (x-t)f(t)dt\) is a contraction mapping with respect to the supremum norm on the space of continuous functions \(C([0,1])\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 2)}
Given a constant symmetric tensor $V_{ij}$, calculate the product $V_{ij}V^{jk}$ by contracting the $j$ index. Express your answer as a matrix multiplication result.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{V^2}
Calculate the average number of steps required for a knight, starting at a corner of an 8x8 chessboard, to return to its starting corner when making random but legal moves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
Determine the values of $x \in \mathbb{R}$ for which the series \( \sum_{n=2}^{\infty} \frac{n^x}{(n+1)^2 \ln (n+1)} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1)}
Evaluate the limit: $$\lim_{n\to \infty}\left[(n+1)\int_0^1x^n\ln(1-x)\,dx\right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}
Evaluate the limit: \[ \lim_{n \to \infty} \int_{0}^{n} \left(1 - \frac{x}{n}\right)^{n} e^{x/2} \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the radius of convergence, \( R \), of the series \( \sum\limits_{n=0}^{\infty} \dfrac{(n!)^{k+2}\cdot x^n}{((k+2)n)!} \), where \( k \) is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(k+2)^{k+2}}
Compute the total differential of the function \( g(x,y) = f(f(x,y), f(y,x)) \) at the point \((1,1)\), given that \( f(1,1) = 1 \), \( \partial_x f(1,1) = 1 \), and \( \partial_y f(1,1) = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\,dx + 4\,dy}
Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{36 - 4\pi\sqrt{3}}{27}}
Squares of a chessboard are painted in 3 colors - white, gray, and black - in such a way that adjacent squares sharing a side differ in color; however, a sharp change in color (i.e., adjacency of white and black squares) is prohibited. Find the number of such colorings of the chessboard (colorings that are identical upon rotating the board by 90 and 180 degrees are considered different).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{33}}
Two trains are 80 kilometers apart and are traveling toward each other on the same track, each at a speed of 25 kilometers per hour. A bee starts at the first train and flies back and forth between the two trains until they are 100 meters apart. The bee completes 36 rounds, landing 18 times on each train. Assuming the bee has no acceleration or deceleration and takes no extra time when turning, calculate the average speed of the bee.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{270}
Through the right focus \( F_{2} \) of the ellipse \(\frac{x}{2} + y^{2} = 1\), draw a chord \(\mathrm{AB}\) with an inclination angle of \(45^\circ\). Then, what is the length \(|AB|\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{4 \sqrt{2}}{3}\)}
Find the complex number $z$ that maximizes the expression $|z-w| - |z-w^2|$, where $z$ lies on the ellipse defined by $|z-4| + |z+4| = 10$, and $w$ is a non-real cube root of unity. Assume $z = x + iy$, and the ellipse's equation is $\frac{x^2}{25} + \frac{y^2}{9} = 1$. Determine the value of $z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3i}
Evaluate the limit for any integer $n > 1$:

$$\lim_{x \to \infty} \left[-x^{n+1} e^{-x}\right]_{x=0}^{x=\infty}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
What is the probability that at 10:00 AM on December 31, the Absent-Minded Scientist has exactly two packs of tablets in the cabinet, given that he starts with one pack of 10 tablets and orders a new pack whenever he sees only one tablet left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{10}}
Determine the number of minutes before Jack arrives at the park that Jill arrives at the park, given that they are 2 miles apart, Jill cycles at a constant speed of 12 miles per hour, and Jack jogs at a constant speed of 5 miles per hour.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Suppose X, Y, and Z are independent normal random variables. X has a mean of 5 and variance of 16, Y has a mean of 7 and variance of 25, and Z has a mean of 10 and variance of 9. Find the probability that the average of X and Z is greater than Y.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5357}
Find the value of \( \lim_{x \to 0} f(x) \) if the function \( f(x)=\lim_{n\to\infty} \frac{(1-\cos(1-\tan(\frac{\pi}{4}-x)))(1+x)^n+\lambda\sin((n-\sqrt{n^2-8n})x)}{x^2(1+x)^n+x} \) is continuous at \( x=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Estimate the number of planes in an airline's fleet, given that on his fifteenth flight, the Absent-Minded Scientist flew on a plane named "Siméon Denis Poisson" for the second time, with no repetitions on previous flights.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Calculate the volume enclosed by the hyperboloid \(-x^2-y^2+z^2=1\) and the plane \(z=2\) using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4\pi}{3}}
Determine the rank of the first homology group of the double branched cover of a knot $K$ in $S^3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Compute the norm of the bounded linear functional \( f \) defined by \( f(x) = x'(t_0) \) for all \( x \) in the space \( C'[a,b] \), where \( C'[a,b] \) is the space of all continuously differentiable functions on \([a,b]\) with the norm \( ||x|| = \max_{a \leq t \leq b} |x(t)| + \max_{a \leq t \leq b} |x'(t)| \). Here, \( t_0 \) is a fixed point in \([a,b]\). What is the norm of \( f \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For a random variable $X$ with differential entropy $h(X)$, determine the change in differential entropy when $X$ is multiplied by a negative constant $a$. Express your answer in terms of $h(X)$ and $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{h(X) + \log |a|}
If $x_0=x_1=1$, and for $n\geq1$

$x_{n+1}=\frac{x_n^2}{x_{n-1}+2x_n}$,

find a formula for $x_n$ as a function of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$x_n = \frac{1}{(2n-1)!!}$

}
In space, there are four pairwise skew lines \( l_{1}, l_{2}, l_{3}, l_{4} \), such that no three of them are parallel to the same plane. A plane \( P \) intersects these lines at points \( A_{1}, A_{2}, A_{3}, A_{4} \), forming a parallelogram. Determine the number of lines that the centers of such parallelograms trace out.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Estimate the number of iterations of Newton's method needed to find a root of \( f(x) = \cos(x) - x \) to within \( 10^{-100} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
If 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}}
Calculate the line integral \( \oint_C P\,dx + Q\,dy \) over the ellipse \( \frac{x^2}{25} + \frac{y^2}{36} = 1 \), where the vector fields are given by:
\[
P = \frac{-y}{(x-1)^2 + y^2}, \quad Q = \frac{x-1}{(x-1)^2 + y^2}
\]
Determine the value of the integral, considering that the vector field is undefined at the point \((1,0)\) inside the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Calculate the volume of the solid formed by the intersection of the paraboloid given by the function \( f(x,y) = 3-x^2-y^2 \) and the plane \( z=1-2x-2y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\pi}
Determine the smallest integer $m$ such that in any distribution of 2011 stickers among 33 friends, if every pair of friends is missing at least $m$ stickers, then there is at least one sticker that none of the friends have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1890}
Evaluate \( \lim_{x\to \infty} \frac{ \cot^{-1} (\sqrt {x+1} -\sqrt x)}{\sec^{-1} \left(\frac{2x+1}{x-1}\right)^x} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the optimal asymptotic behavior of the ratio \( \frac{\|(-\Delta)^{\frac{s}{2}}F_d\|_{L^2}^2}{\|(-\Delta)^{\frac{s}{2}}f\|_{L^2}^2} \) as \( d \to \infty \), where \( F_d = f^{\otimes d} \) and \( f \in H^s(\mathbb{R}) \) with \( \|f\|_{L^2} = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d^s}
Determine the number of labeled trees on 9 vertices where exactly three vertices have a degree of 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{317520}
Determine the least ordinal \( \alpha \) such that \( V_{\alpha} \) satisfies the axioms of power, pairing, and infinity simultaneously, given that \( V_{\omega} \) does not satisfy the axiom of infinity, \( V_{\omega+1} \) does, and \( V_{\omega+\omega} \) satisfies all three axioms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\omega + \omega}
Find the minimum constant \( c \) such that for any positive integer \( m \) and a \( kn \)-regular graph \( G \) with \( m \) vertices, the vertices of \( G \) can be colored with \( n \) different colors so that the number of monochrome edges is at most \( cm \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{k}{2}}
Suppose $v_1, v_2, \ldots, v_n$ are $n$ i.i.d. continuous random variables with the range $[\underline{v}, \bar{v}]$. Determine the limit of the expectation of the minimum of these random variables, $E[\min(v_1, v_2, \ldots, v_n)]$, as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\underline{v}}
Let $a_1,a_2,\ldots,a_{2017}$ be the $2017$ distinct complex numbers which satisfy $a_i^{2017}=a_i+1$ for $i=1,2,\ldots,2017$. Compute$$\displaystyle\sum_{i=1}^{2017}\frac{a_i}{a_i^2+1}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2016}
Evaluate the line integral \( \oint_C \boldsymbol{F} \cdot d\boldsymbol{l} \) for the vector field \( \boldsymbol{F} = x\hat{e}_x + z\hat{e}_y + x\hat{e}_z \), where the contour \( C \) consists of the following segments: a straight line from the origin to \((0, 0, b)\); a straight line from \((0, 0, b)\) to \((0, b, 0)\); and a straight line from \((0, b, 0)\) back to the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{b^2}{2}}
Evaluate the limit \( \lim_{R\rightarrow \infty} \int_{C_R} \frac{6z^6 + 5z^5}{z^6 + z^5 + 10}dz \), where \( C_R=\{z \in \mathbb{C} : |z|=R \} \) for \( R>0 \), using the Residue theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi i}
Determine the splitting field of the polynomial \(x^8 + x^6 + x^5 + x^4 + x^3 + x^2 + 1\) over the finite field \(\mathbb{F}_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{F}_{2^6}}
A car is equipped with one spare tire. The lifetimes of the four tires at the start of a long-distance journey are independent identically distributed exponential random variables with a mean of 5000 miles. The spare tire has an exponential lifetime with a mean of 1000 miles. Calculate the expected number of miles that can be covered without needing to visit a tire shop.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1875}
Find the derivative of \( \frac{\partial{A_{ij}\vec{x}}}{\partial{A_{ij}}} \), where \( A_{ij}\vec{x} \) is a vector whose \( i^{th} \) entry is given by \( A_{i1}x_1 + A_{i2}x_2 + \ldots \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x_j \vec{e}_i}
Evaluate the integral \( \int_D \frac{e^{1/w^2}}{w-2}dw \), where \( D \) is the annulus centered at the origin with inner radius 1 and outer radius 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i e^{1/4}}
Find the derivative of \( \text{Tr}(S^T SUU^T) \) with respect to the matrix \( U \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 S^T S U}
Determine the values of the parameter \( \alpha \in \mathbb{R} \) for which the function \( f: (0,1) \rightarrow \mathbb{R} \), defined by \( f(x) = \frac{x-1}{x^{\alpha} \ln(x)} \), is integrable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1)}
Calculate the residue of the function \( \frac{e^{2/z}}{(z+2)^3} \) at the point \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8e}}
Find the residue of the function \( \frac{(z^6-1)^2}{z^5(2z^4-5z^2+2)} \) at \( z=0 \) without directly computing the 5th derivative using the residue formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{21}{8}}
Determine the maximum possible length of a path that includes all vertices in an Eulerian semi-graceful graph with $n$ vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n-1}
Let $T: P_2(\mathbb{R}) \to P_2(\mathbb{R})$ be a linear transformation defined by the following extensions: $T(x) = x-2$, $T(x^2-x) = x+2x^2$, and $T(x-2+x^2) = -1-x^2$. Compute $T(2+x^2)$. Provide your answer as a polynomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5x^2 + 5x - 5}
What is the characteristic function of the Dirac delta function?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Two circles \( \Omega_1 \) and \( \Omega_2 \) have centers \( O_1 \) and \( O_2 \) with radii 3 and 4, respectively. Points \( A \) and \( B \) lie on circles \( \Omega_1 \) and \( \Omega_2 \), such that segment \( AB \) is tangent to both circles and intersects \( O_1O_2 \). Given that \( O_1O_2 = 25 \), find the area of quadrilateral \( O_1AO_2B \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Evaluate the limit: \[ \lim_{x\rightarrow0} \frac{(4^{\arcsin(x^2)} - 1)(\sqrt[10]{1 - \arctan(3x^2)} - 1)}{(1-\cos\tan6x)\ln(1-\sqrt{\sin x^2})} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit \( \lim_{n\to\infty} \int_{0}^{1} \frac{ne^{-\sqrt{nx}} \sin{x}}{(1+x^{2})\arctan{x}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the infinite series: $$\sum_{n=1}^{\infty}\frac{g(n)}{n(n+1)}$$ where the function $g(n)$ is defined as follows: $$ g(n) = \begin{cases}0&n=1\\g(z)& n = 2z\\g(z) - 1&n = 2z + 1\end{cases} $$ Find the value of this summation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - 2\ln 2}
For all positive integers $n$, denote by $\sigma(n)$ the sum of the positive divisors of $n$ and $\nu_p(n)$ the largest power of $p$ which divides $n$. Compute the largest positive integer $k$ such that $5^k$ divides\[\sum_{d|N}\nu_3(d!)(-1)^{\sigma(d)},\]where $N=6^{1999}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
8
}
A rectangular playing field of size \( 13 \times 2 \) is to be completely covered using an unlimited number of dominoes of sizes \( 2 \times 1 \) and \( 3 \times 1 \). The dominoes must cover the field without gaps or overlaps, and no domino should extend beyond the playing field. Additionally, all dominoes must be oriented the same way, with their long sides parallel to each other. Determine the number of such coverings possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{257}
Let $B$ be an orthogonal matrix with $\det(B) < 0$. Determine the range of possible values for $\det(B-I)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 0]}
8. You have been kidnapped by a witch and are stuck in the *Terrifying Tower*, which has an infinite number of floors, starting with floor 1, each initially having 0 boxes. The witch allows you to do the following two things:

- For a floor  $i$ , put 2 boxes on floor  $i+5$ , 6 on floor  $i+4$ , 13 on floor  $i+3$ , 12 on floor  $i+2$ , 8 on floor  $i+1$ , and 1 on floor  $i$ , or remove the corresponding number of boxes from each floor if possible.
- For a floor  $i$ , put 1 box on floor  $i+4$ , put 3 boxes on floor  $i+3$ , 6 on floor  $i+2$ , 5 on floor  $i+1$ , and 3 on floor  $i$ , or remove the corresponding number of boxes from each floor if possible.

At the end, suppose the witch wishes to have exactly  $n$  boxes in the tower. Specifically, she wants them to be on the first 10 floors. Let  $T(n)$  be the number of distinct distributions of these  $n$  boxes that you can make. Find  $\displaystyle\sum_{n=1}^{15} T(n)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In an acute triangle \(ABC\), the altitudes from \(A\), \(B\), and \(C\) intersect at the orthocenter \(H\). Let \(S\) be the midpoint of the circular arc \(BHC\) on the circumcircle of triangle \(BCH\). If the segments \(AS\) and \(AH\) are equal in length, determine the measure of angle \(\angle BAC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Determine the dimension of \( \frac{m}{m^2} \) over \( k \), where \( m = m_P(V) \) and \( V = V(x^2-y^3, y^2-z^3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the equation \( xf(x) - yf(y) = (x-y)f(x+y) \) for all real numbers \( x \) and \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = ax + b}
Find an irrational number $x$ such that $x^x$ is rational. Provide a specific example of such a number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}^{\sqrt{2}}}
Determine the largest interval around \(x=0\) where the function \(f(x) = x^5 - 5x + 3\) is invertible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
There are twenty individuals numbered $1,2,...,20$. Each individual randomly and independently chooses 10 others from this group to call. Let $X$ be the total number of calls (incoming and outgoing) involving Individual 1, and let $Y$ be the number of calls between Individual 1 and Individual 2. Calculate the expected value of $X$ given that $Y=1$, denoted as $E(X|Y=1)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{759}{38}}
An equilateral triangle with side length 3 is divided into nine smaller triangular cells. Initially, zeros are written in these cells. In one move, you can choose two numbers located in neighboring cells (sharing a side) and either add one to both or subtract one from both. Determine the value of \( n \) such that after several moves, the cells contain consecutive natural numbers \( n, n+1, \ldots, n+8 \) in some order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the local order of convergence of the iteration procedure given by \( x_{n+1} = x_{n} - p \cdot \frac{f(x_{n})}{f'(x_{n})} \) for a function \( f \in C^{p+1}(\mathbb{R}) \) with a \( p \)-fold root at \( x^* \), where \( p \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the dimension of $H^0(T, \mathcal{L})$, where $T$ is a complex torus of dimension 2 and $\mathcal{L}$ induces a $(m,n)$-polarization on $T$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{mn}
The integers $0 \leq a < 2^{2008}$ and $0 \leq b < 8$ satisfy the equivalence $7(a+2^{2008}b) \equiv 1 \pmod{2^{2011}}$. Then $b$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Three balls are randomly and independently thrown into buckets labeled with positive integers such that the probability of a ball being thrown into bucket \(i\) (for \(i = 1, 2, \ldots\)) is \(2^{-i}\). It is not allowed for a bucket to contain more than one ball. The probability that the three balls fall into buckets labeled in an arithmetic sequence is given by \(\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. Find \(p+q\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Compute the \( \mathbb{C} \)-dimension of the vector space \( \mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \), where \( \mathbb{R} \) is the field of real numbers and \( \mathbb{C} \) is the field of complex numbers. Consider \( \mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \) as a \( \mathbb{C} \)-vector space via \( a(b\otimes c) := ab \otimes c, \) for \( a, b, c \in \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Semicircle \( \Gamma \) has diameter \( \overline{AB} \) of length14. Circle \( \Omega \) lies tangent to \( \overline{AB} \) at a point \( P \) and intersects \( \Gamma \) at points \( Q \) and \( R \) . If
 \( {QR} = 3\sqrt{3} \) and \( \angle {QPR} = {60}^{ \circ  } \) ,then the area of \( \bigtriangleup {PQR} \) is \( \frac{a\sqrt{b}}{c} \) ,where \( a \) and \( c \) are relatively prime positive integers,and
 is a positive integer not divisible by the square of any prime. What is \( a + b + c \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
122}