inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the conditional expectation \( \mathrm{E}(A|B) \) where \( B \) is the number of bats leaving a cave at the time of a nuclear explosion, following a geometric distribution with mean \( \lambda \). Each bat contributes an independent exponentially distributed amount of wind movement with mean \( \frac{1}{\mu} \). Let \( A \) be the total amount of wind movement at the time of the explosion. Assume no wind movement is accumulated if no bats are flying.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{B}{\mu}}
In triangle $ABC$ with $AB=8$ and $AC=10$, the incenter $I$ is reflected across side $AB$ to point $X$ and across side $AC$ to point $Y$. Given that segment $XY$ bisects $AI$, compute $BC^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Let $r, s,$ and $t$ be the distinct roots of $x^3- 2022x^2 + 2022x + 2022.$ Compute $$\frac{1}{1-r^2} + \frac{1}{1-s^2} + \frac{1}{1-t^2}.$$ 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \frac{2025}{2023} \]}
A workshop has 11 workers, of which 5 are fitters, 4 are turners, and the remaining 2 master workers can act as both fitters and turners. If we need to select 4 fitters and 4 turners to repair a lathe from these 11 workers, there are __ different methods for selection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{185}
Assume $a_n > 0$ and $\lim\limits_{n \to \infty}\dfrac{a_1+a_2+\cdots+a_n}{n}=a<+\infty$. Find the limit \( \lim\limits_{n \to \infty}\dfrac{a_1^p+a_2^p+\cdots+a_n^p}{n^p} \) where $p > 1$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There is an oil tank in the shape of a right frustum that can hold 190L of oil. If the lengths of its two bases are 60cm and 40cm respectively, find its depth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75 \text{ cm}}
Find the maximum value of the expression \[|1+x|+|1+xy|+|1+xy^2|+\cdots+|1+xy^{2013}|-1007|1+y|\] given that \(|x| = |y| = 1\) for complex numbers \(x\) and \(y\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014\sqrt{2}}
What is the minimal Hamming distance of the concatenated word $R$, where $R$ is formed by concatenating a word $M$ of size $n^2$ bits (organized into an $n \times n$ matrix with $n > 3$) and the encoded main diagonal $B$ using a Hamming code with a minimal distance of 3, given that $M$ is unencoded with a Hamming distance of 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
已知$A = \{ x \vert x ^ { 2 } - 4 x + 3 < 0 , x \in R \}$,

$B = \{ x \vert 2 ^ { 1 - x } + a \leq 0 , x ^ { 2 } - 2 ( a + 7 ) x + 5 \leq 0 , x \in R \}$,

若A⊆B，则实数a的取值范围是
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4≤a≤-1.}
Typhoon "Haima" moves northward at a speed of 25 km/h. The observation station is located at point A on the sea. At 9 a.m., the typhoon's center was observed to be at point B, southeast of the station. At 10 a.m., the typhoon's center was observed to be at point C, in a south-southeast direction 75° from the station. At this time, the distance between the observation station and the typhoon's center, AC, is __________ km.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25\sqrt{2}}
Evaluate the limit \( \lim_{n \to \infty} n\left(\left(2x^{\frac{1}{n}}-1\right)^{n}-x^2\right) \) for \( x \ge 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-x^2 (\ln x)^2}
已知集合 A=\{1,2,\cdots,6\}，映射 f:A\rightarrow A 为一一映射. 则满足对于任意的 x\in A，f^{(12)}(x)=x 的映射 f 的个数为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{576}
Given the basis \(B = \{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3, \mathbf{e}_4\}\) for \(\mathbb{R}^4\), where \(\mathbf{e}_1 = (1, 0, 0, 0)\), \(\mathbf{e}_2 = (1, 1, 0, 0)\), \(\mathbf{e}_3 = (1, 1, 1, 0)\), and \(\mathbf{e}_4 = (1, 1, 1, 1)\), and the dual basis \(B^\perp = \{\mathbf{e}^1, \mathbf{e}^2, \mathbf{e}^3, \mathbf{e}^4\}\), where \(\mathbf{e}^1 = (1, -1, 0, 0)\), \(\mathbf{e}^2 = (0, 1, -1, 0)\), \(\mathbf{e}^3 = (0, 0, 1, -1)\), and \(\mathbf{e}^4 = (0, 0, 0, 1)\), find the dot product \(\mathbf{u} \cdot \mathbf{v}\) where \(\mathbf{u} = \mathbf{e}^1 - \mathbf{e}^2 + \mathbf{e}^3 - \mathbf{e}^4\) and \(\mathbf{v} = (1, 2, 3, 4)\). Use the covariant components of \(\mathbf{v}\) and the contravariant components of \(\mathbf{u}\) to compute the dot product.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
There are $2022$ distinct integer points on the plane. Let $I$ be the number of pairs among these points that are exactly $1$ unit apart. Find the maximum possible value of $I$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3954}
A function $S(m, n)$ satisfies the initial conditions $S(1, n) = n$ and $S(m, 1) = 1$, along with the recurrence relation $S(m, n) = S(m - 1, n)S(m, n - 1)$ for $m \geq 2$ and $n \geq 2$. Determine the largest integer $k$ such that $2^k$ divides $S(7, 7)$. \( \text{Provide your answer as an integer.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{370}
Calculate the remainder when \(787^{777}\) is divided by \(100\). Use modular arithmetic to determine the tens digit efficiently.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the line integral \( E = \int_\gamma \frac{-y \cdot dx + x \cdot dy}{x^2 + y^2} \) for the curve \( \gamma \), which is the parabola \( y = 2x^2 - 1 \) from \((1, 1)\) to \(\left(\frac{1}{2}, -\frac{1}{2}\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{2}}
Calculate the line integral of the vector field \( F(x,y) = (\sin(x) + 3x^2, 2x - e^{-y^2}) \) along the path \( \gamma_1(t) = (\cos(t), \sin(t)) \) for \( 0 \leq t \leq \pi \). The path is the upper half of the unit circle from \((1,0)\) to \((-1,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi - 2}
How many ways are there to fill in a three by three grid of cells with $0$’s and $2$’s, one number in each cell, such that each two by two contiguous subgrid contains exactly three $2$’s and one $0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
8
}
Compute the integral \( \int_S \omega \) over the set \( S = \{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2=1, z\geq 0\} \), where the differential form \( \omega \) is defined as follows: \( \omega = \frac{dydz}{x} \) when \( x \neq 0 \), \( \omega = \frac{dzdx}{y} \) when \( y \neq 0 \), and \( \omega = \frac{dxdy}{z} \) when \( z \neq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Evaluate \( \lim_{x\rightarrow 0}\frac{(1+x)^{\frac{1}{x}}+(1+2x)^{\frac{1}{2x}}+(1+3x)^{\frac{1}{3x}}-3e}{9x} \) using the fact that \( \lim_{x\to 0}\left(\frac{(1+kx)^{\frac{1}{kx}}-e}{x}\right)= \frac{-ke}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{e}{3}}
Inside a grid rectangle with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut along the boundaries of the cells (the hole does not contain boundary cells). If you cut this figure along all the horizontal lines of the grid, you get 20 strips, each 1 cell in width. How many strips will you get if, instead, you cut it along all the vertical lines of the grid? (Note: A $1 \times 1$ cell is also considered a strip!)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Chip and Dale play the following game. Chip starts by splitting $222$ nuts between two piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $222$. Then Chip moves nuts from the piles he prepared to a new (third) pile until there will be exactly $N$ nuts in any one or two piles. When Chip accomplishes his task, Dale gets an exact amount of nuts that Chip moved. What is the maximal number of nuts that Dale can get for sure, no matter how Chip acts? (Naturally, Dale wants to get as many nuts as possible, while Chip wants to lose as little as possible).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{111}
6. Three identical cones with vertex $A$ and vertex angle $\frac{\pi}{3}$ touch each other externally. Each of them touches internally a fourth cone with vertex at point $A$. Find the vertex angle of the fourth cone. (The vertex angle of a cone is the angle between its generators in the axial section.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}+2\arcsin\frac{1}{\sqrt{3}}}
In how many ways can the nine digits from 1 to 9 be placed in a $3 \times 3$ grid so that the sums of the rows and the sums of the columns are all equal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Suppose Lucy picks a letter at random from the extended set of characters 'ABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789'. What is the probability that the letter she picks is in the word 'MATHEMATICS123'?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{36}}
Given a triangle $ABC$ with $O$ as the incenter. Find the angle $A$ if the circumradii of triangles $ABC$ and $BOC$ are equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
An object with an initial temperature $T_0$ is placed in a room with a constant temperature of 30°C. After 10 minutes, the object's temperature is -10°C, and after another 10 minutes, it is -5°C. Using Newton's law of cooling, determine the initial temperature $T_0$ of the object.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{110}{7}}
Let $g_n(x) = \frac{\sin^2(nx)}{n \sin x}$. Find the limit:

$$\lim_{n\to\infty} \max_{x\in R} g_n(x)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{\pi}}
Write out the interval where the function $y = -(x - 1)^2$ is monotonically increasing.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1]}
Determine the number of ring homomorphisms from \( \mathbb{Z}[x,y] \) to \( \mathbb{F}_2[x]/(1+x+x^2+x^3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
实数$x _ { 1 } , x _ { 2 } , \cdots , x _ { 2 0 0 1 }$满足$\sum _ { k = 1 } ^ { 2 0 0 0 } \vert x _ { k } - x _ { k + 1 } \vert = 2 0 0 1 ,$，令$y _ { k } = \frac { 1 } { k } ( x _ { 1 } + x _ { 2 } + \cdots +$ $x _ { k } )$ k=1,2,⋯,2001.求$\sum _ { k = 1 } ^ { 2 0 0 0 } \vert y _ { k } - y _ { k + 1 }$ 的最大可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
Given natural numbers \(A\) and \(B\) such that \(A \le B\), you repeatedly toss a fair coin to decide the next step. If the result is heads, decrement \(A\) by 1. If the result is tails and \(A < B\), increment \(A\) by 1; otherwise, do nothing. Determine the expected number of coin flips required for \(A\) to reach 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2B - A)(A - 1)}
The number 52f3b is divisible by 7, 13, and 89. What is the value of f * 2 * 3 * b?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Evaluate the limit: \[ \lim_{x \to 0}\left[\frac{1}{\log (x+ \sqrt{1+x^2})}-\frac{1}{\log (1-x)}\right] \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the asymptotic complexity of the recurrence relation \( T(n) = T\left(\frac{3}{5} n + 1\right) + n^2 \) using Big O notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^2)}
Given that $f(x)= \lim\limits_{n\to\infty} n(x^{1/n} - 1)$, find the value of $f(xy)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln(xy)}
Three projective planes $\pi_1$, $\pi_2$, and $\pi_3$ intersect such that $\pi_1 \cap \pi_2 \cap \pi_3$ is a point, $\dim\,(\pi_1 \cap \pi_2) = 0$, $\dim\,(\pi_1 \cap \pi_3) = 1$, and $\dim (\pi_2 \cap \pi_3) =1$. Determine the dimension of $\pi_1 \lor \pi_2\lor \pi_3$. \(\boxed{\ }\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Suzie flips a fair coin 6 times. The probability that Suzie flips 3 heads in a row but not 4 heads in a row is given by \(\frac{m}{n}\), 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{19}
Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that$$f^{f^{f(x)}(y)}(z)=x+y+z+1$$for all $x,y,z \in \mathbb{N}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x)=x+1}
A random number selector can only select one of the nine integers $1, 2, ..., 9$, and it makes these selections with equal probability. Determine the probability that after $n$ selections ($n>1$), the product of the $n$ numbers selected will be divisible by $10$. Please calculate the probability when n = 2 and give the result in the form of \frac{m}{n}, and then provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
Find the mass percentage of O in Al2(CO3)3. What is the mass percentage of O in this compound?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61.54\%}
Immediately after harvest, the water content in 1 ton of tomatoes was $99 \%$. By the time of sale, the water content decreased by $4 \%$ (after drying). What is the total weight of the tomatoes now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200 \text{ kg}}
A positive integer sequence has its first term as 8 and its second term as 1. From the third term onwards, each term is the sum of the two preceding terms. What is the remainder when the 2013th term in this sequence is divided by 105?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once, and the European teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have won?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
What is the order of the symmetry group of a cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Find a solution to the differential equation \((x + 1 + f(−x)) \cdot (1 − f′(x)) = x + 1\) for \(-1 \leq x \leq 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2x - 2}
Let $L \ge 2$ be an integer, with $a_0 = 10$, $a_L = 2$, and integers $a_1, \ldots, a_{L-1} \ge 2$ such that their sum is 36. Find the maximum value of $\sum_{i=0}^{L-1} a_i a_{i+1}$ over all possible values of $L$ and $a_1, \ldots, a_{L-1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{556}
Let $M$ be a subset of $\{1,2,3... 2011\}$ satisfying the following condition: For any three elements in $M$, there exist two of them $a$ and $b$ such that $a|b$ or $b|a$.Determine the maximum value of $|M|$ where $|M|$ denotes the number of elements in $M$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The maximum value of |M| is 21.

}
Task 4.5. Write the largest eight-digit number in which all even digits are present. (Even digits: $0,2,4,6,8$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99986420}
It is known that for the inequality involving \( x \)
$$
\frac{\mathrm{e}^{x}}{x^{3}} - x - a \ln x \geqslant 1
$$
to hold for any \( x \in (1, +\infty) \), determine the range of values for the real number \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \leq -3}
Determine the set of values of $x$ for which the infinite series $\sum_{k=0}^{\infty} E_k(x)$, where $E_0(x) = x$ and $E_{n+1}(x) = e^{E_n(x)}$, converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Given a positive arithmetic geometric sequence {a_n} where the sum of the first n terms is S_n, and a_{n+1}/a_n < 1, if a_3 + a_5 = 20 and a_2 * a_6 = 64, calculate the value of S_6. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{126}
In the diagram, \( P Q = 19 \), \( Q R = 18 \), and \( P R = 17 \). Point \( S \) is on \( P Q \), point \( T \) is on \( P R \), and point \( U \) is on \( S T \) such that \( Q S = S U \) and \( U T = T R \). Find the perimeter of \(\triangle P S T\). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
The coordinates of the vertices of a triangle are $A(p, q), B(q, r), C(r, p)$, where $p<q<r$. What are the bounds of the angles at each vertex?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ < \alpha, \beta < 90^\circ}
Seven standard dice are glued together to make a solid. The pairs of faces of the dice that are glued together have the same number of dots on them. How many dots are on the surface of the solid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
a trailer carries 3 , 4 and 6 crates on a trip . each crate weighs no less than 120 kg . what is the maximum weight of the crates on a single trip ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{720}
A mouse has a wheel of cheese which is cut into  $2018$  slices. The mouse also has a  $2019$ -sided die, with faces labeled  $0,1,2,\ldots, 2018$ , and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on  $k$ , and the mouse has at least  $k$  slices of cheese remaining, then the mouse eats  $k$  slices of cheese; otherwise, the mouse does nothing. What is the expected number of seconds until all the cheese is gone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}
A variable parabola with latus rectum \( l \) touches a fixed parabola \( y^2 = 4bx \), with the axes of the two parabolas being parallel. Determine the latus rectum of the parabola that is the locus of the vertex of the moving parabola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4b - l}
Find the area of an equiangular octagon with side lengths 1, 2, 2, 4, 1, 2, 2, 4, in that order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10+9\sqrt{2}}
Find the minimum positive integer  $k$  such that there exists a function  $f$  from the set  $\Bbb{Z}$  of all integers to  $\{1, 2, \ldots k\}$  with the property that  $f(x) \neq f(y)$  whenever  $|x-y| \in \{5, 7, 12\}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \(a\) be a root of the polynomial \(x^3 + x^2 + 1\) in some extension of the field \(\mathbb{Z}_2\). Determine the multiplicative inverse of \(a+1\) in this extension.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^2}
How many 6-tuples  $ (a_1,a_2,a_3,a_4,a_5,a_6)$  are there such that each of  $ a_1,a_2,a_3,a_4,a_5,a_6$  is from the set  $ \{1,2,3,4\}$  and the six expressions
\[ a_j^2 \minus{} a_ja_{j \plus{} 1} \plus{} a_{j \plus{} 1}^2\]
for  $ j \equal{} 1,2,3,4,5,6$  (where  $ a_7$  is to be taken as  $ a_1$ ) are all equal to one another?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Let \( P(x) \) and \( Q(x) \) be distinct polynomials of degree 2020 with non-zero coefficients. Suppose they have \( r \) common real roots (counting multiplicity) and \( s \) common coefficients. Determine the maximum possible value of \( r + s \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020}
The brothers found a hoard of gold and silver. They divided it in such a way that each received 100 kg. The eldest received the most gold - 30 kg - and a fifth of all the silver. How much gold was in the hoard?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50 \text{ kg}}
Given a positive integer \( n \) that is a multiple of 5, and the least common multiple of \( n \) and \( 5! \) equals 5 times the greatest common divisor of \( n \) and \( 10! \), find the number of \( n \) that satisfy this condition. Express your answer as a single integer count.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
12 chess players take part in a tournament. Every player plays twice with each of his opponents. How many games are to be played?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
A rhombus has side length $85$ and diagonals of integer lengths. What is the sum of all possible areas of the rhombus?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20664}
Find the minimum value of the expression \( \large \dfrac{y}{x + 2} + \dfrac{x}{y + 2} + \dfrac{y^2 + 3x^2 + 4}{8xy \cdot (xy + 2)} \) for \( 1 < x \le y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In triangle $ABC$, the orthocenter is $H$. Given $AB = 84$, $AC = 32\sqrt{3}$, and $\angle BAC = 60^\circ$, points $D, E, F$ are on $AH, BH, CH$ respectively such that $7AD = AH$, $7BE = BH$, and $7CF = CH$. Calculate the value of $AB \cdot CF + BC \cdot AD + AC \cdot BE$. \(\boxed{\text{Your answer here}}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1152}
Find the prime numbers \( p \) for which there exist \( x, y \in \mathbb{N}^{*} \) such that

\[ x\left(y^{2} - p\right) + y\left(x^{2} - p\right) = 5 p \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2, 3, 7}
A basic computer, printer, monitor, and keyboard have a total price of $2,500. If the same printer, monitor, and keyboard had been purchased with an enhanced computer, the total price would be $600 more than the price of the basic computer setup. Additionally, in the enhanced setup, the price of the printer would be 1/6 of the total, the price of the monitor would be 1/5 of the total, and the price of the keyboard would be 1/8 of the total. What was the price of the basic computer in the original setup?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$975.83}
A person can swim in still water at 4 km/h. If the speed of water is 2 km/h, how many hours will the man take to swim back against the current for a certain distance, if it takes him 4 hours to do so?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Evaluate the limit: $$\lim\limits_{n \to \infty} \int_0^n \frac{1}{1 + n^2 \cos^2 x} \, dx$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $G$ be the centroid of $\triangle ABC$. Determine the position of point $P$ in the plane of $\triangle ABC$ such that $PA \cdot AG + BP \cdot BG + CP \cdot CG$ is minimized. Express this minimum value in terms of the side lengths of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3} (a^2 + b^2 + c^2)}