inclusionAI/Ring-lite-rl-data huggingface dataset:
What is the % change in the area of a rectangle when its length increases by 30 % and its width decreases by 30 %?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9\%}
While playing a video game, Ned lost thirteen lives but collected a certain number of power-ups, each giving him five extra lives. After these events, he ended up with seventy lives. If we let x be the initial number of lives he had and y be the number of power-ups collected, how many lives did Ned have at the start?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83}
A plane-parallel glass plate is normally illuminated by a thin beam of light. Behind the plate, at some distance from it, there is an ideal mirror (with a reflection coefficient of one). The plane of the mirror is parallel to the plate. It is known that the intensity of the beam passing through this system is 256 times less than the intensity of the incident beam. Assume that the reflection coefficient at the glass-air interface is constant regardless of the direction of the beam. Ignore the absorption and scattering of light in the air and glass. Find the reflection coefficient at the glass-air interface under these conditions. (10 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.75}
Evaluate the double integral $\int\int 2xy \, dA$ over the region bounded by $y = 1 - x^2$, $y = 1 + x$, and $y = 0$ in the $xy$ plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{12}$}
The dollar is now worth  $\frac{1}{980}$  ounce of gold. After the  $n^{th}$  7001 billion dollars bailout package passed by congress, the dollar gains  $\frac{1}{2{}^2{}^{n-1}}$  of its  $(n-1)^{th}$  value in gold. After four bank bailouts, the dollar is worth  $\frac{1}{b}(1-\frac{1}{2^c})$  in gold, where  $b, c$  are positive integers. Find  $b + c$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{506}
Let  $T$  denote the  $15$ -element set  $\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}$ . Let  $S$  be a subset of  $T$  in which all six digits  $1,2,\ldots ,6$  appear and in which no three elements together use all these six digits. Determine the largest possible size of  $S$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
The natural numbers from 1951 to 1982 are arranged in a certain order one after another. A computer reads two consecutive numbers from left to right (i.e., the 1st and 2nd, the 2nd and 3rd, etc.) until the last two numbers. If the larger number is on the left, the computer swaps their positions. Then the computer reads in the same manner from right to left, following the same rules to change the positions of the two numbers. After reading, it is found that the number at the 100th position has not changed its position in either direction. Find this number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1982}
In a grid of numbers and letters, starting from the letter A in the middle of the grid, how many different paths can spell out "AMC8"? Each path must move to an adjacent (up, down, left, or right) letter or number only once (no diagonal moves allowed). Provide your answer as a single integer (number of paths).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{248}
Example 9 Given a positive integer $k$, when $x^{k}+y^{k}+z^{k}=1$, find the minimum value of $x^{k+1}+y^{k+1}+z^{k+1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{-\frac{1}{k}}}
In an electric circuit, two resistors are connected in parallel. One resistor has a resistance of 9 ohms, and the other has a resistance that results in a combined resistance of 4.235294117647059 ohms. What is the resistance of the first resistor?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the limit: $$ \lim_{x\to0} \sum_{n=1}^\infty \frac{\sin x}{4+n^4x^4} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{8}}
Determine the number of solutions to the equation $X^{10} - I = 0$ in $M_2(\mathbb{R})$, the set of $2 \times 2$ real matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Solve the recurrence relation \( T(n) = n \cdot T(n/2)^2 \). Determine the asymptotic behavior of \( T(n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta\left(\frac{4^n}{n}\right)}
Let $X \subset \mathbb{P}^3_\mathbb{C}$ be a smooth algebraic surface of degree $d$, and let $L \subset X$ be a line on this surface. Compute the self-intersection number $L^2$ of the line $L$ on the surface $X$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 - d}
Find the result of dividing \(2^{2^k} - b^{2^k}\) by the product \((a+b)(a^2+b^2)(a^4+b^4) \cdots (a^{2^{k-1}}+b^{2^{k-1}})\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a - b}
Let \( f(n) = \frac{n}{3} \) if \( n \) is divisible by 3, and \( f(n) = 4n - 10 \) otherwise. Find the sum of all positive integers \( c \) such that \( f^5(c) = 2 \). (Here, \( f^5(x) \) means \( f(f(f(f(f(x))))) \).)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{748}
Determine the conditional expectation \( \mathbb{E}(X|\mathcal{G}) \) where \( X(\omega) = \cos(\pi \omega) \) and \( \mathcal{G} \) is the \( \sigma \)-algebra of countable and co-countable subsets of \((0,1)\). Assume \( \lambda \) is the Lebesgue measure restricted to \((0,1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a party attended by $2015$ guests among any $5$ guests at most $6$ handshakes had been exchanged. Determine the maximal possible total number of handshakes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1007 * 1008
}
Determine the cardinality of the set of all root paths in an infinite complete infinitary tree where the branching factor k is infinite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Determine \( \phi'(0) \) for the function \( \phi(x) = \int_{0}^{x}\sin\left(\frac{1}{t}\right)dt \), where \( \phi: [0, \infty) \to \mathbb{R} \). Note that the function is undefined at \( t=0 \) and the limit does not exist at that point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \( \int_{25\pi/4}^{53\pi/4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7\pi}{4}}
Evaluate the surface integral \( \int_S { \vec{D} \cdot} d\vec{s} \) over the surface of the cube defined by the vector field \( \vec{D} = 2y\vec{a_x} +  xz\vec{a_y} +z\vec{a_z} \), where the cube's coordinates are from \(x, y, z \in (0, 2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the limit: \[ \lim_{n \to \infty} \sum_{k = 1}^{n^2+1}\frac{1}{\sqrt{n^{2} + k}}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the limit of the function \( g_n(x) = \frac{1}{\sqrt{2\pi n^{-5}}}\exp\left(-\frac{(x+\frac{1}{n})^2}{2n^{-5}}\right) \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta(x)}
Let $T = \left\{(\lambda_n)_{n} \in l^2 : \sum_{n=1}^{\infty} \frac{\lambda_n}{\sqrt{n}} = 0\right\}$. Determine $T^{\perp}$, the orthogonal complement of $T$ in $l^2$. Assume $\mathbb{K} = \mathbb{R}$. Provide your answer in terms of a description of the subspace $T^{\perp}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0\}}
Determine the cardinality of the set of equivalence classes \( P(\mathbb{N} \times \mathbb{N})/_R \), where \( R \subseteq P(\mathbb{N} \times \mathbb{N})^2 \) is defined by the relation \( \langle r,s\rangle \in R \Leftrightarrow (\exists \pi : \mathbb{N} \to \mathbb{N} \forall x,y \in \mathbb{N} (\langle x,y\rangle \in r \leftrightarrow \langle\pi(x),\pi(y)\rangle \in s)) \). Here, \( \pi \) is a bijection, and \( A/_R \) denotes the set of equivalence classes of \( A \) under \( R \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
How many ways, without taking order into consideration, can 2002 be expressed as the sum of 3 positive integers (for instance, $1000+1000+2$ and $1000+2+1000$ are considered to be the same way)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{334000}
Source: 1976 Euclid Part B Problem 1
-----
Triangle  $ABC$  has  $\angle{B}=30^{\circ}$ ,  $AB=150$ , and  $AC=50\sqrt{3}$ . Determine the length of  $BC$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50\sqrt{3}}
Let the number  $x$ . Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent  of the number  $x$  (for example,  $x\cdot x=x^{2}$ ,  $x^{2}\cdot x^{2}=x^{4}$ ,  $x^{4}: x=x^{3}$ , etc). Determine the minimal number of operations needed for calculating  $x^{2006}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
3. The continuation of the height $B H$ of triangle $A B C$ intersects the circumscribed circle around it at point $D$ (points $B$ and $D$ lie on opposite sides of line $A C$). The degree measures of arcs $A D$ and $C D$, not containing point $B$, are $60^{\circ}$ and $90^{\circ}$, respectively. Determine in what ratio the segment $B D$ is divided by side $A C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}:1}
An ant is walking on the edges of an icosahedron of side length  $1$ . Compute the length of the longest path that the ant can take if it never travels over the same edge twice, but is allowed to revisit vertices.

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Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{175}
Find the radius of convergence and the convergence at the end points of the series: $$\sum_{n=1}^\infty(2+(-1)^n)^nx^n$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find all prime numbers $ p,q$ less than 2005 and such that $ q|p^2 \plus{} 4$, $ p|q^2 \plus{} 4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 2), (5, 29), (29, 5)}
A number $n$ is called multiplicatively perfect if the product of all the positive divisors of $n$ is $n^2$. Determine the number of positive multiplicatively perfect numbers less than $100$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Determine the power of the group of languages A, where A consists of languages L such that L = L* and L is defined over the alphabet Σ = {1}.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Determine the largest natural number \( n \) such that

\[ 4^{995} + 4^{1500} + 4^{n} \]

is a square number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2004}
Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from $1$ to $50$ uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability $\frac{3}{5}$. If he flips heads, he adds $1$ to his score. A player wins the game if their score is higher than the other player's score. 

The probability that Dukejukem defeats Trollner to win WASS equals $\frac{m}{n}$, where $m$ and $n$ are coprime positive integers. Compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{751}
Calculate the expected value \( \mathbb{E}(X^3) \) for the stick breaking problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Determine the number of non-isomorphic quotient groups of the symmetric group $S_4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
radha bought a watch for rs . 144 and got a percentage of profit equal to the cost price of the watch . what is the cost price of the watch ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
In the given figure, hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Triangle  $ABC$  has side lengths  $AB=12$ ,  $BC=25$ , and  $CA=17$ . Rectangle  $PQRS$  has vertex  $P$  on  $\overline{AB}$ , vertex  $Q$  on  $\overline{AC}$ , and vertices  $R$  and  $S$  on  $\overline{BC}$ . In terms of the side length  $PQ=w$ , the area of  $PQRS$  can be expressed as the quadratic polynomial
\[\text{Area}(PQRS)=\alpha w-\beta\cdot w^2\]
Then the coefficient  $\beta=\frac{m}{n}$ , 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{161}
How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
求函数$f ( x ) = \sqrt { \frac { 1 } { 4 } x ^ { 4 } - 2 x ^ { 2 } - 4 x + 1 3 } + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 }$的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac { 7 } { 2 }$}
Rectangle \(ABCD\) has area 2016. Point \(Z\) is inside the rectangle and point \(H\) is on \(AB\) so that \(ZH\) is perpendicular to \(AB\). If \(ZH : CB = 4 : 7\), what is the area of pentagon \(ADCZB\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1440}
Robert read a book for 10 days. He read an average of 25 pages per day for the first 5 days and an average of 40 pages per day for the next 4 days, and read 30 more pages on the last day. Calculate the total number of pages in the book.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{315}
Petrov writes down the odd numbers: $1, 3, 5, \ldots, 2013$, and Vasechkin writes down the even numbers: $2, 4, \ldots, 2012$. Each of them calculates the sum of all digits of all their numbers and tells the result to their classmate Masha. Masha subtracts Vasechkin's result from Petrov's result. What is the final result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1007}
In the given figure, \(ABCD\) is a rectangle and triangles \(\triangle ABE\) and \(\triangle CDF\) are right triangles. The area of triangle \(\triangle ABE\) is \(150 \mathrm{~cm}^{2}\) and the lengths of segments \(AE\) and \(DF\) are 15 cm and 24 cm, respectively. What is the length of segment \(CF\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
设 $a_{n}$ 是与 $\sqrt{\frac{n}{2}}$ 的差最小的整数， $b_{n}$ 是与 $\sqrt{2 n}$ 的差最小的整数，记 $\left\{\frac{1}{a_{n}}\right\}$ 的前 $n$ 项和为 $S_{n},\left\{\frac{1}{b_{n}}\right\}$的前 $n$ 项和为 $T_{n}$ ，则 $2 T_{100}-S_{100}$ 。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the sides of a triangle with an area of 84, given that the sides are three consecutive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13, 14, 15}
Three positive reals  $x , y , z $  satisfy  $x^2 + y^2 = 3^2 
y^2 + yz + z^2 = 4^2 
x^2 + \sqrt{3}xz + z^2 = 5^2 .$  
Find the value of  $2xy + xz + \sqrt{3}yz$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A teacher must divide 221 apples evenly among 403 students. What is the minimal number of pieces into which she must cut the apples? (A whole uncut apple counts as one piece.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{611}
By dividing  $2023$  by a natural number  $m$ , the remainder is  $23$ . How many numbers  $m$  are there with this property?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $n$ be a positive integer, $n \ge 2$, and put $\theta = 2 \pi / n$. Define points $P_k = (k,0)$ in the $xy$-plane, for $k = 1, 2 , \dots, n$. Let $R_k$ be the map that rotates the plane counterclockwise by the angle $\theta$ about the point $P_k$. Let $R$ denote the map obtained by applying, in order, $R_1$, then $R_2, \dots$, then $R_n$. For an arbitrary point $(x,y)$, find, and simplify, the coordinates of $R(x,y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{The coordinates of $R(x,y)$ are $(x+n,y)$.}
The cells of a  $2021\times 2021$  table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate  $(1, 1)$ , contains the number  $0$ . For every other cell  $C$ , we consider a route from  $(1, 1)$  to  $C$ , where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell  $C$ . For example, the cell with coordinate  $(2, 1)$  contains  $1 = 1 + 0$ , the cell with coordinate  $(3, 1)$  contains  $3 = 2 + 0 + 1$ , and the cell with coordinate  $(3, 2)$  contains  $7 = 3 + 0 + 1 + 3$ . What is the last digit of the number in the cell  $(2021, 2021)$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the derivative of the specified order.

\[ y = \frac{\log_{3} x}{x^{2}}, \quad y^{(IV)} = ? \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{120 \ln x - 154}{x^6 \ln 3}}
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$78$}
一个四位数，它的个位数字与百位数字相同，如果把这个四位数的数字顺序颠倒过来（即千位数字与个位数字互换，百位数字与十位数字互换），所得的新数减去原数得7812，求原来的四位数．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1979}
Calculate the value of \((2\vec{a}+\vec{b})\cdot[(\vec{a}\times\vec{b})\times(\vec{a}-2\vec{b})]\) given \(\vec{a} = \frac{\hat{i}-2\hat{j}}{\sqrt{5}}\) and \(\vec{b} = \frac{2\hat{i}+\hat{j}+3\hat{k}}{\sqrt{14}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. Find the sum of $d$ and $e$.

$ \begin{array}{ccccccc}     & 7 & 4 & 2 & 5 & 8 & 6 \\+    & 8 & 2 & 9 & 4 & 3 & 0 \\\hline1    & 2 & 1 & 2 & 0 & 1 & 6 \\\end{array} $
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the derivative of the function \( f(M) = \text{Tr}(M\log M - M) \), where \( M \) is a positive definite matrix in \( \mathbb{S}_+^n \) and \( \log(M) = \sum_{i=1}^{n}\log(\lambda_i)v_iv_i^T \) with \((\lambda_i, v_i)\) as eigenpairs of \( M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log M}
On Monday, Millie puts a quart of seeds, $25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\text{Friday}$}
Find the largest real  $ T$  such that for each non-negative real numbers  $ a,b,c,d,e$  such that  $ a\plus{}b\equal{}c\plus{}d\plus{}e$ : \[ \sqrt{a^{2}\plus{}b^{2}\plus{}c^{2}\plus{}d^{2}\plus{}e^{2}}\geq T(\sqrt a\plus{}\sqrt b\plus{}\sqrt c\plus{}\sqrt d\plus{}\sqrt e)^{2}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{30}}{30 + 12\sqrt{6}}}
Arrange 3 volunteer teachers to teach in 6 schools, with at most 2 people per school. How many different distribution plans are there? (Answer with a number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
Inside a circle with radius $6$ lie four smaller circles with centers $A$, $B$, $C$, and $D$. These circles touch each other as shown. The point where the circles with centers $A$ and $C$ touch each other is the center of the large circle. Calculate the area of quadrilateral $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
$f(x)=\sum_{n=0}^{\infty} 2^{-n} \cos (n x)$, 记 $A=\int_{0}^{2 \pi} f^{2}(x) \mathrm{d} x$, 则 [10A] $=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73}
A problem-solving robot with an initial IQ balance of 25 solves a set of questions with point values ranging from $1$ to $10$. Each time it correctly solves a problem, the robot’s IQ balance decreases by the point value of the problem and then increases by 1. If the robot's IQ balance becomes less than the point value of a problem, it fails to solve that problem. What is the maximum score the robot can achieve?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Determine the number of homomorphisms from $(\mathbb{Z} / p) [x] / (x^2 - x)$ to $\mathbb{Z} / q$, where $p$ and $q$ are primes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A train is travelling at a certain speed and crosses a pole in 6 seconds. The length of the train is 200.016 meters. What is the speed of the train in kmph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120.0096}
Let \( {a}_{0},{a}_{1},\ldots ,{a}_{19} \in  \mathbb{R} \) and
 \( P\left( x\right)  = {x}^{20} + \mathop{\sum }\limits_{{i = 0}}^{{19}}{a}_{i}{x}^{i},x \in  \mathbb{R}. \)
 If \( P\left( x\right)  = P\left( {-x}\right) \) for all \( x \in  \mathbb{R} \) ,and
 \( P\left( k\right)  = {k}^{2}, \)
 for \( k = 0,1,2,\ldots ,9 \) then find
 \( \mathop{\lim }\limits_{{x \rightarrow  0}}\frac{P\left( x\right) }{{\sin }^{2}x} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$-(9!)^2 + 1$}
Find all $t$ such that $x-t$ is a factor of $6x^2+13x-5.$

Enter your answer as a list separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{5}{2}, \dfrac{1}{3}}
Roll a die twice. Let $X$ be the maximum of the two numbers rolled. Which of the following numbers is closest to the expected value $E(X)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.5}
Find all natural numbers \( n \) such that the polynomial \((X^2+1)^{6n+2}+X^4+1\) divides the polynomial \(X^2+X+1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}