inclusionAI/Ring-lite-rl-data huggingface dataset:
Given a parabola $C$: $y^{2}=4x$ with a point $M(4,-4)$ on it. Points $A$ and $B$ are two moving points on the parabola such that $\overrightarrow{MA} \cdot \overrightarrow{MB}=0$. Determine the maximum distance from point $M$ to line $AB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \sqrt {5}}
What is the term for a partially ordered set that is isomorphic to its own dual?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{self-dual}
Let $F$ and $E$ be fields with $F < E$. Suppose $V$ is a vector space over $F$ with $\mathrm{dim}_F(V) = n$. Consider $V^E = V \otimes_F E$, viewing $V$ as a vector space over $E$. What is $\mathrm{dim}_E(V^E)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
A and B play a game on a $2^{100} \times 100$ grid. They take turns filling in symbols in the first row, with A going first. In each round, A chooses an empty cell in the first row and fills it with an "$\times$"; then, B chooses an empty cell in the first row and fills it with an "○". Once all the cells in the first row are filled, they start filling in the symbols in the second row, and so on, until all the cells in the grid are filled.

A's goal is to have as many distinct rows as possible in the grid, while B's goal is to have as few distinct rows as possible. If both use their optimal strategies, how many distinct rows will there be in the grid in the end?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{50}}
Given that $F\_1$ and $F\_2$ are the two foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > b > 0)$, and $P$ is a point on the hyperbola such that $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2} = 0$ and $|\overrightarrow{PF\_1}| \cdot |\overrightarrow{PF\_2}| = 2ac (c$ is the semi-focal distance$)$, find the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5} + 1}{2}}
A circle passes through vertices \(A\), \(B\), and \(C\) of a parallelogram \(ABCD\) with sides \(AB = 5\) and \(BC = 2\) and intersects the line containing the diagonal \(BD\) at point \(E\). It is known that \(BE = 8\). Find \(BD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{29}{8}}
Elijah is painting his L-shaped living room, which consists of a main rectangle and an adjacent square. The main rectangular part of the room has two walls that are 5 meters wide and two walls that are 3 meters wide, while the square part of the room has walls that are 4 meters wide. The height of the room is 2.5 meters. Elijah needs enough masking tape to cover the entire width of the walls and windows, as well as tape off the door frames located on one of the 3-meter walls.

He decides to apply two layers of masking tape in order to ensure proper coverage. There are 3 windows in the living room with dimensions of 1.5 meters by 1 meter and a door frame that is 2 meters high and 1 meter wide. Elijah can order the exact amount of masking tape he needs. Considering the dimensions of the room, doors, and windows, how many meters of masking tape does Elijah need to order?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
A sphere of radius \( r \) touches all the lateral faces of a triangular pyramid at the midpoints of the sides of its base. The segment connecting the apex of the pyramid with the center of the sphere is divided into two equal parts by the point of intersection with the base of the pyramid. Find the volume of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{r^3 \sqrt{6}}{4}}
Given that the real numbers $a$, $b$, and $c$ form an arithmetic sequence, and the projection point $M$ of point $P$($-3$, $0$) on the moving line $ax + by + c = 0$ ($a$ and $b$ are not simultaneously zero). If the coordinates of point $N$ are ($2$, $3$), find the range of $|MN|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[5 - \sqrt{5}, 5 + \sqrt{5}]}
Determine the maximum possible value of \[\frac{\left(x^2+5x+12\right)\left(x^2+5x-12\right)\left(x^2-5x+12\right)\left(-x^2+5x+12\right)}{x^4}\] over all non-zero real numbers  $x$ .

*2019 CCA Math Bonanza Lightning Round #3.4*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{576}
Let $n$ be a positive integer such that no matter how $10^n$ is expressed as the product of two positive integers, at least one of these two integers contains the digit 0. Find the smallest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, if $a_2a_4a_6a_8=120$ and
$$\frac {1}{a_{4}a_{6}a_{8}}+ \frac {1}{a_{2}a_{6}a_{8}}+ \frac {1}{a_{2}a_{4}a_{8}}+ \frac {1}{a_{2}a_{4}a_{6}}= \frac {7}{60},$$
find the value of $S_9$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {63}{2}}
Given an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$, where the left and right vertices are $A$ and $B$ respectively, the right focus is $F$, $P$ is a point on the ellipse, the line $AP$ intersects the line at point $M$, the angle bisector of $\angle PFB$ intersects the line $x=a$ at point $N$. If $PF\perp AB$, the area of triangle $MAB$ is $6$ times the area of triangle $NFB$, then the eccentricity of the ellipse $C$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Determine the values of \( \alpha \in \mathbb{R} \) such that the function \((x,y) \mapsto (1-xy)^{\alpha}\) is integrable on the region \((0,1)\times (0,1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, \infty)}
Let \(T_1\) be a triangle with side lengths \(2011\), \(2012\), and \(2013\). For \(n \geq 1\), if \(T_n = \triangle ABC\) and \(D, E\), and \(F\) are the points of tangency of the incircle of \(\triangle ABC\) to the sides \(AB\), \(BC\), and \(AC\), respectively, then \(T_{n+1}\) is a triangle with side lengths \(AD, BE\), and \(CF\), if it exists. Find the perimeter of the last triangle in the sequence \(\left(T_n\right)\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1637}
A company employs 3 workers, each with a tenure following an exponential distribution with a mean of 1 year before quitting. The time to hire a replacement worker is also exponentially distributed with a mean of 0.1 year, independently for each position. The daily revenue based on the number of workers is as follows: 3 workers - $1000, 2 workers - $800, and 0 or 1 worker - $500. Calculate the company's expected long-run average daily revenue.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{943.28}
Determine the sum of the infinite series $$\sum_{n=1}^{\infty} \frac{f(n)}{n(n+1)}$$, where $f(n)$ represents the number of ones in the binary representation of the integer $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \(a > 1\) be a positive real number and \(n \geq 2\) be a natural number. The equation \(\lfloor a x \rfloor = x\) has exactly \(n\) different solutions. Find the range of values for \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \frac{1}{n} \leq a < 1 + \frac{1}{n-1}}
Cathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Cathy happy, but RRRGR will not. The probability that Kathy will be happy is $\frac{m}{n}$, 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{$157$}
Let $A B C$ be an acute scalene triangle with circumcenter $O$ and centroid $G$. Given that $A G O$ is a right triangle, $A O=9$, and $B C=15$, let $S$ be the sum of all possible values for the area of triangle $A G O$. Compute $S^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
The power function $y=(m^2-m-1)x^{1-m}$ is decreasing for $x \in (0, +\infty)$. Find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the value of $$\sum_{k=1}^{2011} \frac{k-1}{k!(2011-k)!}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{2009\left(2^{2010}\right)+1}{2011!}$}
Let $w = e^{i\cdot \pi/2022}$. Determine the value of the sum $T = w^1 + w^3 + w^5 + ... + w^{2021}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a fluid with a velocity field and density, calculate the rate of change of density in the Lagrangian frame, where the initial density \( \rho_0 = 2 \) and \( A = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
7. 

In triangle $A B C$, the bisector $A L$ ( $L \in B C$ ) is drawn, and $M$ and $N$ are points on the other two bisectors (or their extensions) such that $M A=M L$ and $N A=N L, \angle B A C=50^{\circ}$.

Find the measure of $\angle M A N$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Given that point $P\left(x,y\right)$ is located in the second quadrant, and $y\leqslant x+4$, where $x$ and $y$ are integers, write down the coordinates of a point $P$ that satisfies the above conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1,3)}
How many solutions does the equation \( \frac{\lfloor x \rfloor}{\{ x \}} = \frac{2007x}{2008} \) have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2006}
Given the line equation ax + by + c = 0, where ab < 0 and bc < 0, determine the quadrant that the line does not pass through. Express your answer as the name of the quadrant (e.g., First, Second, Third, Fourth).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Fourth}
Given the sequence $\left\{a_{n}\right\}$ satisfying $a_{1}=1$ and $a_{n+1}=a_{n}+\frac{1}{2 a_{n}}$, find $\lim _{n \rightarrow \infty}\left(a_{n}-\sqrt{n}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Higher Secondary P4

If the fraction $\dfrac{a}{b}$ is greater than $\dfrac{31}{17}$ in the least amount while $b<17$, find $\dfrac{a}{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{6}}
Evaluate the limit: $$\lim_{n \to \infty} \frac{1}{n^3} \sum_{i=1}^n \left(e^{\frac{2i}{n}} + \frac{4i^2}{n^2}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the line integral \( \int_C \vec{B} \cdot d\vec{r} \), where \( \vec{B} = \frac{\cos(\alpha)}{p^2}e_p + \left(\frac{\sin(\alpha)}{p^2}+\frac{1}{ap}\right)e_\alpha + 0e_z \) and the curve \( C \) is parameterized by \( \vec{r} = (b\cos(s),b\sin(s),b) \) with \( b>0 \) and \( s: 0\rightarrow2\pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi}{a}}
Calculate the mass of a plate with surface density \( \mu = \frac{x}{y^5} \), bounded by the curves \( \frac{x^2}{16} + y^2 = 1 \), \( \frac{x^2}{16} + y^2 = 3 \), \( y = \frac{x}{4} \), and \( x = 0 \) where \( y \geq \frac{x}{4} \) and \( x \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A positive integer is called powerful if every prime in its prime factorization is raised to a power greater than or equal to $2$. How many positive integers less than 100 are powerful?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Simplify the expression into a closed form: $$\frac{1}{6}\sum_{n=0}^{\infty} \sum_{k=0}^{n} \left[{n \choose k}\left(\frac{1}{3}\right)^{k}\left(\frac{1}{2}\right)^{n-k} (2+5k+3(n-k))\right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Each cell of a \(3 \times 3\) square contains a natural number. All the numbers are distinct and not equal to one. It is known that the number in each cell is a divisor of the product of all the numbers in the cells adjacent to it (by side). Find the maximum possible number of prime numbers among those written in the cells.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the probability that if the digits $0,1,2, \ldots, 9$ are placed in random order into the empty places in the row of digits

$$
5-383-8-2-936-5-8-203-9-3-76
$$

the resulting number will be divisible by 396.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
There are two people each wearing red and yellow clothes, and one person wearing blue clothes. Now, these five people are to be arranged in a row, with the requirement that people wearing the same color clothes cannot be adjacent. The total number of different arrangements is ___.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Circle $\omega_1$ has radius $7$ and center $C_1$. Circle $\omega_2$ has radius $23$ and center $C_2$ with $C_1C_2 = 34$. Let a common internal tangent of $\omega_1$ and $\omega_2$ pass through $A_1$ on $\omega_1$ and $A_2$ on $\omega_2$, and let a common external tangent of $\omega_1$ and $\omega_2$ pass through $B_1$ on $\omega_1$ and $B_2$ on $\omega_2$ such that $A_1$ and $B_1$ lie on the same side of the line $C_1C_2$. Let $P$ be the intersection of lines $A_1A_2$ and $B_1B_2$. Find the area of quadrilateral $PC_1A_2C_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{345}
One day, School A bought 56 kilograms of fruit candy at 8.06 yuan per kilogram. A few days later, School B also needed to buy the same 56 kilograms of fruit candy, but it happened that there was a promotional event, and the price of fruit candy was reduced by 0.56 yuan per kilogram. Additionally, they received 5% extra fruit candy for free. How much less did School B spend compared to School A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51.36}
Evaluate the integral: $$\int_{-\infty}^\infty \frac{\cos(x)}{x+i}\,dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi i e^{-1}}
Let $S$ be the set of all positive integers $n$ such that each of the numbers $n + 1$, $n + 3$, $n + 4$, $n + 5$, $n + 6$, and $n + 8$ is composite. Determine the largest integer $k$ with the following property: For each $n \in S$, there exist at least $k$ consecutive composite integers in the set \{$n, n + 1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9$\}.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
将正九边形的每个顶点等概率地涂上红、蓝两种颜色之一, 则存在三个同色的顶点构成锐角三角形的概率为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{503}
Find an isomorphism between the localization of the ring $\mathbb{Z}_{12}$ at the set $S=\{4,8\}$ and a familiar ring.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_3}
Let $\mathbb{N}_0$ be the set of non-negative integers. Consider a triple $(f, a, b)$, where $f$ is a function from $\mathbb{N}_0$ to $\mathbb{N}_0$, and $a, b \in \mathbb{N}_0$ that satisfies the following conditions:

- $f(1) = 2$
- $f(a) + f(b) \leq 2\sqrt{f(a)}$
- For all $n > 0$, $f(n) = f(n-1)f(b) + 2n - f(b)$

Find the sum of all possible values of $f(b + 100)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10201}
Suppose that $Y$ is a positive continuous random variable with probability density function (pdf) $f_Y(t) = t f_X(t)$, where $f_X(t)$ is the pdf of a positive continuous random variable $X$. Given that $\operatorname{var}(Y) = 2$, determine $\operatorname{var}(X)$. \( \text{Provide your answer as a numerical value.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine whether the series \( \sum\limits_{n=1}^\infty q^{\sqrt{n}} \) converges for \( q > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 < q < 1}
Let $K$ be an algebraically closed field of characteristic $0$. Determine the polynomial condition for $a \in K$ such that the polynomial $f(x) = x^3 - 3ax + 1$ has distinct roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4a^3 - 1}
Calculate the line integral of the vector field along the curve \( y = x^2 \) from \( x = 0 \) to \( x = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Compute the limit: $$\lim_{n \to \infty} \int_0^{n^2} e^{-x^2}n\sin\frac{x}{n} \, dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the chromatic number of a graph $H$ where the vertices represent non-empty subsets of $[n]$, and two vertices are connected if their corresponding subsets have a non-empty intersection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}}
Suppose X, Y, and Z are independent exponential random variables with rate \( \lambda \). Calculate the probability that \( X \le 2Y \le 3Z \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{18}{55}}
Determine the number of real solutions to the differential equation \( y'' = -4yy' \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Each edge of a cube is increased by $50 \%$. The percent of increase of the surface area of the cube is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
125
}
Find the minimal polynomial of the linear transformation $T_p: M_{2\times 2}^{\mathbb{C}} \to M_{2\times 2}^{\mathbb{C}}$ defined by $T_p(X) = P^{-1}XP$, where $P = \begin{pmatrix} i & 2 \\ -1 & -i \end{pmatrix}$. Determine the minimal polynomial of $T_p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 - 1}
Let $G = \langle g_1, g_2 \rangle$ be a group, and let $H = \langle g_1 g_2 g_1^{-1} g_2^{-1} \rangle$ be a subgroup of $G$. Determine the normal subgroup of $G$ generated by $H$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[G, G]}
What is the term used to describe a topological space where every proper filter can be refined by a Cauchy filter, which is a stronger condition than being totally bounded?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{compact}
Compute the limit: $$\lim_{n\to\infty} \frac{1}{\log(n)}\sum _{k=1}^n \frac{\cos (\sin (2 \pi \log (k)))}{k}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{J_0(1)}
What is the least positive integer value of $x$ such that $(3x)^2 + 3 \cdot 29 \cdot 3x + 29^2$ is a multiple of 43?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Compute $i^{-100}+i^{-99}+i^{-98}+\cdots+i^{-1}+i^0+i^1+\cdots+i^{99}+i^{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Rabbits Peter and Pauline have three offspring--Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$204$}
Let $M$ be the set of those positive integers which are not divisible by $3$. The sum of $2n$ consecutive elements of $M$ is $300$. Determine $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let \( \mathbb{Z}_{\ge 0} \) denote the set of nonnegative integers. Define a function \( f:\mathbb{Z}_{\ge 0} \to\mathbb{Z} \) with \( f(0)=1 \) and \[ f(n)=512^{\left\lfloor n/10 \right\rfloor}f\left(\left\lfloor n/10 \right\rfloor\right) \] for all \( n \ge 1 \). Determine the number of nonnegative integers \( n \) such that the hexadecimal (base \( 16 \)) representation of \( f(n) \) contains no more than 2500 digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10010}
Given $0 \leqslant \alpha \leqslant \frac{\pi}{2}, 0<\beta \leqslant \frac{\pi}{2}$, find the minimum value of $\cos ^{2} \alpha \sin \beta+\frac{1}{\sin \beta}$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The altitudes of an acute-angled triangle \(ABC\) drawn from points \(B\) and \(C\) are extended to intersect with the circumcircle at points \(B_1\) and \(C_1\). It turned out that the segment \(B_1C_1\) passes through the center of the circumcircle. Find the angle \(BAC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
If a square is divided into parts by three lines, there is always a polygon among the resulting parts whose diameter is not less than $\sqrt{13}$. What is the minimum side length of the square? (The diameter of a polygon is the distance between its two farthest vertices.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $V = \mathbb{R}^n$ with a non-standard inner product defined by a metric matrix $G$. Let $L$ be the linear operator on $V$ represented by the matrix $A$. Find the matrix representation of the adjoint operator $L^A$ in terms of the matrix $A$ and the metric matrix $G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{G^{-1} A^\top G}
In triangle \( ABC \), the measure of angle \( \angle B \) is \( 16^\circ \). Let \( I \) be the incenter and \( E \) be the excenter relative to side \( BC \). If \( IE = 2 \times AC \), calculate the measure of angle \( \angle ACB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132^\circ}
Tian Ji and the King of Qi had a horse race. Tian Ji's top horse is better than the King of Qi's middle horse but worse than the King of Qi's top horse. Tian Ji's middle horse is better than the King of Qi's bottom horse but worse than the King of Qi's middle horse. Tian Ji's bottom horse is worse than the King of Qi's bottom horse. Now, if a horse is randomly selected from each side for a race, what is the probability that the King of Qi's horse wins? Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$, $k \geq 7$, and for which the following equalities hold:

$$d_7 = 2d_5 + 1\text{ and }d_7 = 3d_4 - 1$$

[i]Proposed by Mykyta Kharin[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2024}
The line $L_{1}$: $ax+(1-a)y=3$ and $L_{2}$: $(a-1)x+(2a+3)y=2$ are perpendicular to each other, find the values of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-3"]}
A train of a certain length is running at a speed of 55 km/hr. It crossed a platform of length 520 m in 64.79481641468682 seconds. What is the length of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{470}