inclusionAI/Ring-lite-rl-data huggingface dataset:
Let  $ABCDEF$  be a regular hexagon with side length 10 inscribed in a circle  $\omega$ .  $X$ ,  $Y$ , and  $Z$  are points on  $\omega$  such that  $X$  is on minor arc  $AB$ ,  $Y$  is on minor arc  $CD$ , and  $Z$  is on minor arc  $EF$ , where  $X$  may coincide with  $A$  or  $B$  (and similarly for  $Y$  and  $Z$ ). Compute the square of the smallest possible area of  $XYZ$ .

*Proposed by Michael Ren*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7500}
For a permutation  $P = (p_1, p_2, ... , p_n)$  of  $(1, 2, ... , n)$  define  $X(P)$  as the number of  $j$  such that  $p_i < p_j$  for every  $i < j$ . What is the expected value of  $X(P)$  if each permutation is equally likely?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H_n}
Ben wants to pick 56 sugar snap peas. At his current rate of picking, all will be picked in seven minutes. How long, in minutes, would it take Ben to pick a different number of sugar snap peas if it takes him 9 minutes to pick 72 sugar snap peas?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
(2013•Shanghai) For a function $g(x)$ defined on the interval $I$, denote $g(I) = \{y | y = g(x), x \in I\}$. Given a function $y = f(x)$ with domain $[0, 3]$ that has an inverse function $y = f^{-1}(x)$, and it is known that $f^{-1}([0, 1)) = [1, 2)$, $f^{-1}((2, 4]) = [0, 1)$. If the equation $f(x) - x = 0$ has a solution $x_0$, then $x_0 =$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Define a monic irreducible polynomial with integral coefficients to be a polynomial with leading coefficient 1 that cannot be factored, and the prime factorization of a polynomial with leading coefficient 1 as the factorization into monic irreducible polynomials. How many not necessarily distinct monic irreducible polynomials are there in the prime factorization of $\left(x^{8}+x^{4}+1\right)\left(x^{8}+x+1\right)$ (for instance, $(x+1)^{2}$ has two prime factors)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
What is the smallest number of sides that a polygon (not necessarily convex) with an odd number of sides can have, if it can be divided into parallelograms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $A=\{x|x\geqslant 1\}$, $B=\{x|-1 \lt x \lt 2\}$. Find the intersection $A\cap B$ and express it in set notation, specifying the conditions for x.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{x|1\leqslant x \lt 2\}}
Given a geometric sequence ${a_n}$ with sum of the first $n$ terms $S_n$, and the equations $a_3=2S_2+5$ and $a_4=2S_3+5$, find the common ratio $q$ of this sequence. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given a hyperbola with the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, its asymptotes intersect the parabola $y^2 = 4x$ at two points A and B, distinct from the origin O. Let F be the focus of the parabola $y^2 = 4x$. If $\angle AFB = \frac{2\pi}{3}$, then the eccentricity of the hyperbola is ________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{21}}{3}}
There are $110$ guinea pigs for each of the $110$ species, arranged in a $110 \times 110$ array. Find the maximum integer $n$ such that, no matter how the guinea pigs are arranged, we can always find a column or a row of $110$ guinea pigs containing at least $n$ different species.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Of 64 players on a cricket team, 37 are throwers. The rest of the team is divided so some are left-handed and the rest are right-handed. Assuming that all throwers are right-handed, there are total 55 right-handed players. What is the ratio of left-handed players to the rest of the team (excluding throwers)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2}
Let $X$ be the set consisting of twenty positive integers $n, n+2, \ldots, n+38$. Determine the smallest value of $n$ such that any three numbers $a, b, c \in X$, not necessarily distinct, can form the sides of an acute-angled triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{92}
Seven cards numbered $1$ through $7$ lay stacked in a pile in ascending order from top to bottom ($1$ on top, $7$ on bottom). A shuffle involves picking a random card [i]of the six not currently on top[/i], and putting it on top. The relative order of all the other cards remains unchanged. Find the probability that, after $10$ shuffles, $6$ is higher in the pile than $3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3^{10} - 2^{10}}{2 \cdot 3^{10}}}
If the function \( f(x) = \cos(n x) \cdot \sin\left(\frac{4}{n} x\right) \) (where \( n \in \mathbb{Z} \)) has a period of \( 3\pi \), then what is the set of possible values for \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{\pm 2, \pm 6\}}
Determine the number of ordered 9-tuples of positive integers \((a_{1}, a_{2}, \cdots, a_{9})\) (elements in the array can be the same) that satisfy the following condition: for any \(1 \leq i < j < k \leq 9\), there exists a different \(l (1 \leq l \leq 9)\) such that
$$
a_{i} + a_{j} + a_{k} + a_{l} = 100.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A pedestrian left point $A$ for a walk at a speed of $v$ km/h. After he had walked 6 km away from $A$, a cyclist left $A$ following him at a speed that was 9 km/h faster than the pedestrian's speed. When the cyclist caught up with the pedestrian, they turned back and returned together to $A$ at a speed of 4 km/h. At what value of $v$ will the pedestrian's total walking time be minimized?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 \text{ km/h}}
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and it satisfies $S_n = (-1)^n \cdot a_n - \frac{1}{2^n}$. Define $b_n = 8a_2 \cdot 2^{n-1}$. If for any $n \in \mathbb{N}^*$, it always holds that $\lambda b_n - 1 > 0$, then the range of values for the real number $\lambda$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{1}{2}, +\infty\right)}
The differential equation of the family of circles whose center lies on the $x$-axis and passes through the origin is $\ldots$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y^2 - x^2 - 2xy\frac{dy}{dx} = 0}
In quadrilateral $ABCD$, $\angle{BAD} \cong \angle{ADC}$ and $\angle{ABD} \cong \angle{BCD}$, $AB = 8$, $BD = 10$, and $BC = 6$. The length $CD$ may be written in the form $\frac {m}{n}$, 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{69}
Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is \(m\) times the area of the square. The original answer is in \(\frac{k}{nm}\) format, where \(m\) is given in the problem. Please find the value of \(k+n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
How many non-empty subsets of $\{1,2,\dots,11\}$ are there such that the product of its elements is the cube of an integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Consider a linear transformation $B$ that maps a four-dimensional subspace $L$ of a five-dimensional vector space $V$ to a vector space $W$. If the intersection of the kernel of $B$ with $L$ is 0-dimensional, what are the possible dimensions of the image of $L$ under $B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Expand the trinomial $(x^2+x+1)^n$. For $n=1, 2, 3, \ldots$, the expansion yields the generalized Pascal's triangle as shown below: 
$$(x^2+x+1)^0 = 1$$
$$(x^2+x+1)^1 = x^2 + x + 1$$
$$(x^2+x+1)^2 = x^4 + 2x^3 + 3x^2 + 2x + 1$$
$$(x^2+x+1)^3 = x^6 + 3x^5 + 6x^4 + 7x^3 + 6x^2 + 3x + 1$$
$$(x^2+x+1)^4 = x^8 + 4x^7 + 10x^6 + 16x^5 + 19x^4 + 16x^3 + 10x^2 + 4x + 1$$
...
Observing the relationships between the coefficients of the polynomials, we can construct the generalized Pascal's triangle as shown. The construction rule is: the first row is 1, and each number in the following rows is the sum of the three numbers above it (if any of these three numbers do not exist, count them as 0), and the $k^{th}$ row consists of $2k+1$ numbers. If in the expansion of $(1+ax)(x^2+x+1)^5$, the coefficient of the $x^8$ term is 75, then the real number $a$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In an arithmetic sequence {a_n}, there are two terms a_m and a_k such that a_m = 1/k and a_k = 1/m (where m, k are positive integers and m!= k). Find the sum of the first mk terms of this sequence. Express your answer as a formula involving m and k.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{mk+1}{2}}
Find all values of \( m \) for which any solution of the equation

\[ 2019 \cdot \sqrt[3]{3.5x - 2.5} + 2018 \cdot \log_{2}(3x - 1) + m = 2020 \]

belongs to the interval \([1; 3]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \in [-8072; -2017]}
In a commercial football tournament, five teams participated. Each team was supposed to play exactly one match with each of the other teams. Due to financial difficulties, the organizers canceled some games. As a result, all teams ended up with a different number of points, and none of the teams had zero points in the points column. What is the minimum number of games that could have been played in the tournament if a win awarded three points, a draw one point, and a loss zero points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 \text{ games}}
Find the number of primes  $p$  between  $100$  and  $200$  for which  $x^{11}+y^{16}\equiv 2013\pmod p$  has a solution in integers  $x$  and  $y$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Brittany is older than Rebecca by 3 years, who is 25. Brittany goes on a 4-year vacation, during which she celebrates her birthday 3 times due to crossing the International Date Line. Additionally, there is a leap year during her vacation that affects her age calculation. How old is Brittany when she returns from her vacation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
If the fractional equation $\frac{2k}{x-1}-\frac{3}{1-x}=1$ has a root, then $k=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{3}{2}}
Once, Alice found herself in a magical forest where two amazing twin brothers, Tralalala and Trulala, lived. Tralalala always lied on Mondays, Tuesdays, and Wednesdays, but told the truth all other days of the week. Trulala always lied on Thursdays, Fridays, and Saturdays, but told the truth all other days of the week. One day, Alice met both brothers (it was impossible to distinguish who was who by appearance), and they declared:

First: "I lie on Saturdays."

Second: "I will lie tomorrow."

First: "I also lie on Sundays."

Determine which day of the week this meeting could have taken place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Wednesday}
Two numbers \( x \) and \( y \) satisfy the equation \( 280x^{2} - 61xy + 3y^{2} - 13 = 0 \) and are respectively the fourth and ninth terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
A factory paints cubes in 6 colors (each face in its own color, with a fixed set of colors). How many different types of cubes can be produced?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the value of $\displaystyle \int^{2021}_{-1}g(x)dx$ given that $g:\mathbb{R}-\{0\} \rightarrow \mathbb{R}$ satisfies the functional equation $g(x)\cdot g(y)=2g(xy)-g\bigg(\frac{2020}{x}\bigg)\cdot g\bigg(\frac{2020}{y}\bigg)\forall x,y\in \mathbb{R}-\{0\}$ and $g(2020)=1$, $g(-3)=-1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020}
During the Tree Planting Festival, students from a class are planting trees. There are three different types of saplings, labeled as A, B, and C, available in the nursery, with an ample supply of each type. They need to plant 5 saplings in a row in 5 pits, with the condition that the same type of sapling cannot be adjacent, and only type A saplings can be planted in the first and the fifth pits. Find the total number of planting methods available. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In $\triangle ABC$, $AB=17$, $AC=25$, and $BC=28$. Points $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively, and $P$ is a point on $\overline{BC}$. Let $Q$ be the second intersection point of the circumcircles of $\triangle BMP$ and $\triangle CNP$. It is known that as $P$ moves along $\overline{BC}$, line $PQ$ passes through some fixed point $X$. Compute the sum of the squares of the distances from $X$ to each of $A$, $B$, and $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{878}
Let $A,B,C$ be angles of a triangle, where angle $B$ is obtuse, and \begin{align*}
\cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\
\cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9}.
\end{align*}There are positive integers $p$, $q$, $r$, and $s$ for which \[ \cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s}, \]where $p+q$ and $s$ are relatively prime and $r$ is not divisible by the square of any prime.  Find $p+q+r+s$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{222}
Let $k = 2^6 \cdot 3^5 \cdot 5^2 \cdot 7^3 \cdot 53$. Let $S$ be the sum of \( \frac{\text{gcd}(m,n)}{\text{lcm}(m,n)} \) over all ordered pairs of positive integers \((m, n)\) where \(mn = k\). If $S$ can be written in simplest form as \( \frac{r}{s} \), compute \( r + s \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{895}
There was only enough room for some people on a raft if no one wore life jackets. There were 7 fewer people who could fit on the raft if everyone wore life jackets. If 8 people on the raft needed life jackets, 17 people could fit on the raft. How many people could fit on the raft if no one wore life jackets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
For the quadratic function $y=x^{2}-4x+7$, when $-1 \lt x \lt 4$, the range of $y$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\leqslant y < 12}
Forty children were holding hands in a circle. Among them, 22 were holding a boy's hand and 30 were holding a girl's hand. How many girls were in the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Mrs. Evans has 385 crayons, 95 pieces of paper, and 210 pencils to give to her students. She wants to make sure that each student receives exactly 5 crayons, 1 piece of paper, and 2 pencils. What is the largest prime number of students she can have in her class while ensuring every student receives the exact same number of items?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73}
求函数$\ln ( \frac { s } { s + 9 } )$的拉普拉斯逆变换。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac { 1 } { t } ( e ^ { - 9 t } - 1 )$}
There is a garden with 3 rows and 2 columns of rectangular flower beds, each measuring 6 feet long and 2 feet wide. Between the flower beds, as well as around the garden, there is a 1-foot wide path. What is the total area of the path in square feet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78}
Find all integer solutions $(x, y)$ to the Diophantine equation $x^{6} = y^{2} + 53$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{(3, 26), (3, -26), (-3, 26), (-3, -26)}}
Given the probability function for selecting a digit d as log(d+1) - log(d), find the set of numbers that has a combined probability equal to twice the probability of selecting the digit 2. Express your answer as a set of numbers enclosed in curly brackets, with numbers separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{4,5,6,7,8\}}
If the sum of all binomial coefficients in the expansion of $$( \sqrt {x}- \frac {1}{2x})^{n}$$ is 512, then the constant term in this expansion is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{21}{2}}
Find the set of values for the real number a such that the quadratic inequality ax^2 - ax + 1 < 0 has no solution. Express your answer as an interval, including the endpoints if they are part of the solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0,4]}
空间中点集定义如下: $A_{n}=\left\{\left.(x, y, z) \in \mathbf{R}^{3}|3| x\right|^{n}+|8 y|^{n}+|z|^{n} \leq 1\right\}, A=\bigcup_{n=1}^{\infty} A_{n}$, 求由 $A$ 中的点组成的图形的体积。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given points $A(-6, -1)$, $B(1, 2)$, and $C(-3, -2)$, find the coordinates of vertex $M$ of the parallelogram $ABMC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(4, 1)}
Calculate the flux of the vector field $F(x,y,z) = \langle 9x, 9y, 4 \rangle$ across the part of the paraboloid $z = x^2 + y^2$ that lies between the planes $z = 0$ and $z = 4$. Assume the outward orientation for the closed surface, and take into account that the normal vector $n$ points downwards.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128\pi}
Given the expression
\[
a_1 \cdot 5^1 + a_2 \cdot 5^2 + a_3 \cdot 5^3 + a_4 \cdot 5^4 + a_5 \cdot 5^5 + a_6 \cdot 5^6
\]
where \(a_1, a_2, a_3, a_4, a_5, a_6 \in \{-1, 0, 1\}\), count the number of expressions that are negative.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{364}
After lunch, there are dark spots with a total area of $S$ on a transparent square tablecloth. It turns out that if the tablecloth is folded in half along any of the two lines connecting the midpoints of its opposite sides or along one of its two diagonals, the total visible area of the spots becomes $S_{1}$. However, if the tablecloth is folded in half along the other diagonal, the total visible area of the spots remains $S$. What is the smallest possible value of the ratio $S_{1}: S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/3}
已知 $\triangle A B C$ 为椭圆 $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ 的内接三角形，且 $A B$ 过点 $P(1,0)$ ，则 $\triangle A B C$ 的面积的最大值为 ? 请给出最大值的答案格式为 $\frac{m\sqrt{n}}{k} $的格式，请计算 $m + n+k$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Given that $a, b \in \mathbb{R}$, $a \neq 0$, and the curves $y = \frac{a + 2}{x}$ and $y = ax + 2b + 1$, if the two curves have at least one common point in the interval $[3, 4]$, find the minimum value of $a^2 + b^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{100}}
Given a triangle \(ABC\). Points \(K, L,\) and \(M\) are placed on the plane such that triangles \(KAM, CLM,\) and \(KLB\) are all congruent to triangle \(KLM\). What inequality sign should be placed between the perimeter of triangle \(KLM\) and the semiperimeter of triangle \(ABC\)?

The vertices of the triangles are listed arbitrarily: for example, one cannot assert that in the congruent triangles \(KAM\) and \(CLM\), point \(K\) corresponds to point \(C\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\geq}
Calculate the improper integral $\displaystyle \int ^{\infty}_{0}\frac{x^a}{(x+b)^2}\mathrm{d}x$ where $|a|<1$ and $b>0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$$\frac{\pi a b^{a-1}}{\sin(\pi a)}$$}
If $\triangle A_1A_2A_3$ is equilateral and $A_{n+3}$ is the midpoint of line segment $A_nA_{n+1}$ for all positive integers $n$, then the measure of $\measuredangle A_{44}A_{45}A_{43}$ equals
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^{\circ}}
For a Monty Hall problem with $n$ doors and $t$ cars, what is the minimum number of goats that the host must reveal to make switching choices a better strategy, assuming $n > 3$ and $t > 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
已知 $a 、 b 、 c$ 是三个不全相等的实数且满足 $a=a b+c 、 b=b c+a 、 c=c a+b$. 则 $a+b+c=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Suppose that $x$ and $y$ are real numbers that satisfy the system of equations

$2^x-2^y=1$
$4^x-4^y=\frac{5}{3}$

Determine $x-y$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A book has 14 chapters. Beth read the first 4 chapters in 6 hours, and she took an additional 10 minutes break between each chapter. If she continues reading at the same pace and taking breaks, in how many hours will she be able to finish reading the remaining chapters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16.5}
A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. Additionally, for this four-digit number, one prime factor minus five times another prime factor is equal to two times the third prime factor. What is this number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1221}
Gina chooses what she and her sister will watch on Netflix some times as often as her sister does. If her sister watches a total of 24 shows on Netflix per week, and each show is 50 minutes long, Gina gets to choose 900 minutes of Netflix. What is the ratio of the number of times Gina chooses to the number of times her sister chooses?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:4}
Through every three vertices of a cube, located at the ends of each trio of edges meeting at one vertex, a plane is drawn. Find the volume of the solid bounded by these planes if the edge length of the cube is $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^3}{6}}
Given \( m \) and \( n \) as natural numbers (\( m, n \in \mathbb{N} \)), find the minimum value of \( 12^m - 5^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A train of length 100 m is going with a speed of 10 m/sec. Another train of 150 m long is going with a speed of 15 m/sec. The distance between the two trains is some distance. The second train will take 60 seconds to cross the first train. What is the distance between the two trains?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Find the limit of the following expression:

$$\lim_{n \to \infty} \frac{\sum\limits_{i = 0}^{n + 1} \binom{n + 1}{i}^3}{\sum\limits_{i = 0}^n \binom{n}{i}^3}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}