inclusionAI/Ring-lite-rl-data huggingface dataset:
Assume the function $f : \mathbb{R} \rightarrow \mathbb{R}$ is continuously differentiable on $\mathbb{R}$. Given that $f(0) = 0$ and $f(x)f'(x) \leq 2$ for all $x \in \mathbb{R}$, determine the largest possible value of $f(4)$. \( \text{Provide your answer in the form } \boxed{\text{value}}. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In triangle $ABC$, $AL$ and $BM$ are the angle bisectors. The circumcircles of triangles $ALC$ and $BMC$ intersect again at point $K$, which lies on side $AB$. Determine the measure of angle $ACB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Let $ABC$ be a triangle with side lengths $AB=7$, $BC=8$, $AC=9$. Draw a circle tangent to $AB$ at $B$ and passing through $C$. Let the center of the circle be $O$. The length of $AO$ can be expressed as $\frac{a\sqrt{b}}{c\sqrt{d}}$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a,c)=gcd(b,d)=1$ and $b,d$ are not divisible by the square of any prime. Find $a+b+c+d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76}
Evaluate the expression \( \frac{u^{2n+1}\ln(u)}{2n+1} \bigg|^{u=1}_{u=0} \) at \( u=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \( \int_0^{\infty}{xI(2x)e^{-x^2}\,dx} \), where \( I(x) = \frac{1}{\pi}\int_0^{\infty}{e^{x\cos\theta}\,d\theta} \) is a Bessel function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{e}{2}}
Determine the number of solutions to the polynomial congruence \( x^4 - 5x - 6 \equiv 0 \pmod{100^{100}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the generator of the kernel of the evaluation homomorphism \( \phi: \mathbb{R}[x] \to \mathbb{R} \) defined by \( f(x) \to f(1+\sqrt{2}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x - 1 - \sqrt{2}}
Calculate the numerical solution at \(x=1.2\) for the initial value problem \(y''=\frac{2}{x}y'-\frac{2}{x^2} y - \frac{1}{x^2}\), with \(y(1)=0\) and \(y'(1)=1\), using the Modified Euler's method. Use a step length of \(h=0.2\) and ensure the solution is accurate to 6 decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.220000}
Determine the ring to which the quotient ring \( R[X,Y,Z,T]/(X-Y^2,T-Y^4,T^3-Z) \) is isomorphic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{R[Y]}
Determine the number of elements in the quotient module \( \mathbb{Z}[1/6]/(2020)\mathbb{Z}[1/6] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{505}
Let $n$ be an even positive integer. Alice and Bob play a game where Alice chooses a set $S$ containing $m$ integers and announces it to Bob. Bob then chooses $i \in \{1,2,\dots, n\}$ that has not been chosen and sets the value of $v_i$ to either $0$ or $1$. After all $v_1, v_2, \dots, v_n$ have been set, the expression \( E = v_1 \cdot 2^0 + v_2 \cdot 2^1 + \dots + v_n \cdot 2^{n-1} \) is calculated. Determine the minimum $m$ such that Alice can always ensure that $E \in S$ regardless of Bob's choices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n/2}}
Find all positive integers \( n \) such that there exists a sequence \((x_k, y_k)\) defined by either \((P(x_k), Q(y_k)) = (x_k + 1, x_k^2 + 1)\) or \((Q(x_k), P(y_k)) = (x_k^2 + 1, x_k + 1)\), starting with \((x_1, y_1) = (1, 3)\), for which \( x_n = y_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Consider the system of equations:
\[x_{1}+x^2_{2}+x^3_{3}+x^4_{4}+x^5_{5} =5\]
and
\[x_{1}+2x_{2}+3x_{3}+4x_{4}+5x_{5}=15\]
where \(x_{1},x_{2},x_{3},x_{4},x_{5}\) are positive real numbers. Determine the number of solutions this system has.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( S = \{1, 2, 3, 4\} \). A sequence of \( n \) terms \( a_1, a_2, \ldots, a_n \) has the property that for any non-empty subset \( B \) of \( S \), there exist \( |B| \) consecutive elements in the sequence that exactly form the set \( B \). Determine the minimum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the value of $k$ such that the equation \( \left(\binom{n}{2}-\binom{k+1}{3}\right)x=\binom{k+1}{2}-1 \) has no solution for \( x \in \mathbb{N} \), where \( n \in \mathbb{N} \) and \( n > 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the $2$-adic valuation of $x^2 - 3$ in the field extension $\mathbb{Q}[x]/(x^3 - 3)$, given that $x$ has $2$-adic valuation $0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the asymptotic order of the function defined by the recurrence relation:
\[ T(n) = n + \sum_{k=1}^n [T(n-k) + T(k)] \]
with the initial condition \( T(1) = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(3^n)}
Find the limit of the expression \[ c = \frac{\sqrt{n-1}\Gamma\left(\frac{n-1}{2}\right)}{\sqrt{2}\Gamma\left(\frac{n}{2}\right)} \] as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the values of \( \rho \in \mathbb{R} \) for which the function \( f(x) = x + e^{\rho|x|} \) is convex on \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, \infty)}
Let $k=\mathbb{Q}(X_1,X_2,X_3,\ldots,X_{n^2})$ and $M=\mathcal{M}_{n,n}(k)$ denote the $k$-algebra of $n \times n$ matrices with coefficients in $k$. Let $A \in M$ be defined by \[ A=\begin{pmatrix} X_1 & X_2 & X_3 & \ldots & X_n \\ X_{n+1} & X_{n+2} & X_{n+3} & \ldots & X_{n+n} \\ X_{2n+1} & X_{2n+2} & X_{2n+3} & \ldots & X_{2n+n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ X_{(n-1)n+1} & X_{(n-1)n+2} & X_{(n-1)n+3} & \ldots & X_{n^2} \end{pmatrix} \] and let $C=\{ B \in M \mid AB=BA \}$ be the commutant of $A$. Determine the dimension $d$ of $C$ as a $k$-vector space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Determine the smallest positive constant \(c\) such that for any simple graph \(G = (V, E)\), if \(|E| \geq c|V|\), then \(G\) contains two cycles with no common vertex, and one of them contains a chord.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Solve the exponential equation \(\frac{e^{2x}}{x+1}=\frac{4e^{\frac{5}{2}}}{5}\) without using the Lambert W function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.62}
Let $A$ be an integral domain with field of fractions $K$. If $M$ is a finitely-generated $A$-submodule of $K$ that is locally free of rank $r$, determine the dimension of the $K$-vector space $M \otimes_A K$. \( \text{Provide your answer as a single integer.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the surface integral \( \iint_S z^2 \, dS \) where \( S \) is the surface of the cube defined by \([-1,1] \times [-1,1] \times [-1,1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{40}{3}}
Let $n$ be a positive integer. Find the minimum value of \( \sum_{k=1}^n \frac{x^k_k}{k} \), where $x_1, x_2, \ldots, x_n$ are positive real numbers such that \( \sum_{k=1}^n \frac{1}{x_k} = n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sum_{k=1}^n \frac{1}{k}}
Determine the number of continuous functions \( f: [0, 1] \to \mathbb{R} \) that satisfy the integral equation:
\[ \int_0^1 f(x) \, dx = \frac{1}{3} + \int_0^1 f^2(x^2) \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that \(f(x)= \lim_{n\to \infty}\dfrac{\left(1- \cos \left(1- \tan \left(\dfrac {\pi}{4}-x\right)\right)\right)(x+1)^n+ \lambda\sin((n - \sqrt{n^2 -8n})x)}{x^2(x+1)^n+x}\) is continuous at \(x=0\), find the value of \(f(0) + 2\lambda\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the number of field homomorphisms from a finite field with nine elements to a finite field with four elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
If six points are placed either along the boundary or inside of a unit square, such that the minimum distance of each of these points from the remaining five points is the same, what is the numerical value of this distance?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the approximate number of lattice points inside a regular hexagon with radius $r$ on a hexagonal lattice where the distance between adjacent points is 1 cm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3r^2}
What is the Kleisli category of the monad \( \mathbb{T} : \mathbf{Set} \to \mathbf{Set} \) associated with the Lawvere theory for the category of sets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbf{Set}}
In a Markov chain modeling a population's genotype distribution, two individuals with genotype Aa reproduce, resulting in various states: (AA, AA), (AA, Aa), (Aa, Aa), (AA, aa), (Aa, aa), and (aa, aa), with the first and last states being absorbing. If the process starts in the state (Aa, Aa), calculate the probability that it will eventually reach the state (AA, AA). Assume each child receives one allele from each parent with a 50% chance of getting either allele.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Suppose $X$ and $Y$ are two independent non-negative random variables such that $E[X] = E[Y] = 1$. Determine the value of $\sup_X \{ \inf_Y \{P(X > Y)\}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Susan is presented with six boxes \( B_1, \ldots, B_6 \), each of which is initially empty, and two identical coins of denomination \( 2^k \) for each \( k = 0, \ldots, 5 \). Compute the number of ways for Susan to place the coins in the boxes such that each box \( B_k \) contains coins of total value \( 2^k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
32
}
Determine the real numbers \(a\) for which the series \(\sum \frac{\sin(ka)}{\log(k)}\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Find the limit of the sequence defined by \(S_n= n\,p^n \sum_{k=n}^{+\infty} \frac{1}{k}\, \binom{k-1}{n-1} \, (1-p)^{k-n}\) where \(0<p<1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p}
Using the Newton-Raphson method, determine the maximum number of iterations required to find the root of the equation $g(x) = 0$ within the interval $(1, 2)$ to a precision of 6 decimal places after rounding. The function $g(x)$ satisfies $|g'(x)| \ge 4$ and $|g''(x)| \le 3$ for all $x$ in the interval, and the method converges for all initial approximations in $(1, 2)$. What is the maximum number of iterations needed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the weight of the topological space $X = (0,1) \times (0,1)$ with the dictionary order topology.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
Three bells begin to ring simultaneously. The intervals between strikes for these bells are,respectively, \( \frac{4}{3} \) seconds, \( \frac{5}{3} \) second and
 2 seconds. Impacts that coincide in time are perceived as one. How many beats will be heard in 1 minute? (Include first and
 last.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
85
}
Determine the number of isomorphism classes of function fields of degree 3 over the rational function field $F = \mathbb{C}(t)$, given that they are ramified only at the points 1 and -1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the rank of the set of functions from \(\omega\) to \(\omega\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\omega + 2}
Evaluate the integral \( \int_{-\infty}^{\infty} dk \left[ \Im\left( \frac{1}{(\omega+i\eta+k)(\omega+i\eta+k+q)} \right) \right] \), where \( \eta \) is an infinitesimal parameter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of zeros of the function \( f(z) = z^5 + 2z^3 - z^2 + 2z + 5 \) within the region \( \{ z \in \mathbb{C} : \text{Re}(z) > 0, \text{Im}(z) > 0 \} \) using the Argument Principle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the volume bounded by the surface \( \sqrt{x} + \sqrt{2y} + \sqrt{3z} = 1 \) in the region where \( x, y, z \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{540}}
The 83 trainees from the Animath workshop each choose an activity for the free afternoon from 5 activities offered. The activities are: Beach, Laser Tag, Movie, Romanesque Church, and Shopping. The following conditions apply:

- Shopping was at least as popular as Laser Tag.
- Movie tickets are sold in batches of 6.
- At most 5 students go to the beach.
- At most one student visited a Romanesque church.

When writing the list of activities in alphabetical order, the number of students corresponding to each activity is written. How many different lists of numbers can be written?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3570}
In a basis for $\Bbb R^d$ that includes the vector with the first $k$ coordinates equal to $1$ and the vectors with $1$ as the first component and $-1$ as the $i$th component for $2 \leq i \leq k$, determine the number of linearly independent vectors such that the sum of their first $k$ components is zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d - 1}
Determine the closure of a given set $B$ in $\mathbb{R}^n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\overline{B}}
Let $G = A_5$ and $H = \langle (1,2,3,4,5) \rangle$. If $g \in N_G(H)$ is an element of order $5$, compute the order of the group $H \langle g \rangle$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given a nilpotent matrix $B$ with a minimal polynomial $\lambda^6$ and the largest Jordan block of size 6, determine the rank of $B^4$. It is known that the difference between the nullity of $B^k$ and $B^{k-1}$ indicates that there are 6 Jordan blocks of size at least $k \times k$. What is the rank of $B^4$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Solve for \(x\) in the equation: \[ \frac{60_{10}!}{12_2!} = 5^x \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Compute
$ \lim\limits_{h\to0}\frac{\sin(\frac{\pi}{3}+4h)-4\sin(\frac{\pi}{3}+3h)+6\sin(\frac{\pi}{3}+2h)-4\sin(\frac{\pi}{3}+h)+\sin(\frac{\pi}{3})}{h^{4}}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{\sqrt{3}}{2}$}
Evaluate the integral \( \iint \frac{y^3}{x^2} \, dA \) over the region bounded by the curves \( x=2y,\; x=y,\; x=y^2,\; x=2y^2 \) using a change of variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{16}}
Given that \(\mathbb{P}(\limsup_n A_n)=0.7\) and \(\mathbb{P}(\liminf_n B_n)=0\), find \(\mathbb{P}(\limsup_n(A_n\cap B_n))\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the period of the function $g(x)$ that satisfies the condition $g(m-x) = g(n+x)$ for any positive integers $m$ and $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a $2 \times 2019$ table, fill each cell with a number such that the top row contains 2019 distinct real numbers, and the bottom row contains the same numbers in a different order. Each column must contain two different numbers whose sum is rational. What is the maximum number of irrational numbers that can be in the first row?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
Identify the abstract algebraic concept that generalizes the idea of combining vectors end-to-start while preserving their origins.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{abelian group}
Evaluate the limit \( \lim_{{n\to\infty}}\frac{\int_{0}^{1}\left(x^2-x-2\right)^n\,dx}{\int_{0}^{1}\left(4x^2-2x-2\right)^n\,dx} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit: \[ \lim_{n \to \infty} \int_{0}^{1} \left[x^n + (1-x)^n \right]^{1/n} \, \mathrm{d}x \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
Determine the value of \( k \) such that the limit \( \lim_{x\to 0} \frac{\sqrt[5]{x+\sqrt[3]{x}}-\sqrt[3]{x+\sqrt[5]{x}}}{x^{k}} = A \) exists, where \( A \) is a non-zero constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{15}}
Find the spectrum of the operator $T = S + S^*$, where $S$ is the unilateral shift operator on $\mathcal{l}^2$ and $S^*$ is its adjoint.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-2, 2]}
For a group $G$, if $ba = a^6b$, simplify the expression $a^ib^ja^kb^lb^{-j}a^{-i}b^{-l}a^{-k}$. What is the simplified form?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^{i(1 - 6^l) + k(6^j - 1)}}
Triangle $ABC$ has integer side lengths, with $BC = 100 \times 7 - 4$. Given that $\angle ABC$ is a right angle, let $r$ and $s$ denote the inradius and semiperimeter of $\triangle ABC$, respectively. Find the perimeter of the triangle $ABC$ that minimizes $\frac{s}{r}$. \(T = 7\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2378}
Determine the number of lines that are both normal and tangent to the curve \( y^2 = x^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
It is given a triangle $ABC$ whose circumcenter is $O$ and orthocenter $H$.\nIf $AO=AH$ find the angle $\\hat{BAC}$ of that triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The angle \( \hat{BAC} \) is \( 60^\circ \).

}
What is the maximum number of distinct prime factors that a number less than $2^{64}$ can have? In other words, find the largest value of $\omega(n)$ for $n < 2^{64}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A car starts at a position \( b \), where \( b \) is normally distributed as \( N(0, 6) \). The car then drives a distance \( d \), which is normally distributed as \( N(0, 5) \). If \( d \) is positive, the car moves up; otherwise, it moves down. Given that \( b \) and \( d \) are statistically independent and the car's final position is \( c = 6 \), find the most likely initial position \( b \) of the car.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{36}{11}}
175 schoolchildren arrived at a health camp. Some children know each other, and some do not. It is known that any six schoolchildren can be accommodated in two three-person rooms such that all the schoolchildren in one room will know each other. What is the minimum number of pairs of acquainted schoolchildren that could have come to the camp?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7569}
Compute the depth of the polynomial ring $k[x,y]$ with respect to the ideal $(x,y^2)$, where $k$ is a field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A fair 20-sided die, numbered from 1 to 20, is rolled 20 times. The results of the rolls are summed. Determine the number below which 99% of the sums will fall.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{270}
Let $X=l^2(\mathbb{Z})$ be the Hilbert space of all real sequences $x= (x_{k})_{k}$ $(k\in \mathbb{Z})$ for which the norm $\|x\|= \left( \sum_{k=-\infty}^{\infty} |x_{k}|^{2}\right)^{\frac{1}{2}}$ is finite. Denote by ${e_{k}:k\in\mathbb{Z}}$ the canonical basis in $X$, and define $L: X\longrightarrow X$ by
$$L
\left(\sum_{k=-\infty}^{\infty}x_{k}e_{k}
\right)=\sum_{k=-\infty}^{\infty}(x_{k}-x_{k-1})e_{k}$$
Determine the norm $\|L\|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the triple integral \( \iiint x\ \text{d}x \) over the region bounded by the surfaces \( z = x^2 + y^2 \) and \( 4x + z = -2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4\pi}
Determine the number of ring homomorphisms from the ring \( \mathbb{Z}[X] / (X^{2} + 1) \) to \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of Q-field homomorphisms from the algebraic closure of the rational numbers to the complex numbers that keep the rational numbers fixed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Determine the number of zeros (counting multiplicities) of the polynomial \(f(z) = z^{100} + 8z^{10} - 3z^3 + z^2 + z + 1\) within the closed unit disc \(|z| < 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Calculate the class number of a field using the Minkowski bound of 6 and the ideal norms: 1, 3, 4, and 6.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Currently, Selena’s analog clock says $4{:}00$. Suddenly her clock breaks, so the hour hand moves $12$ times as fast as it normally does, but the minute hand stays the same speed. Find the degree measure of the smaller angle formed by the minute and the hour hand $2024$ minutes from now.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120

}
Given the function for accumulated value over time, \( a(t) = xt^{2} + yt + z \), where \( 100a(2) = 152 \) and \( 200a(4) = 240 \), find the accumulated value at time 8 of an investment of $1600 made at time 6. Use the conditions to determine the constants \( x, y, \) and \( z \), and then calculate the accumulated value at time 8.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-78400}
Find the derivative of the vector expression $A^Tx$ with respect to the matrix $A$, where $A \in \mathbb{R}^{n \times m}$ and $x \in \mathbb{R}^n$. Express your answer as $\frac{d(A^Tx)}{dA}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_m \otimes x^T}
Find the Cumulative Distribution Function (CDF) of the absolute difference between two independent uniform random variables $X_1$ and $X_2$, both distributed on the interval [0, 1].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2z - z^2}
Evaluate the limit \( \lim_{n\to\infty}\frac{a_{12}(n)}{a_{22}(n)} \), where \( a_{ij}(n) = \begin{pmatrix}2 & 1\\ 1 & 0\end{pmatrix}^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \sqrt{2}}
Let $S_n = \sum_{i=1}^n X_i$ be a symmetric random walk. Compute the conditional expectation $E(X_i \mid \mathcal{G}_{n+1})$ for $1 \leq i \leq n$, where $\mathcal{G}_{n} = \sigma(S_{n},S_{n+1},\dots)$. Assume $S_n$ is a martingale.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{S_{n+1}}{n+1}}