inclusionAI/Ring-lite-rl-data huggingface dataset:
For a real number $\alpha > 0$, consider the infinite real sequence defined by $x_1 = 1$ and 
\[ \alpha x_n = x_1 + x_2 + \cdots + x_{n+1} \text{ for } n \ge 1. \] 
Determine the smallest $\alpha$ for which all terms of this sequence are positive reals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Assume $n$ is a positive integer. Considers sequences $a_{0}, a_{1}, \ldots, a_{n}$ for which $a_{i} \in \{1,2, \ldots, n\}$ for all $i$ and $a_{n}=a_{0}$.
Suppose $n$ is odd. Find the number of such sequences if $a_{i}-a_{i-1} \not\equiv i \pmod{n}$ for all $i=1,2, \ldots, n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{

$(n-1)^n-(n-1)$}
Determine the cardinality of the set of continuous functions \( f : [0,1] \rightarrow \mathbb{R}^2 \) such that \( f(0) = a \) and \( f(1) = b \), where \( a \) and \( b \) are distinct points in \( \mathbb{R}^2 \). Express your answer in terms of beth numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\beth_1}
Evaluate the integral \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{dx}{3+2\cos x+\sin x}\) using the Weierstrass substitution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Calculate the moment of inertia about the $z$-axis of the domain $$E=\{(p,\theta,\phi)\,|\,0\le p\le2,0\le\theta\le2\pi,\frac{\pi}{3}\le\phi\le\frac{\pi}{2}\}$$ which is the domain between the cone $z=\sqrt{(x^2+y^2)/3}$ and under the half sphere $z=\sqrt{4-x^2-y^2}$. The density is constant $\rho(x,y,z)=k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{88 \pi}{15}k\)}
Given the joint moment generating function of the time-to-failure random variables of two components, \( M_{X,Y}(s,t)=\frac{1}{(1-3s-2t+6ts)} \), calculate the probability that the machine consisting of these two components will fail after the third hour.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-\frac{5}{2}}}
The light glows for every 16 seconds. Between 1:57:58 am and another time, it glowed 310.5625 times. What was the ending time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:20:47}
Let \( a_{k} \) be the coefficient of \( x^{k} \) in the expansion of
\[ (x+1)+(x+1)^{2}+(x+1)^{3}+(x+1)^{4}+\cdots+(x+1)^{99}. \]

Determine the value of \( \left\lfloor \frac{a_{4}}{a_{3}} \right\rfloor \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Consider fractions of the form \(\frac{n}{n+1}\), where \(n\) is a positive integer. If 1 is subtracted from both the numerator and the denominator, and the resultant fraction remains positive and is strictly less than \(\frac{\alpha}{7}\), determine \(\beta\), the number of these fractions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
$n$ players participate in a chess tournament where each player plays a game with every other player. The winner of a game receives 1 point, the loser receives 0 points, and a draw awards each player 0.5 points. A tournament is said to have property $P(m)$ if, among any $m$ players, there is one player who has defeated the other $m-1$ players, and there is one player who has lost to the other $m-1$ players. For a given $m$ ($m \geq 4$), find the minimum value of $n$, denoted by $f(m)$, such that in any tournament with property $P(m)$, all $n$ players have different scores.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(m) = 2m - 3}
How many multiples of 4 are there between 200 and 500?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
Determine the set \(\bigcap_{x \in A} \bigcup_{n \in \Bbb N} \left(\frac{1}{n}, x-\frac{1}{n^2}\right)\) where \(A=\{ x \in \Bbb R \mid x > \pi\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \pi]}
设整数 $n>4,(x+2 \sqrt{y}-1)^{n}$ 的展开式中 $x^{n-4}$ 与 $x y$ 两项的系数相等, 则 $n$的值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Find the minimum value of $x$ that minimizes the expression $$x+[c+p(x)wy]\left[1-\frac{p(x)y}{y^{\max}} \right]$$ given that $w=0$. Assume all other variables are constants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Suppose \( f(z) = \prod_{n=1}^{17}(z-\frac{\pi}{n}) \) for \( z \in \mathbb{C} \), and let \( \gamma(t) = e^{2it} \) for \( t \in [0, 2\pi] \). If \( \int_{\gamma} \frac{f'(z)}{f(z)}\, dz = \alpha \pi i \), find the value of \( \alpha \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
The pentagon PQRST is divided into four triangles with equal perimeters. The triangle PQR is equilateral. PTU, SUT, and RSU are congruent isosceles triangles. Find the ratio of the perimeter of the pentagon PQRST to the perimeter of the triangle PQR. Express your answer as a ratio in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{3}}
Determine the number of integer solutions to the Diophantine equation \(4x^3 - 3 = y^2\) for \(x > 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Consider digits $A, B, C, D$, with $A \neq 0$, such that $ABCD = (CD)^2 - (AB)^2$. Compute the sum of all distinct possible values of $A + B + C + D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Given the sets $A=\{1, 3, 2m-1\}$ and $B=\{3, m^2\}$; if $B \subseteq A$, find the value of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given a matrix $M \in M_{5}(\mathbb{R})$ with the characteristic polynomial $(x-2)^5$, determine the largest possible dimension of the centralizer of $M$ in $M_{5}(\mathbb{R})$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Evaluate the line integral \( I = \int_\gamma \left(4e^{(y-4x+3)^2}-y\right)dx + \left(-e^{(y-4x+3)^2}+2x\right)dy \), where \( \gamma \) is the curve along the parabola \( y = x^2 \) from \( (1,1) \) to \( (3,9) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Find the minimum value of the integral

$$ \mathcal {I} [y] = \int_{- \infty}^{\infty} \dfrac{1}{2} y'^2 + 1 - \cos y \; \mathrm d x $$

subject to the boundary conditions $y(-\infty) = 0$ and $y(\infty) = 2 \pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The ratio of the length to the width of a rectangle is $5$ : $2$. If the rectangle has a diagonal of length $13$, find the area expressed as $kd^2$ for some constant $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{29} d^2}
Jake is retrieving the notes on a communal board and sorting them into different bins based on their colors and categories. The red notes are complaints, the blue notes are compliments, the green notes are suggestions, and the yellow notes are inquiries. The board has the following distribution of notes:

- Red notes are in 5 rows with 6 notes in each row, and there are 3 more red notes scattered around the board.
- Blue notes are in 4 rows with 7 notes in each row, and another 12 blue notes are placed in various corners of the board.
- Green notes are in 3 triangular arrangements, with each triangle having 1 green note at the top and additional green notes forming the triangle's base: the first triangle has a base of 4 notes, the second triangle has a base of 5 notes, and the third triangle has a base of 6 notes.
- Yellow notes are in 2 diagonal arrangements starting from the top corners of the board: the first diagonal has 5 yellow notes, and the second diagonal has 3 yellow notes. There are also 6 yellow notes in the center of the board, forming a hexagon.

Jake needs to place the notes into the correct bins. How many notes in total are put into the complaints, compliments, suggestions, and inquiries bins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{133}
On an infinite tape, numbers are written in a row. The first number is one, and each subsequent number is obtained from the previous one by adding the smallest nonzero digit of its decimal representation. How many digits are in the decimal representation of the number that is in the $9 \cdot 1000^{1000}$-th position in this row?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3001}
Calculate the work done by the force \( \vec{F} = \sin(x^2)\hat{x} + (3x-y)\hat{y} \) in moving a particle along the path from the origin \( O \) to point \( A = (3,0) \) and then to point \( B = (0,4) \) in straight lines.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Ribamko A.V.

In the lower left corner of a $n \times n$ chessboard, there is a knight. It is known that the minimum number of moves it takes for the knight to reach the upper right corner is equal to the minimum number of moves it takes to reach the lower right corner. Find $n$.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
How many distinct ordered triples of real numbers $(a, b, c)$ exist such that $a, b, c$ are the roots of the cubic equation $x^3 + ax^2 + bx + c = 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the largest \( n \) so that the number of integers less than or equal to \( n \) and divisible by 3 equals the number divisible by 5 or 7 (or both).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
A 19-digit binary string contains all possible 4-digit binary strings exactly once. If this string starts with 1111, what are the last four digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0111}
Find the maximum value of an integer $B$ such that for every 9 distinct natural numbers with the sum of $2023$, there must exist a sum of 4 of these numbers that is greater than or equal to $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{910}
Determine the number of ways to color each rational number either white or red such that for any two distinct rational numbers \( x \) and \( y \), if they satisfy any of the following conditions: (a) \( xy = 1 \), (b) \( x + y = 0 \), or (c) \( x + y = 1 \), then \( x \) and \( y \) are painted different colors. How many such colorings, called "sanferminera" colorings, exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
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). Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \([x]\) denote the greatest integer less than or equal to the real number \(x\). If
\[
a=\frac{\sqrt{6+\sqrt{6+\cdots+\sqrt{6}}}}{2016 \text{ nested radicals}}, \\
b=\frac{\sqrt[3]{6+\sqrt[3]{6+\cdots+\sqrt[3]{6}}}}{2017 \text{ nested radicals}},
\]
then find \([a+b]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the congruence class of \( \sigma(m^2) \) modulo 4 that would make the equation \( \frac{2D(m^2)s(m^2)}{D(p^k)s(p^k)} = (\gcd(m^2, \sigma(m^2)))^2 \) invalid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Quadrilateral $ABCD$ has right angles at $B$ and $C$, $\triangle ABC \sim \triangle BCD$, and $AB > BC$. There is a point $E$ in the interior of $ABCD$ such that $\triangle ABC \sim \triangle CEB$ and the area of $\triangle AED$ is $17$ times the area of $\triangle CEB$. The original answer for \(\tfrac{AB}{BC}\) is in the form \(k + \sqrt{m}\). Please find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
For any integer $n \ge 2$, let $b_n$ be the least positive integer such that, for any integer $N$, $m$ divides $N$ whenever $m$ divides the digit sum of $N$ written in base $b_n$, for $2 \le m \le n$. Find the integer nearest to $\frac{b_{36}}{b_{25}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5394}
In a school there are 1200 students. Each student must join exactly  $k$  clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of  $k$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ k = 23 }
Seven sticks with lengths 2, 3, 5, 7, 11, 13 and 17 inches are placed in a box. Three of the sticks are randomly selected. What is the probability that a triangle can be formed by joining the endpoints of the sticks? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{35}}
For each positive integer, define a function \[ f(n)=\begin{cases}0, &\text{if n is the square of an integer}\\ \\ \left\lfloor\frac{1}{\{\sqrt{n}\}}\right\rfloor, &\text{if n is not the square of an integer}\end{cases}. \] Find the value of $\sum_{k=1}^{200} f(k)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{629}
若整数 a,b,c 满足：0\leq a\leq 10, 0\leq b\leq 10, 0\leq c\leq 10，10\leq a+b+c\leq 20，则满足条件的有序数组 (a,b,c) 共有__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{891}
Given the parabola $y^2=2px$ ($p>0$) and a point $M(1, m)$ on it, the distance from $M$ to its focus is 5. Find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm4}
Given that the graph of a power function passes through the points $(2, 16)$ and $\left( \frac{1}{2}, m \right)$, then $m = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{1}{16}"]}
Determine the dimension of the ring $R[[X]]$, where $R$ is a Noetherian ring.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dim(R) + 1}
Based on a city's rules, the buildings of a street may not have more than $9$ stories. Moreover, if the number of stories of two buildings is the same, no matter how far they are from each other, there must be a building with a higher number of stories between them. What is the maximum number of buildings that can be built on one side of a street in this city?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{511}
Given four points \( A, B, C, \) and \( D \) in space that are not coplanar, each pair of points is connected by an edge with a probability of \( \frac{1}{2} \). The events of whether any two pairs of points are connected are mutually independent. What is the probability that \( A \) and \( B \) can be connected by a spatial path (composed of one edge or a series of edges)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
Let $ABC$ be a triangle and let $P$ be a point in its interior. Suppose $ \angle B A P = 10 ^ { \circ } , \angle A B P = 20 ^ { \circ } , \angle P C A = 30 ^ { \circ } $ and $ \angle P A C = 40 ^ { \circ } $. Find $ \angle P B C $.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Given the elliptic curve \(E\) over \(\mathbb{F}_{11}\) defined by \(y^2 = x^3 + 5\) and the function \(f(x,y)=\frac{2x+y-3}{3x-y+2}\), calculate \(f(\mathcal{O})\) using projective coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
At an athletic facility with 5 tennis courts, doubles teams use a court for a duration that follows an exponential distribution with a mean of 40 minutes. If a team arrives and finds all courts occupied with k other teams waiting, what is the expected waiting time for them to get a court?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8(k + 1)}
Let \( {a}_{1},{a}_{2},\ldots ,{a}_{1000} \) be positive integers whose sum is \( S \) . If \( {a}_{n} \) ! divides \( n \)
 for each \( n = 1,2,\ldots ,{1000} \) , compute the maximum possible value of \( S \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1716
}
Let $a$ be a 28-digit palindromic number that is a multiple of 13. All the digits of $a$ are 1, except for the 13th, 14th, 15th, and 16th digits, which are $A$, $B$, $C$, and $D$ respectively. Determine the minimum possible value of $A + B + C + D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
For each integer $n\geq 2$, let $S_n$ be the sum of all products $jk$, where $j$ and $k$ are integers and $1\leq j<k\leq n$. Find the sum of the $10$ least values of $n$ such that $S_n$ is divisible by $3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{197}
Calculate the expected value of the product of the number of balls in each urn when K balls are uniformly distributed into S urns. For example, if K = 8 and S = 3, one possible distribution is (4, 2, 2), resulting in a product of 16. Determine the expected value for K = 8 and S = 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{112}{9}}
9-8. All digits in the notation of 6-digit natural numbers $a$ and $b$ are even, and in the notation of any number between them, there is an odd digit. Find the largest possible value of the difference $b-a$.

---

Translation:
9-8. All digits in the notation of 6-digit natural numbers $a$ and $b$ are even, and in the notation of any number between them, there is an odd digit. Find the largest possible value of the difference $b-a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{111112}
Evaluate the integral \( \int_C{\omega} \) where \( \omega=\frac{dx \wedge dy}{x^2+y^2} \) and \( C(t_1,t_2)=(t_1+1)(\cos(2\pi t_2),\sin(2\pi t_2)) : I_2 \rightarrow \mathbb{R}^2 - \text{{(0,0)}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi \ln 2}
If the distance from a point $(x_{0}, y_{0})$ on the parabola $x^{2}=28y$ to the focus is $3$ times the distance from the point to the $x$-axis, then $y_{0}=\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{2}}
Define $g_n(x) = 3x^ne^{-x^2}-1$. For $n>1$, let $w_n$ be a real number such that $g_n(w_n)=0$. Determine the value of $\lim\limits_{n\to\infty}\frac{w_n^2}{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of distinct ring homomorphisms from the ring of real numbers \((\mathbb{R}, +, \times)\) to itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $n$ be a positive integer with $k$ digits. A number $m$ is called an \( alero \) of \( n \) if there exist distinct digits \( a_1, a_2, \dotsb, a_k \), all different from each other and from zero, such that \( m \) is obtained by adding the digit \( a_i \) to the \( i \)-th digit of \( n \), and no sum exceeds 9.

For example, if \( n = 2024 \) and we choose \( a_1 = 2 \), \( a_2 = 1 \), \( a_3 = 5 \), \( a_4 = 3 \), then \( m = 4177 \) is an alero of \( n \), but if we choose the digits \( a_1 = 2 \), \( a_2 = 1 \), \( a_3 = 5 \), \( a_4 = 6 \), then we don't obtain an alero of \( n \), because \( 4 + 6 \) exceeds 9.

Find the smallest \( n \) which is a multiple of \( 2024 \) that has an alero which is also a multiple of \( 2024 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10120}
Determine all positive integers relatively prime to all the terms of the infinite sequence \[ a_n=2^n+3^n+6^n -1,\ n\geq 1. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["1"]}
Simplify the following expression in the finite field $\mathbb{F}_{2^4}$ using the irreducible polynomial $f(x) = x^4 + x + 1$:

$$\left( \frac{\alpha^{10} + \alpha^{15}}{\alpha^8+\alpha^5} \right)^2 + \frac{\alpha^{10} + \alpha^{15}}{\alpha^8+\alpha^5} + \alpha^8 + \alpha^5 + \alpha^6$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The high-speed train "Sapsan," approaching a railway station at a speed of \( v = 216 \) km/h, emits a warning sound signal lasting \( \Delta t = 5 \) seconds when it is half a kilometer away from the station. What will be the duration of the signal \( \Delta t_{1} \) from the perspective of passengers standing on the platform? The speed of sound in the air is \( c = 340 \) m/s.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.12 \text{ seconds}}
Let $G$ be a permutation group on a finite set $\Omega$ with orbits $\Omega_1,\ldots,\Omega_k$. A transversal is a set $\{\omega_1,\ldots,\omega_k\}$ with $\omega_j\in\Omega_j$ for each $j$. For a transversal $\tau=\{\omega_1,\ldots,\omega_k\}$ and subset $H\subseteq G$, define
\[H(\tau) = \bigcup_{h\in H} \{\omega_1^h,\ldots,\omega_k^h\}. \]
Let $N(G)$ be the smallest integer such that for some transversal $\tau$ and some $H\subseteq G$ with $|H|=N$, we have $H(\tau)=\Omega$. Given $n=|\Omega|$ and $m=\max_{i=1}^k |\Omega_i|$, determine $N(G)$. It is known that $N(G)\ge m$ and $N(G)\le m\log n$. What is $N(G)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m}
Define a function \( f \), whose domain is positive integers, such that:
$$
f(n)=\begin{cases}
n-3 & \text{if } n \geq 1000 \\
f(f(n+7)) & \text{if } n < 1000
\end{cases}
$$
Find \( f(90) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999}
We see a rod of length $a$ on top of a tower of height $b$. At what distance from the tower do we see this rod at the maximum angle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{b(a + b)}}
已知 $a, b \in \mathbf{N}^{*}, a+b \leq 2024$, 使得 $a b^{2}+b+7$ 整除 $a^{2} b+a+b$ 的解的 $(a, b)$ 有 $\qquad$组.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
For nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Let $g(n)=3^{f(n)}$. Compute
\[\sum_{i=1}^{1000} g(i).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
8000
}
Let $k$ and $n \ge 3$ be two natural numbers. How many strings in $\{1,...,n\}^k$ contain at least one occurrence of $1$ and $2$, or at least one occurrence of $2$ and $3$ or at least one occurrence of $1$ and $3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^k-3(n-2)^k+2(n-3)^k}
If the positive integers \( m \) and \( n \) satisfy \( \frac{(m+n)!}{n!}=5040 \), then the value of \( m!n \) is ________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
For constants $a,\ b,\ c,\ d$ consider a process such that the point $(p,\ q)$ is mapped onto the point $(ap+bq,\ cp+dq)$.
Note : $(a,\ b,\ c,\ d)\neq (1,\ 0,\ 0,\ 1)$. Let $k$ be non-zero constant. All points of the parabola $C: y=x^2-x+k$ are mapped onto $C$ by the process.
Find $a,\ b,\ c,\ d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(a, b, c, d)=(-1, 0, 2, 1)$}
Determine the intrinsic dimension of the set \( M = \{ (x,y,z) \in \mathbb{R}^3 : (a,b,c,d) \in \mathbb{R}^4\} \) given the parameterized functions:
\[ x(a,b,c,d) = ac \]
\[ y(a,b,c,d) = bc + ad \]
\[ z(a,b,c,d) = bd \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}