inclusionAI/Ring-lite-rl-data huggingface dataset:
Evaluate the contour integral \( I = \frac{1}{2\pi i}\oint_{|z|=1}\frac{2(1-\cos(z))e^z}{z^4}dz \) using the residue theorem by finding the coefficient \( a_{-3} \) in the Laurent series expansion around \( z_0 = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine all values of the parameter \( a \) for which there exists a number \( b \) such that the system of equations \( x^{2}+y^{2}+2a(a-x-y)=64 \) and \( y=\frac{7}{(x+b)^{2}+1} \) has at least one solution \( (x, y) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-8, 15]}
Given a triple of digits \((a, b, c)\), where \(a, b, c \in \{0, 1, \dots, 9\}\), you can guess a triple of digits \((m, n, p)\) each turn. You will be informed whether the statement \(a = m\) or \(b = n\) or \(c = p\) is true, but not which specific equality holds. What is the minimum number of turns required to guarantee determining the triple \((a, b, c)\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Use Stokes' theorem to evaluate the line integral \( \int_C{ \vec{F} \cdot \vec{dr}} \), where \( \vec{F}(x, y, z) = \langle 2 x^2 y , \frac{2 x^3}{3}, 2xy \rangle \) and \( C \) is the curve of intersection of the hyperbolic paraboloid \( z = y^2 - x^2 \) and the cylinder \( x^2 + y^2 = 1 \), oriented counterclockwise as viewed from above.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Let $p$ be a prime number. Given a positive integer $n$, an $n \times n$ matrix $A$ is formed with each element $(a_{ij})$ randomly selected, with equal probability, from $[0, 1, \ldots, p - 1]$. Let $q_n$ be the probability that $\text{det}A \equiv 1 \pmod{p}$. Find $\lim_{n \to \infty} q_n$. \( \text{Express your answer as a single value.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{p}}
During a vacation, I dined with my 6 friends under the following conditions: I dined with all 6 friends on 1 day, with every 5 of them on 2 days, with every 4 of them on 3 days, with every 3 of them on 4 days, and with every 2 of them on 5 days. Additionally, each friend was present at 7 dinners and absent at 7 dinners. How many dinners did I have alone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the minimal number of vertices required to triangulate a Klein bottle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the expectation of the random variable $f(w) = 1 - w$ on the probability space $(\Omega, B, \mu)$, where $\Omega = [0,1]$, $\mu$ is the Lebesgue measure, and $B$ is the Borel $\sigma$-algebra. Use the measure $\mu_f$ defined by:

$$\mu_f = \begin{cases} 0 & x < 0 \\ x & 0 \leq x \leq 1 \\ 1 & x > 1 \end{cases}$$

Express the expectation as:

$$E(f) = \int_{\mathbb R}x\mu_f(dx) = \int_{[0,\infty)}x\mu_f(dx) - \int_{(-\infty,0)}x\mu_f(dx)$$

What is the value of $E(f)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Evaluate the integral: $$\int^1_0 \left (\sum^\infty_{n=0}\frac{\left\lfloor 2^nx\right\rfloor}{3^n} \right )^2{\rm d}x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{32}}
Evaluate the integral: \[ \int_{0}^{2\pi} e^{\cos\theta} \cos(n\theta - \sin\theta) \, d\theta \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi}{n!}}
Evaluate the limit: $$\lim_{x \rightarrow 0^+} \frac{\arctan(e^x+\arctan x)-\arctan(e^{\sin x}+\arctan(\sin x))}{x^3}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Through vertex $A$ of parallelogram $ABCD$, a line is drawn that intersects diagonal $BD$, side $CD$, and line $BC$ at points $E$, $F$, and $G$, respectively. Find the ratio $BE:ED$ if $FG:FE=4$. Round your answer to the nearest hundredth if needed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.24}
Determine the maximum value of \( k \) such that any good number can be transformed into any other good number by appending a special number to either end, inserting it between any two digits, or erasing a special number from its representation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the integral: $$\int_0^{\pi/4}\frac{\ln(\tan x)}{\sin 2x}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}
Provide an example of a metric space that is complete but not pre-compact.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
A fair standard six-sided die is tossed three times. Given that the sum of the first two tosses equals the third, what is the probability that at least one "2" is tossed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{8}{15}$
}
Calculate the probability that a sequence \((x_n)_{n \ge 1}\), defined by \(x_1 = x_2 = 1\) and \(x_{n+2} = ax_{n+1} - bx_n\), where \(a\) and \(b\) are distinct numbers chosen randomly from the set \(\{1, 2, \ldots, 10\}\), converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{10}}
Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and point $P$ is a point on the right branch of the hyperbola. $M$ is the incenter of $\triangle PF\_1F\_2$, satisfying $S\_{\triangle MPF\_1} = S\_{\triangle MPF\_2} + \lambda S\_{\triangle MF\_1F\_2}$. If the eccentricity of this hyperbola is $3$, then $\lambda = \_\_\_\_\_\_$.
(Note: $S\_{\triangle MPF\_1}$, $S\_{\triangle MPF\_2}$, $S\_{\triangle MF\_1F\_2}$ represent the area of $\triangle MPF\_1$, $\triangle MPF\_2$, $\triangle MF\_1F\_2$ respectively.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Determine the values of \( x \) for which the series \( \sum_{n=0}^{\infty} \left(\frac{n}{2n+1}\right)^{2n-1}x^n \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-4, 4)}
Find the linear function $f(x)$ such that $f(f(f(\cdots f(x)))) \geqslant 1024x + 1023$, where $f$ is applied multiple times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2x + 1}
A circle passes through the vertices \( A \) and \( B \) of triangle \( ABC \) and intersects its sides \( AC \) and \( BC \) at points \( Q \) and \( N \) respectively. Given \( AQ : QC = 5 : 2 \) and \( CN : NB = 5 : 2 \), find the length of \( AB \) if \( QN = 5\sqrt{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7\sqrt{5}}
A sequence of numbers is defined by $D_0=0$, $D_1=0$, $D_2=1$ and $D_n=D_{n-1}+D_{n-3}$ for $n\geq3$. What are the parities (evenness or oddness) of the triple of numbers $(D_{2021}, D_{2022}, D_{2023})$, where E denotes even and O denotes odd?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
(E, O, E)
}
Given a set \( \Omega_3 = \{1, 2, 3\} \), a collection of subsets \( A_1, A_2, \ldots, A_r \) is called a covering of \( \Omega_3 \) if \( \bigcup_{i=1}^r A_i = \Omega_3 \), \( \emptyset \neq A_i \subset \Omega_3 \), and \( A_i \neq A_j \) for all \( i \neq j \). Denote by \( C_3 \) the count of different coverings of \( \Omega_3 \). Given that \( C_1 = 1 \) and \( C_2 = 5 \), determine the value of \( C_3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{109}
What is the maximum number of moves required to transform one position of a Rubik's cube into another?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the coefficient of \(x^{203}\) in the expansion of \((x-1)(x^2 - 2)(x^3-3)\dots(x^{20} - 20)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Determine the number of countable models of ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Find the range of values for $p$ such that the inequality \(2r^2+(2p+2q-pq)r+2(p^2+q^2+pq)\ge0\) holds for all real numbers $q$ and $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-2, 4]}
What is the size of the largest subset $S'$ of $S = \{2^x 3^y 5^z : 0 \le x,y,z \le 4\}$ such that there are no distinct elements $p,q \in S'$ with $p \mid q$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
设实数 $x_{1}, x_{2}, \cdots, x_{21}$ 满足 $0 \leqslant x_{i} \leqslant 1(1,2, \cdots, 21)$, 求 $\sum_{i=1}^{21} \sum_{k=1}^{21}\left|x_{i}-x_{k}\right|$ 的最大值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{220}
Triangle  $ABC$  has  $AB=25$ ,  $AC=29$ , and  $BC=36$ .  Additionally,  $\Omega$  and  $\omega$  are the circumcircle and incircle of  $\triangle ABC$ .  Point  $D$  is situated on  $\Omega$  such that  $AD$  is a diameter of  $\Omega$ , and line  $AD$  intersects  $\omega$  in two distinct points  $X$  and  $Y$ .  Compute  $XY^2$ .

*Proposed by David Altizio*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
You are given two line segments of length \(2^{n}\) for each integer \(0 \leq n \leq 10\). How many distinct nondegenerate triangles can you form with three of the segments? Two triangles are considered distinct if they are not congruent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Find the number of distinct numbers in the list
\[\left\lfloor \frac{1^2}{1000} \right\rfloor, \ \left\lfloor \frac{2^2}{1000} \right\rfloor, \ \left\lfloor \frac{3^2}{1000} \right\rfloor, \ \dots, \ \left\lfloor \frac{1000^2}{1000} \right\rfloor.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{751}
Brad wanted to set up a lemonade stand to earn some money. He decided to sell lemonade in three different sizes: small, medium, and large. The cost of ingredients and the prices for each size varied. He also considered stand setup costs and the cost of advertising.

Every gallon of lemonade would yield 16 small glasses, 10 medium glasses, or 6 large glasses. Brad estimated the following costs for each size:
- Small: $2.00 per gallon
- Medium: $3.50 per gallon
- Large: $5.00 per gallon

He also decided to charge the following prices for each size:
- Small: $1.00 per glass
- Medium: $1.75 per glass
- Large: $2.50 per glass

Brad made 2 gallons of lemonade for each size, which resulted in:
- 32 small glasses
- 20 medium glasses
- 12 large glasses

During the sales process, several size-specific challenges arose:
- Brad drank 4 small glasses himself at the stand
- His best friend bought 3 medium glasses, but one glass accidentally spilled
- Brad sold all the large glasses except for 2, due to their demand

Furthermore, the expenses involved were:
- Stand setup cost: $15.00
- Advertisement cost: $10.00

Taking into account the various expenses, glasses sold, and unsold glasses of lemonade, calculate the net profit Brad earned from his lemonade stand.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$35.00}
Determine the smallest natural number written in the decimal system with the product of the digits equal to  $10! = 1 \cdot 2  \cdot 3\cdot  ... \cdot9\cdot10$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45578899}
7. Suppose that $x, y$, and $z$ are complex numbers of equal magnitude that satisfy
$$
x+y+z=-\frac{\sqrt{3}}{2}-i \sqrt{5}
$$
and
$$
x y z=\sqrt{3}+i \sqrt{5} \text {. }
$$

If $x=x_{1}+i x_{2}, y=y_{1}+i y_{2}$, and $z=z_{1}+i z_{2}$ for real $x_{1}, x_{2}, y_{1}, y_{2}, z_{1}$, and $z_{2}$, then
$$
\left(x_{1} x_{2}+y_{1} y_{2}+z_{1} z_{2}\right)^{2}
$$
can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1516}
Given the function $f(x)=\log_{2}(x^{2}-ax-3a)$, if it is a decreasing function on the interval $(-\infty, -2]$, then the range of values for $a$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-4,4)}
Tim and Allen are playing a match of tenus. In a match of tenus, the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability $3 / 4$, and in the even-numbered games, Allen wins with probability $3 / 4$. What is the expected number of games in a match?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{16}{3}}
$甲、乙、丙三位同学进行羽毛球比赛，约定赛制如下：累计负两场者被淘汰；比赛前抽签决定首先比赛的两人，另一人轮空；每场比赛的胜者与轮空者进行下一场比赛，负者下一场轮空，直至有一人被淘汰；当一人被淘汰后，剩余的两人继续比赛，直至其中一人被淘汰，另一人最终获胜，比赛结束.经抽签，甲、乙首先比赛，丙轮空.设每场比赛双方获胜的概率都为  \frac{1}{2}  .$
求丙最终获胜的概率.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{7}{16}$}
$5$ people stand in a line facing one direction. In every round, the person at the front moves randomly to any position in the line, including the front or the end. Suppose that $\frac{m}{n}$ is the expected number of rounds needed for the last person of the initial line to appear at the front of the line, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{137}
Compute the dimension of the spline space \(\mathcal{S}_\Delta^{3,2}\) on the interval \([a,b]\) with \(n\) subintervals, where \(\Delta\) denotes the grid on \([a,b]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + 3}
Let $a$ and $b$ be real numbers such that $a + b \ne 0$. Solve the equation
$$\frac{1}{(x + a)^2 - b^2} +\frac{1}{(x +b)^2 - a^2}=\frac{1}{x^2 -(a + b)^2}+\frac{1}{x^2-(a -b)^2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\dfrac{a^2 + b^2}{a + b}$}
A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_n$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10} = .4$ and $a_n\le.4$ for all $n$ such that $1\le n\le9$ is given to be $p^aq^br/\left(s^c\right)$ where $p$, $q$, $r$, and $s$ are primes, and $a$, $b$, and $c$ are positive integers. Find $\left(p+q+r+s\right)\left(a+b+c\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{660}
A function \(g\) is defined by \(g(w)=(3-2i)w^2+c w+d\) for all complex numbers \(w\), where \(c\) and \(d\) are complex numbers. If \(g(2)\) and \(g(-i)\) are both purely imaginary, find the minimum value of \(|c|+|d|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{5}}
Determine the order of the function \( f(n,k) = \sum_{i=1}^{n-1}(n-i)(k-i)\log(k-i) \) in terms of Big O notation, given that \( n < k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^2 k \log k)}
The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length $5{.}$ The numbers in positions $(5, 5), \,(2,4),\,(4,3),$ and $(3, 1)$ are $0, 48, 16,$ and $12{,}$ respectively. What number is in position $(1, 2)?$
\[ \begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
29
}
Determine the Krull dimension of $\mathbb{C}[x,y,z]/I$ where $I=(x^2-yz,xz-x)$ using the fact that $p_1=(x,y)$, $p_2=(x,z)$, and $p_3=(x^2-y,z-1)$ are prime minimal over $I$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a biregular graph with 40 red vertices each of degree 7 and 30 blue vertices each of degree 9, determine the minimum number of additional blue vertices required so that the minimum distance between any two distinct red vertices is 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral \( \int_0^\infty \frac{\sqrt{x}}{x^2+4} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Let \( \mathcal{A} = A_0A_1A_2A_3 \cdots A_{2013}A_{2014} \) be a regular 2014-simplex, meaning the 2015 vertices of \( \mathcal{A} \) lie in 2014-dimensional Euclidean space and there exists a constant \( c > 0 \) such that \( A_iA_j = c \) for any \( 0 \le i < j \le 2014 \). Let \( O = (0,0,0,\dots,0) \), \( A_0 = (1,0,0,\dots,0) \), and suppose \( A_iO \) has length 1 for \( i=0,1,\dots,2014 \). Set \( P=(20,14,20,14,\dots,20,14) \). Find the remainder when \[PA_0^2 + PA_1^2 + \dots + PA_{2014}^2 \] is divided by \( 10^6 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{348595}
What is the maximum number of sides that the projection of a convex polyhedron with $n$ faces can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 n - 4}
Determine the range of real values for \( n \) such that the integral \( I(n) = \int^{\infty}_{0} \frac{\ln(x)}{x^n(1+x)}\, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1)}
In circle $k$ with center $O$, points $A, B, C, D$ are consecutive points on the circumference such that $\angle AOB = \angle BOC = \angle COD = \alpha < 60^\circ$. Let $E$ be the projection of $D$ onto the diameter $AO$, and $F$ be the trisection point of segment $DE$ closer to $E$. If $G$ is the intersection point of $AO$ and $BF$, find the measure of angle $OGD$ as $\alpha$ approaches $60^\circ$. \( \alpha \rightarrow 60^\circ \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Determine the structure of the factor ring \( \mathbb{Z}_3[x]/(x^3+2x+1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{F}_{27}}
In triangle $ABC$, the angle at $C$ is $30^o$, side $BC$ has length $4$, and side $AC$ has length $5$. Let $ P$ be the point such that triangle $ABP$ is equilateral and non-overlapping with triangle $ABC$. Find the distance from $C$ to $ P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\sqrt{41}$}
Find the number of solutions in natural numbers for the equation \(\left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
9. (3 points) Using the digits $0, 1, 2, 3, 4$, the number of even numbers that can be formed without repeating any digit is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{163}
Let  $a=256$ . Find the unique real number  $x>a^2$  such that
\[\log_a \log_a \log_a x = \log_{a^2} \log_{a^2} \log_{a^2} x.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{32}}
已知过点 $(0,1)$ 的直线 $l$ 与曲线 $C: y=x+\frac{1}{x}(x>0)$ 交于两个不同点 $M$ 与 $N$, 求曲线 $C$ 在点 $M 、 N$ 处切线的交点轨迹。请给出该轨迹线段两个端点坐标的和，该和为\frac{m}{n}形式，请给出m+n的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
A pentagon $ABCDE$ is circumscribed about a circle. The angles at the vertices $A{}$, $C{}$ and $E{}$ of the pentagon are equal to $100^\circ$. Find the measure of the angle $\angle ACE$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Evaluate the integral: $$ \frac{\exp\left(-\frac{N}{2}\right)}{2\pi N} \iint \exp\left(-\left( \frac{x^2-2xy-2Ny+2y^2}{2N}\right)\right) \,dx\,dy $$ where $N > 0$. Provide your answer in terms of $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the shortest possible length of an algorithm that generates a harlequin pattern on a Rubik's cube. The pattern should use at most 6 colors, with no more than 2 facelets of the same color, and no adjacent facelets of the same color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the rightmost nonzero digit of $ \frac{100!}{5^{20}}$ (here $ n!=1\cdot2\cdot3\cdot\ldots\cdot
n$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6
}
Find the values of \( x \) for which the distance between the vector \( A = (1, 1, 1) \) and the subspace \( L \) spanned by the vectors \( (1, 0, -1) \) and \( (3, 5, x) \) in \( \mathbb{R}^3 \) is maximal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8}
Evaluate the infinite sum: $$\sum_{n=0}^\infty \binom{2n+2}{n}\left(\frac{2}{3}\right)^n\left(\frac{1}{3}\right)^{n+2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
How many of the first $500$ positive integers can be expressed in the form
\[\lfloor 3x \rfloor + \lfloor 6x \rfloor + \lfloor 9x \rfloor + \lfloor 12x \rfloor\]
where \( x \) is a real number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
An $n$-digit positive integer is cute if its $n$ digits are an arrangement of the set $\{1,2,\ldots,n\}$ and its first $k$ digits form an integer that is divisible by $k$, for $k = 1,2,\ldots,n$. For example 321 is a cute 3-digit integer because 1 divides 3, 2 divides 32, and 3 divides 321. How many cute 6-digit integers are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2
}
The lateral face of a regular quadrangular pyramid forms a $45^{\circ}$ angle with the base plane. Find the angle between the opposite lateral faces.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Let $\triangle A_1B_1C_1$ be an equilateral triangle with an area of $60$. Chloe constructs a new triangle $\triangle A_2B_2C_2$ as follows: 

- She flips a coin. 
  - If it lands on heads, she constructs point $A_2$ such that $B_1$ is the midpoint of $\overline{A_2C_1}$. 
  - If it lands on tails, she constructs $A_2$ such that $C_1$ is the midpoint of $\overline{A_2B_1}$. 

- She performs similar operations for $B_2$ and $C_2$.

What is the expected value of the area of $\triangle A_2B_2C_2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{195}
Assume that $X$ has five path components. If $CX$ is the cone on $X$, determine the relative homology group $H_1(CX,X)$. Provide your answer in terms of an isomorphism to a known group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}^4}
Let $A B C$ be a triangle with $A B=A C=\frac{25}{14} B C$. Let $M$ denote the midpoint of $\overline{B C}$ and let $X$ and $Y$ denote the projections of $M$ onto $\overline{A B}$ and $\overline{A C}$, respectively. If the areas of triangle $A B C$ and quadrilateral $A X M Y$ are both positive integers, find the minimum possible sum of these areas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1201}
Find the original function \( f(x, y) \) given its gradient components: \( \frac{\partial f}{\partial x} = 2xy - y \) and \( \frac{\partial f}{\partial y} = x^2 - x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 y - x y}
Three metal cubes with edges of 3 dm, 4 dm, and 5 dm, along with a metal cylinder of height 7 dm and a base diameter of 4 dm, and a metal rectangular prism of dimensions 2 dm x 3 dm x 6 dm, are melted and formed into a single cube. Find the edge of the new cube.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}