inclusionAI/Ring-lite-rl-data huggingface dataset:
Calculate the value of the series:

\[-50^2 - 2 \cdot 49^2 - 3 \cdot 48^2 - \dots - 0 \cdot 0^2 + 1^2 \cdot 51 + 2^2 \cdot 52 + \dots + 50^2 \cdot 100\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3208325}
Barney washes his kitchen towels once a week. He owns eighteen towels and uses a certain number of towels at a time that he changes to clean towels daily. He missed one week of laundry. The following week, Barney will not have clean towels for 5 days. How many towels does Barney use at a time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral $\iiint_V xe^{-(x^2+y^2+z^2)^2}\,{\rm d} V$ using spherical coordinates, where V is the solid that lies between y=0 and the hemisphere $x^2+y^2+z^2=4$ in the right half space $(y>0)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The function $f(x) = (m^2 - m - 1)x^m$ is a power function, and it is a decreasing function on $x \in (0, +\infty)$. The value of the real number $m$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Suzie flips a fair coin 6 times. The probability that Suzie flips 3 heads in a row but not 4 heads in a row is given by  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Let point O be located inside a regular tetrahedron ABCD. G is the centroid of the tetrahedron. The line OG intersects the faces of the tetrahedron at points A1, B1, C1, D1. Find the value of (A1O/A1G) + (B1O/B1G) + (C1O/C1G) + (D1O/D1G).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the surface integral \( \int_{F} a \cdot dn \) for the vector field \( a = [-z^2 - 2z, -2xz + 2y^2, -2xz - 2z^2]^T \) over the area \( F \) on the cylinder \( x^2 + z^2 = 4 \), which is above the ground plane \( z = 0 \), in front of the plane \( x = 0 \), and between the planes \( y = 0 \) and \( y = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{136}{3}}
Some unit-radius circles are drawn on the plane, and we color the center of each circle blue. We mark some points on the circumference of the circles in red such that exactly 2 red points lie on each circumference. What is the maximum number of blue points if there are a total of 25 colored points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Four students participate in a competition where each chooses one question from two options, A and B. The rules result in the following point system: 21 points for correct A, -21 points for incorrect A, 7 points for correct B, and -7 points for incorrect B. If the total score of the four students is 0, calculate the number of different scoring situations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
A competition consists of \( n \) true/false questions. After analyzing the responses of 8 participants, it was found that for any two questions, exactly two participants answered "True, True"; exactly two participants answered "False, False"; exactly two participants answered "True, False"; and exactly two participants answered "False, True". Determine the maximum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the values of $n$ for which the determinant of the matrix defined by $a_{i,j} = (2i+3j)^3$ is zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 5}
Compute the determinant of the  $n\times n$  matrix  $A=(a_{ij})_{ij}$ ,  $$ a_{ij}=\begin{cases}
(-1)^{|i-j|} & \text{if}\, i\ne j,
2 & \text{if}\, i= j.
\end{cases} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + 1}
Determine the value of $R$ that minimizes the residuals in the following model:
\[
\tan(\Delta RA)=\frac{\left(\frac{R}{2115}\right) \cos(14.174550 - 0.003488 t) \sin(0.814907 - 0.000468 t) - \sin(15.04 t)}{\left(\frac{R}{2115}\right) \cos(14.174550 - 0.003488 t) \cos(0.814907 - 0.000468 t) - \cos(15.04 t)}
\]
where $t$ is a variable of time, and all other variables are constants. Find the value of $R$ that minimizes the values of $\Delta RA$ across all times $t$. Assume $R_E$ and $\lambda$ are constants such that $R_E \cos \lambda = 2115$. Provide the value of $R$ that achieves this minimum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2115}
A sequence of distinct circles  $\omega_1, \omega_2, \cdots$  is inscribed in the parabola  $y=x^2$  so that  $\omega_n$  and  $\omega_{n+1}$  are tangent for all  $n$ .  If  $\omega_1$  has diameter  $1$  and touches the parabola at  $(0,0)$ , find the diameter of  $\omega_{1998}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3995}
a, b, and c are partners who have invested in a business venture. The profit distribution ratio for a is 2/3, while b and c split the remaining 1/3 equally. The partnership agreement states that the profit distribution ratios will change after a specified period. After the first year, a's ratio will drop by 10%, while b and c's ratios will both increase by 5%.

Additionally, each partner has a different investment period in the business. a invested in the business for 2 years, whereas b invested for 3 years, and c invested for 4 years.

a's income is increased by Rs. 200 when the rate of profit rises from 5% to 7%. Calculate the initial capital invested by a.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15,000}
128 students represent x percent of the boys at a school. The boys at the school make up a certain percentage of the total school population of 160 students. What percentage of the total school population do the boys make up?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8000/x}
Find four distinct 3-digit numbers such that the sum of any three of them is divisible by the fourth number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100, 200, 300, 600}
In triangle  $ABC$  we have  $AB=36$ ,  $BC=48$ ,  $CA=60$ . The incircle of  $ABC$  is centered at  $I$  and touches  $AB$ ,  $AC$ ,  $BC$  at  $M$ ,  $N$ ,  $D$ , respectively.  Ray  $AI$  meets  $BC$  at  $K$ . The radical axis of the circumcircles of triangles  $MAN$  and  $KID$  intersects lines  $AB$  and  $AC$  at  $L_1$  and  $L_2$ , respectively. If  $L_1L_2 = x$ , compute  $x^2$ .

*Proposed by Evan Chen*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{720}
平面上给出 10 个点,任何三点都不共线,作 4 条线段,每条线段连接平面上的两个点。这些线段是任选的,且这些线段都有相同的被选的可能性.由这些线段中的某三条线段构成以给定 10 点中三点为顶点的三角形的概率为 $\frac{m}{n}$, 其中 $m, n$ 为互质的正整数, 求 $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{489}
A  $9 \times 9$  square consists of  $81$  unit squares. Some of these unit squares are painted black, and the others are painted white, such that each  $2 \times 3$  rectangle and each  $3 \times 2$  rectangle contain exactly 2 black unit squares and 4 white unit squares. Determine the number of black unit squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Given an angle \(\theta\), consider the polynomial
$$
P(x)=\sin (\theta) x^{2}+(\cos (\theta)+\tan (\theta)) x+1.
$$

Given that \(P\) only has one real root, find all possible values of \(\sin (\theta)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0, \frac{\sqrt{5}-1}{2}}
The notation $\overline{abc}$ represents a three-digit number in decimal form. If $\overline{abc}$ is equal to the sum of all two-digit numbers formed by the digits $a, b, c$, find all the three-digit numbers that satisfy this condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132, 264, 396}
In triangle \(ABC\), \(AB = AC = a\). An equilateral triangle \(BCD\) is constructed outward from side \(BC\). For what value of \(A\) is \(AD\) the longest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^ightarrow}
Let $a = e^{4 \pi i/5}$ and $b = e^{2 \pi i/17}.$  Compute the value of the product
\[(a + b)(a + b^{16})(a^2 + b^2)(a^2 + b^{15})(a^3 + b^8)(a^3 + b^9)(a^4 + b^4)(a^4 + b^{13}).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\overline{AB}$, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{384}
How to find a bound on \begin{align} \sup_{v \in \mathbb{R}^n, a^2 I \preceq A }\| v^T(vv^T+A)^{-1} \| \end{align} where $A$ is symmetric $n\times n$ matrix?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{2a}\)}
$已知点F_1是椭圆C：x^2/4+y^2/3=1的左焦点，Q是椭圆C上的任意一点，A(\frac{1}{2},1)．$
$求|QF_1|+|QA|的最大值；$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$4+ \frac{\sqrt{5}}{2}$}
在以正2n+1边形之顶点为顶点的三角形中，求锐角三角形的个数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{有$\frac{n(n+1)(2n+1)}{6}$个}
Let $F$ be the right focus of the hyperbola $E$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 (a>0, b>0)$, $A$ and $B$ be the left and right vertices of the hyperbola $E$, respectively. Point $P$ is a moving point on the hyperbola $E$ other than $A$ and $B$. The line $l: x=t$ intersects the line perpendicular to $AP$ passing through $F$ at point $Q$. Given that points $B$, $P$, and $Q$ are collinear, find the maximum value of $\frac{t}{a}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{4}}
Let $S_{1}, S_{2}, \ldots, S_{10}$ be the first ten terms of an arithmetic progression (A.P.) of positive integers. If $S_{1} + S_{2} + \ldots + S_{10} = 55$ and $\left(S_{10} - S_{8}\right) + \left(S_{9} - S_{7}\right) + \ldots + \left(S_{3} - S_{1}\right) = d$, find $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
What is the largest possible order of the finite field needed to contain all the roots of an irreducible polynomial \( f(x) \) of degree 11 over \( \mathbb{F}_5 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5^{11}}
Given that \(\mathrm{G}\) is the centroid of \(\triangle \mathrm{ABC}\), and the equation \(\sqrt{7 \mathrm{GA}} \sin A + 3 \overrightarrow{\mathrm{GB}} \sin B + 3 \sqrt{7 \mathrm{GC}} \sin C = \mathbf{0}\) holds, find \(\angle \mathrm{ABC}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
A certain intelligence station has four different passwords, $A$, $B$, $C$, and $D$. Each week, one of these passwords is used, and each week the password is chosen at random and equally likely from the three passwords that were not used in the previous week. If password $A$ is used in the first week, what is the probability that password $A$ will also be used in the seventh week? (Express your answer as a simplified fraction.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{61}{243}}
Nathan has discovered a new way to construct chocolate bars, but it's expensive! He starts with a single $1 \times 1$ square of chocolate and then adds more rows and columns from there. If his current bar has dimensions $w \times h$ (where $w$ is the number of columns and $h$ is the number of rows), then it costs $w^2$ dollars to add another row and $h^2$ dollars to add another column. What is the minimum cost to get his chocolate bar to size $20 \times 20$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5339}
Evaluate the limit \( \lim_{{n\to\infty}} \int_{0}^{1} \frac{n e^{-\sqrt{n x}} \sin(x)}{(1+x^2) \arctan(x)} \, dx \) using the dominated convergence theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the extremum of \( f\left( t\right)  = {\int }_{1}^{t}\frac{\ln x}{x + t}{dx}\left( {t > 0}\right) \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0
}
Determine the number of integers \( 2 \leq n \leq 2016 \) such that \( n^n - 1 \) is divisible by \( 2, 3, 5, \) and \( 7 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
设 $a, b$ 为不超过 12 的正整数，满足：存在常数 $C$ ，使得 $a^{n}+b^{n+9} \equiv C(\bmod 13)$ 对任意正整数 $n$ 成立. 求所有满足条件的有序数对 $(a, b)$. 请提供所有满足条件的数对中所有元素的和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Calculate the expected number of calls needed to fill a bingo card with 25 squares, assuming numbers are called randomly without replacement from a set of 75 numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{950}{13}}
A particle of unit mass moves along the $x$-axis with a Lagrangian given by $L= \frac{1}{12}\dot x^4 + \frac{1}{2}x \dot x^2-x^2$. The force $Q=\dot x^2 \ddot x$ acts on the particle in the $x$-direction. Given the initial conditions $x(0)=1$ and $\dot x(0)=1$, determine the value of $\dot x$ at $x=\frac{1}{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\sqrt{5}}
Given that
$\frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!}$
 find the greatest integer that is less than $\frac N{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{137}
Evaluate the limit: $$\lim_{x \to \infty} \frac{ex^{x+1}-x(x+1)^x}{(x+1)^x} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find $a \in \mathbb{R}$, such that solutions of the system $$\dot x = 3x + y + 1; \dot y = 6x+2y+a$$ are bounded when $-\infty < t< \infty$ and determine if they are stable or not.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a, b, c$ (not necessarily distinct) are in the same subset, then $a \neq b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the degree of the field extension \( \mathbb{C}(e_{1}, e_{2}, e_{3})(x_{1}+2x_{2}+3x_{3})/\mathbb{C}(e_{1}, e_{2}, e_{3}) \), where \( e_{1}=x_{1}+x_{2}+x_{3} \), \( e_{2}=x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3} \), and \( e_{3}=x_{1}x_{2}x_{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $|a|<\frac{\pi}{2}$.
Evaluate
\[\int_0^{\frac{\pi}{2}} \frac{dx}{\{\sin (a+x)+\cos x\}^2}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\sec{a}}{1+\sin{a}}$}
A train of a certain length moving at a speed of 72 kmph takes 69.99440044796417 seconds to cross another train of length 300 m, moving at 36 kmph in the same direction. What is the length of the first train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{399.944}
A unit circle is centered at $(0, 0)$ on the $(x, y)$ plane. A regular hexagon passing through $(1, 0)$ is inscribed in the circle. Two points are randomly selected from the interior of the circle, and horizontal lines are drawn through them, dividing the hexagon into at most three pieces. The probability that each piece contains exactly two of the hexagon's original vertices can be written as \[ \frac{2\left(\frac{m\pi}{n}+\frac{\sqrt{p}}{q}\right)^2}{\pi^2} \] for positive integers $m$, $n$, $p$, and $q$ such that $m$ and $n$ are relatively prime and $p$ is squarefree. Find $m+n+p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD},BC=CD=43$, and $\overline{AD}\perp\overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{194}
Evaluate the surface integral \( \iint_{S} [ \nabla \phi \times \nabla \psi] \cdot \mathbf{n} \, dS \) where \( \phi=(x+y+z)^2 \) and \( \psi=x^2-y^2+z^2 \). The surface \( S \) is the curved surface of the hemisphere \( x^2+y^2+z^2=1 \) with \( z \ge 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi}
Evaluate the limit: $$\lim_{m\to\infty}\left(\sqrt[m]{m+1}\cdot\sqrt[m^2+m]{\binom{m}{1}\binom{m}{2}\cdots\binom{m}{m}}\right) $$ where $$ \binom{m}{j}=\frac{m!}{(m-j)!j!}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
Let $a, b, c$, and $d$ be real numbers such that $a^2 + b^2 + c^2 + d^2 = 3a + 8b + 24c + 37d = 2018$. Evaluate $3b + 8c + 24d + 37a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1215}
Evaluate the integral \( \frac{i}{4-\pi}\int_{|z|=4}\frac{1}{z\cos{z}}dz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In this diagram, $AB$ and $AC$ are the equal sides of an isosceles triangle $ABC$, in which an equilateral triangle $DEF$ is inscribed. Designate angle $BFD$ by $a$, angle $ADE$ by $b$, and angle $FEC$ by $c$. The relationship between $a$, $b$, and $c$ is such that $k(b + c) = 2a$. Please find the value of $k + m + n$, where $m$ and $n$ are the coefficients of $b$ and $c$, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the number of possible combinations of real and complex roots for a 7th degree polynomial, considering the possibilities of distinct and repeated roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \(T: \mathbb{R}^4 \to \mathbb{R}^4\) be a linear transformation such that the null space of \(T\) is \(\{(x,y,z,w) \in \mathbb{R}^4 : x+y+z+w=0\}\) and the rank of \((T-4I)\) is 3. If the minimal polynomial of \(T\) is \(x(x-4)^a\), find the value of \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the monomial $-3x^{2}y$, what are its coefficient and degree? Express your answer as two integers, where the first integer is the coefficient and the second integer is the degree.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-3, 3)}
Determine the number of subgroups of order 3 in the symmetric group $S_6$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
In the land of knights and liars (knights always tell the truth, liars always lie), ten people were given different numbers from 1 to 10. Each person was then asked, "Is your number divisible by 2?" Three people answered affirmatively. To the question "Is your number divisible by 4?" six people answered affirmatively. To the question "Is your number divisible by 5?" two people answered affirmatively. What numbers did the liars receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2, 5, 6, 10}
Calculate the expected time it takes for a monkey typing random capital letters to produce the sequence "ABBA". Assume the problem is modeled as a Markov Process. Provide the precise expected time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{457002}
Here is the pseudocode of an algorithm. If the output value of $y$ is 20,  
what is the input value of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Evaluate the triple integral \( J = \iiint_{Q} \frac{1}{x^2+y^2+z^2} \,dx \,dy \,dz \) where \( Q \) is the region between the spheres \( x^2+y^2+z^2 = 1 \) and \( x^2+y^2+z^2 = 9 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32\pi}
The amount of a claim that a car insurance company pays out follows an exponential distribution. By imposing a deductible of $d$, the insurance company reduced the expected claim payment by $10\%$. Calculate the percentage reduction in the variance of the claim payment after the deductible is applied.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1\%}
In a game, there are three indistinguishable boxes; one box contains two red balls, one contains two blue balls, and the last contains one ball of each color. To play, Raj first predicts whether he will draw two balls of the same color or two of different colors. Then, he picks a box, draws a ball at random, looks at the color, and replaces the ball in the same box. Finally, he repeats this; however, the boxes are not shuffled between draws, so he can determine whether he wants to draw again from the same box. Raj wins if he predicts correctly; if he plays optimally, what is the probability that he will win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{6}}
Calculate the norm of the operator \( A:L^{1}[1,2] \rightarrow L^{1}[0,1] \), defined by \( Af(x) = \int_{1}^{2} \frac{f(t)}{x+t} \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Find the sum of the infinite series $$\sum_{n=1}^\infty \frac{n}{(1+x)^{2n+1}}$$ and determine the interval of convergence for $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{x+1}{x^2(x+2)^2}}
Given a joint probability density function $f_{X,Y}(x,y)$, find the probability $P(Y > 3X)$. Provide your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Consider the ten numbers $ar, ar^2, ar^3, \ldots, ar^{10}$. If their sum is 18 and the sum of their reciprocals is 6, determine their product.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
Determine the largest open interval $I$ containing $x = 0$ on which there exists a unique solution $y(x)$ to the initial value problem:
\[ (\sin(x)-1)y''' + (x^2-x)y'' + \frac{1}{x-1}y' + x^5y = e^{x^3} \]
with initial conditions $y(0) = 1$, $y'(0) = 5$, and $y''(0) = 2$. What is the largest open interval $I$ containing $x = 0$ on which there exists a unique solution $y(x)$ to this problem?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( -\frac{3\pi}{2}, 1 \right)}
Let $n$ be a positive integer. A frog starts on the number line at 0 . Suppose it makes a finite sequence of hops, subject to two conditions:
- The frog visits only points in $\left\{1,2, \ldots, 2^{n}-1\right\}$, each at most once.
- The length of each hop is in $\left\{2^{0}, 2^{1}, 2^{2}, \ldots\right\}$. (The hops may be either direction, left or right.)

Let $S$ be the sum of the (positive) lengths of all hops in the sequence. What is the maximum possible value of $S$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4^{n}-1}{3}}
Let $T$ be a linear transformation from $M(n, \mathbb{R})$ to $M(n, \mathbb{R})$ defined by $T(X) = (A - I)X$, where $A$ is a non-zero nilpotent matrix of order $n \times n$. Determine the trace of the transformation $T$. \( \text{(Express your answer as a single number.)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-n^2}