inclusionAI/Ring-lite-rl-data huggingface dataset:
Given that for each \(a \in (0,1)\), the limit \(\lim_{h \to 0^+} \int_{h}^{1-h} t^{-a} (1-t)^{a-1} dt\) exists and is denoted by \(g(a)\), and that \(g(a)\) is differentiable on \((0,1)\), find the value of \(g'\left(\frac{1}{2}\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the values of \( n \) for which the integral \( \int_0^\infty \frac{\sin(x+x^2)}{x^n} \, \mathrm{d}x \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 2)}
Given $N$ sets, each containing between 1 and 100 elements, determine the number of ways to select one element from each set such that no two selected elements are the same. For example, if the sets are $\{1, 2, 3\}$, $\{4\}$, and $\{1, 2\}$, the answer is 4, corresponding to the arrangements $(1,4,2)$, $(2,4,1)$, $(3,4,1)$, and $(3,4,2)$. Find the total number of such arrangements for the given sets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $\xi_{1},\xi_{2},\ldots$ be a sequence of independent random variables such that $P(\xi_{i} = 2^i) = \frac{1}{2^i}$ and $P(\xi_{i} = 0) = 1 - \frac{1}{2^i}$ for $i \geq 1$. Find the almost sure value of the limit $\lim_{n\to \infty}(\xi_{1}+\cdots+\xi_{n})/n$. Apply the Law of Large Numbers to determine the limiting value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the minimum number of rounds $N$ required such that you would accept to play the following game: You choose a finite sequence of natural numbers $(a_0, a_1, \ldots, a_n)$. In each round, Discriminant the cat gives you a natural number $\beta$, and you respond with the value of $\sum_{k=0}^{n} a_{k} \beta^{k}$. The game ends after $N$ rounds or when Discriminant guesses your sequence. For what values of $N$ would you accept to play this game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit: \[ \lim_{n\to\infty} \int_0^n \left( 1- \frac{x}{n}\right)^n \log \left(2+\cos\left( \frac{x}{n}\right)\right)\, dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log 3}
Find a $3 \times 3$ matrix $A$ that is invariant under any rotation, i.e., satisfies $AR = RA$ for any rotation matrix $R$. What is the form of such a matrix?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda I}
Consider a $2024 \times 2024$ grid of unit squares. Two distinct unit squares are adjacent if they share a common side. Each unit square is to be coloured either black or white. Such a colouring is called $\textit{evenish}$ if every unit square in the grid is adjacent to an even number of black unit squares. Determine the number of $\textit{evenish}$ colourings.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^2024
}
Let \( \mathcal{H} = l^2(\mathbb{N}) \). If \( N \geq 1 \) and \( L: \mathcal{H} \rightarrow \mathbb{F} \) is defined by \( L(\{\alpha_n\}) = \alpha_N \), find a vector \( h_0 \in \mathcal{H} \) such that the linear functional \( L(h) = \langle h, h_0 \rangle \) holds for all \( h \in \mathcal{H} \), where \( \langle \cdot, \cdot \rangle \) denotes the inner product on \( l^2(\mathbb{N}) \) with \( \langle x, y \rangle = \sum_{n\geq1} x(n)\overline{y(n)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e_N}
In a scalene triangle \( ABC \), one angle equals the difference between the other two angles, and one angle is twice another angle. The angle bisectors of angles \( A \), \( B \), and \( C \) intersect the circumcircle of the triangle at points \( L \), \( O \), and \( M \) respectively. Given that the area of triangle \( ABC \) is 32, find the area of triangle \( LOM \), rounding to the nearest whole number if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Calculate the limit: $$\lim_{x\to\infty} \left(\int_0^1 t^{-tx} dt\right)^{\frac{1}{x}}$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\frac{1}{e}}}
Squares $ABCD$ and $EFGH$ have a common center and $\overline{AB}\parallel \overline{EF}$. The area of $ABCD$ is $2016$, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on a side of $IJKL$. Find the difference between the largest and smallest possible integer values of the area of $IJKL$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
840
}
Evaluate the definite integral \( \int_0^{\pi} \cos(ie^{ix}) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Determine the signature of the quadratic form \( f(x, y, z) = xy + yz + xz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 2)}
Determine the lower bound for the index of the commutator subgroup of a non-abelian p-group using the conjugacy class equation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2}
Find the integer closest to the integral \( \int_{0}^{6} x^{\{\sqrt{x}\}} \, dx \), where \( \{x\} \) denotes the fractional part of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a circle, 101 nonzero integers are arranged such that each integer is greater than the product of the two integers that follow it in a clockwise direction. Determine the maximum number of negative integers that can be present among these 101 integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
$A$ and $B$ move uniformly along two straight paths intersecting at right angles in point $O$. When $A$ is at $O$, $B$ is $500$ yards short of $O$. In two minutes they are equidistant from $O$, and in $8$ minutes more they are again equidistant from $O$. Then the ratio of $A$'s speed to $B$'s speed is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2:3$}
Let \(p\) be a prime number and \((G,\cdot)\) be a group with \(p^3\) elements. We denote by \(\operatorname{C}(x)\) the centralizer of \(x\in G\). If \(|\operatorname{Z}(G)|=p\), determine the number of distinct centralizers \(\operatorname{C}(x)\) for \(x\in G\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p + 2}
Calculate \( \lim_{R \rightarrow \infty} \int\int_{D_R} \frac{x^2 \cos y}{x^2+y^2+1} \, dx \, dy \), where \( D_R = \{(r\cos \theta, r\sin \theta):0\leq r\leq R,0\leq \theta\leq \frac{\pi}{R^2} \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Solve the trigonometric equation for $x$ in degrees: $$\sin 84^\circ\cdot \sin(54^\circ-x)=\sin126^\circ\cdot \sin x.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
Compute the integral \( I = \int_0^\infty \frac{\sin^4 x}{x^3} \, dx \) using the Laplace transform method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Given a tetrahedron \( ABCD \) with edge \( AB \) as the diagonal of the base of a quadrilateral pyramid, and edge \( CD \) parallel to the other diagonal of this base with its ends on the lateral edges of the pyramid, find the smallest possible volume of the pyramid if the volume of the tetrahedron is \( V \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4V}
Given a linear transformation \( T \) defined by \( T(f(t)) = f(3t - 2) \), find the determinant of \( T \) with respect to the basis \( \{1, t, t^2\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Determine the closure of the set \( W = \{ f \in C[0,1] \mid f(0) = 0 \} \) with respect to the \( \|\cdot\|_1 \) norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{L^1[0,1]}
Let $p = 2027$ be the smallest prime greater than $2018$, and consider the polynomial $P(X) = X^{2031} + X^{2030} + X^{2029} - X^5 - 10X^4 - 10X^3 + 2018X^2$. Define $\mathrm{GF}(p)$ as the integers modulo $p$, and $\mathrm{GF}(p)(X)$ as the set of rational functions with coefficients in $\mathrm{GF}(p)$. Let $D: \mathrm{GF}(p)(X) \to \mathrm{GF}(p)(X)$ be a function such that for any $f, g \in \mathrm{GF}(p)(X)$ with $g \neq 0$, \[ D\left(\frac{f}{g}\right) = \frac{D(f) \cdot g - f \cdot D(g)}{g^2} \] and for any nonconstant polynomial $f$, $D(f)$ is a polynomial with degree less than that of $f$. If the number of possible values of $D(P(X))$ can be expressed as $a^b$, where $a$ and $b$ are positive integers with $a$ minimized, find $ab$. \(\text{Proposed by Brandon Wang}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2027}
Two players each have a randomly shuffled deck of 52 cards. They compare the cards in order, one at a time, checking if there is a match in rank (suit doesn't matter). What is the probability that there is no match in rank for any of the 52 card comparisons?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-4}}
Two pulleys with diameters of 80 mm and 200 mm are connected by a belt that is 1500 mm long. Calculate the distance between the axes of the pulleys, assuming the belt is tight. Provide your answer in millimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{527}
Evaluate the limit: \[ \lim_{n\to\infty} \int_0^\infty \frac{n\sin x}{1+n^2x^2}\, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Jeffrey, being from Tennessee, sees 10s everywhere he looks. If he assigns to each of his 10000 lovers a unique integer ID number from 1 to 10000, how many of them will have the sequence "10" in their ID?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Find the limit of the bounded solution $u(x,t)$ of the heat equation $u_t = u_{xx}$ with the initial condition $u(x,0) = \frac{e^{2x}-1}{e^{2x}+1}$ as $t \to \infty$, specifically for $x = 1$. Express your answer as a single value or expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the conditional expectation \( \mathbb{E}\left[ \int_0^1 (W_t+1)^2 \, dt \big | \mathscr{F}_0 \right] \), where \( W_t \) is a standard Brownian motion and \( \mathscr{F}_0 \) is the sigma-algebra generated by \( W_0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Evaluate the contour integral \( \int_{|z|=2} z^n(1 - z)^m\ dz \) where \( m \) and \( n \) are negative integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of distinct ways to fill a 3 × 3 grid with 0s and 1s, given that the center cell must contain a 0 and two grids are considered identical if one can be transformed into the other through a symmetry operation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Determine the values of the real number $p$ for which the integral \[ I_p = \int_{0}^{1}\frac{r^{p+1}}{(1-r^{2})^{\frac{p}{2}-\frac{1}{2}}}dr \] converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 3)}
Determine the radius of convergence of the series \(\sum\limits_{n=0}^\infty 3^nz^{n!}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the size of the conjugacy class of the permutation \((123)(456)\) in the alternating group \(A_7\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{280}
Determine the minimum possible degree of a polynomial function $f(z)$ on the complex plane, given that $f(z) = \delta$ whenever $z^3 = \gamma$ and $z^3 = \gamma$ has exactly 3 distinct solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the real values of \( a \) (with \( a \neq -1 \)) for which the following limit holds:

\[
\lim_{n \to \infty} \frac{1^a + 2^a + \cdots + n^a}{(n+1)^{a-1} \left[\sum_{i=1}^n (na+i)\right]} = \frac{1}{60}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the condition on a linear operator $A$ such that the operator $e^{tA}$ is norm-preserving for all $t \le 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A^* = -A}
Evaluate the improper integral \(\int_{0}^{\infty} \left(e^{-\frac{1}{x^2}}-e^{-\frac{4}{x^2}}\right) dx\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\pi}}
Evaluate the limit without using L'Hospital's rule: \[ \lim_{x \to 2} \frac{\sqrt{17 - 2x^{2}}\sqrt[3]{3x^{3} - 2x^{2} + 8x - 5} - 9}{(x - 2)^{2}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{11}{3}}
In a Cartesian coordinate system, you can move from \((a,b)\) to \((a+1,b)\), \((a,b+1)\), or \((a+1,b+1)\), but the path must not form a right angle at any point. Determine the number of ways to move from \((0,0)\) to \((5,5)\) following these rules.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83}
Given the joint probability density function:
\[
f_{X,Y}(x,y) = \begin{cases} 
ye^{-xy-y} & \text{if } x > 0, y > 0 \\
0 & \text{otherwise}
\end{cases}
\]
Calculate the distribution of the product $XY$. Assume $X$ and $Y$ are dependent random variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-z}}
Evaluate the sum \( \sum_{n=1}^{2019} \lambda(n) \left\lfloor \frac{2019}{n} \right\rfloor \), where \( \lambda(n) = (-1)^{\Omega(n)} \) and \( \Omega(n) \) is the number of prime factors of \( n \) counted with multiplicity. Use the properties of \( \lambda(n) \) to find the value of the sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
In a 4D spacetime, a signal spreads uniformly in all directions at an unknown velocity. What is the minimum number of spacetime points required to determine the origin of the signal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the number of possible Beta-reductions for the lambda calculus expression \((\lambda x.\lambda y.y)(\lambda x.x)((\lambda x.x)(\lambda y.y))\). Consider the order of evaluation in your calculation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Simplify the infinite series: $$\sum_{i=0}^\infty \frac{4^i}{(i!)^2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_0(4)}
Evaluate the integral: $$\int_{-\pi/2}^{\pi/2}\sin(\tan(\tan(x)))\tan(x)dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Suppose $X$ and $Y$ are independent random variables, each uniformly distributed over the interval $(0,1)$. Calculate the expected value $\mathbb{E}[X | \max \{X, Y\}]$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
Determine the order of the smallest subgroup of the symmetric group $S_4$ that contains the set of permutations \( \{ (12)(34), (14)(32), (24), (31) \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let \(u(x,y)=x^3+ax^2y+bxy^2+2y^3\) be a harmonic function and \(v(x,y)\) be its harmonic conjugate. If \(v(0,0)=1\), find \(|a+b+v(1,1)|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
How many ordered pairs of sets(A,B)have the properties:
\( A \subseteq  \{ 1,2,3,4,5,6\} \)
 \( B \subseteq  \{ 2,3,4,5,6,7,8\} \)
 \( A \cap  B \) has exactly3elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
720.

}
Determine the supremum of \(|q(z_1,z_2,z_3)|\) over the set \(\{|z_i| = 1, i = 1,2,3\}\), where \(q(z_1,z_2,z_3) = z_1^2z_2^2z_3^2 - z_1^6 - z_2^6 - z_3^6\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{13}}
Evaluate \( \frac{\partial^6 f}{\partial x^4 \partial y^2}(0,0) \) for the function \( f(x,y) = \frac{1}{1-x^2y} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Calculate the surface integral \( \iint_S F \cdot dS \) for the vector field \( F(x, y, z) = xi + yj + zk \), where \( S \) is the triangular surface with vertices \((0, 1, 1), (1, 0, 1), (1, 1, 0)\) and \( n \) is the unit normal to \( S \) pointing towards the origin. Use the parameterization \( r(u, v) = (1-v)i + (u+v)j + (1-u)k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Determine the closure of the subspace $X = \mathrm{lin}\{e_n : n \in \mathbb{Z}^+\}$ in $l^\infty$, where $e_j$ is the sequence with a 1 in the $j$-th position and 0 elsewhere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c_0}
Determine the number of linearly independent matrices in a commuting family of $4 \times 4$ complex matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Calculate the degree of the field extension \([\mathbb{Q}(\sqrt[n]{m}):\mathbb{Q}]\), where \(m\) is an integer with a prime factorization of the form \(m = p \prod p_i^{n_i}\), and \(p \neq p_i\). Assume \(n\) is a natural number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Consider a Markov chain where each day there is a 0.2 probability of rain in the morning. The probability of rain in the afternoon given it rained in the morning is 0.6, and the probability of rain in the afternoon given it did not rain in the morning is 0.3. John goes to the office in the morning and leaves in the afternoon, and he has a total of 3 umbrellas. Let \( X_n \) be the number of umbrellas in his office at the \( n \)-th night. Given the transition probability matrix:
\[
P_{transition} = \begin{pmatrix}0.92 & 0.08 & 0 & 0 \\ 0.24 & 0.68 & 0.08 & 0 \\ 0 & 0.24 & 0.68 & 0.08 \\ 0 & 0 & 0.24 & 0.76 \end{pmatrix}
\]
What is the proportion of time John has 0 umbrellas in his office at night?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{40}}
Square \( ABCD \) has center \( O \). Points \( P \) and \( Q \) are on \( AB \), \( R \) and \( S \) are on \( BC \), \( T \) and \( U \) are on \( CD \), and \( V \) and \( W \) are on \( AD \), so that \( \triangle APW \), \( \triangle BRQ \), \( \triangle CTS \), and \( \triangle DVU \) are isosceles and \( \triangle POW \), \( \triangle ROQ \), \( \triangle TOS \), and \( \triangle VOU \) are equilateral. What is the ratio of the area of \( \triangle PQO \) to that of \( \triangle BRQ \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:1}
What linear algebra operation transforms a matrix $V$ such that $V'^T M V' = I$, where $I$ is the identity matrix, given that $V^T M V = E$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{V' = V E^{-1/2}}
Given two distributions $p$ and $q$ over a common finite set $X$ with a total variation distance $TV(p, q) = \epsilon$, and a function $f: X \to \mathbb{R}$ such that $0 \leq \sum_{x \in X} f(x) p(x) \leq 1$ and $0 \leq \sum_{x \in X} f(x) q(x) \leq 1$, determine the tightest upper bound on the expression \( \Big|\sum_x f(x)(p(x)-q(x))\Big| \) in terms of $\epsilon$. Consider how the condition $TV(p, q) = \epsilon$ can be used to refine the bound beyond the trivial upper bound of 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of positive solutions to the equation \( \cos(97x) = x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
In the binomial expansion of \((1+\sqrt{7})^{207}\), determine the value of \(k\) for which the term \(C_{207}^{k}(\sqrt{7})^{k}\) is maximized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Estimate the value of $\sqrt{90} - \sqrt{88}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["0.11"]}
Given a list of prime numbers under 100,000, determine the last prime number that appears in the decimal expansion of pi.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99991}
Determine the smallest integer $k$ such that the semigroup $C_n$ is equal to the union of the $k$-th powers of the set $A$, where $A = \{\begin{bmatrix}2\\1  \end{bmatrix},\begin{bmatrix}3\\2  \end{bmatrix},\cdots, \begin{bmatrix}n\\n-1  \end{bmatrix}\}$ and $C_n$ is the semigroup of all order-preserving and decreasing transformations in the full transformations semigroup $T_n$. \[ C_n = \bigcup\limits_{i=1}^{k} A^k \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n-1}
Determine the 4-torsion points of the elliptic curve $E: y^2 = x^3 - 5$ over the field of rational numbers $\mathbb{Q}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathcal{O}}
Determine whether the proportion of finite topologies of size \( n \) that are \( T_{0} \) approaches 1 or 0 as \( n \) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a Markov chain \( X_n \) with state space \( S = \{1, 2, 3, 4\} \) and transition matrix \( P \) as follows:
\[
P = \begin{pmatrix}
  \frac{1}{4} & 0 & \frac{1}{2} & \frac{1}{4} \\
  \frac{1}{4} & \frac{1}{4} & \frac{1}{2} & 0 \\
  0 & \frac{2}{3} & \frac{1}{3} & 0 \\
  0 & \frac{1}{2} & 0 & \frac{1}{2}
\end{pmatrix}
\]
Calculate \( EX_{\tau} \), where \( \tau = \inf \{ n : X_n = 2 \text{ or } X_n = 3 \} \), given that the initial state is \( X_0 = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{3}}
Calculate the entropy for a random variable with a delta function probability density function given by $\Pr(X=x) = \delta(x-x_0)$. What is the entropy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}
Given the monthly demand for the Instant Pie Maker is \(D(x,y)= \frac{1}{100}ye^{xy/800} \text{ units}\), where \(x\) dollars are spent on online advertisements and \(y\) dollars are spent on social media marketing, and the spending is defined as \(x=15+t^{1/2}\) and \(y=t\cos(t)\) dollars, find the rate at which the demand is changing with respect to time 16 months from now. Round your answer to 3 decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.016}
Let \( \omega \) be the incircle of triangle \( {ABC} \) . Line \( {l}_{b} \) is parallel to side \( {AC} \) and
 tangent to \( \omega \) . Line \( {l}_{c} \) is parallel to side \( {AB} \) and tangent to \( \omega \) . It turned ou
 that the intersection point of \( {l}_{b} \) and \( {l}_{c} \) lies on circumcircle of \( {ABC} \)
 Find all possible values of \( \frac{{AB} + {AC}}{BC} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2
}
Two players, A and B, alternately draw from the set \( S = \{0, 1\} \) with replacement. A player wins by drawing two consecutive zeros. Player A draws first. What is the probability that player A wins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{5}}
Determine the number of elements in the quotient ring \( \Bbb Z_5[i]/\langle 1+i\rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}