inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the inverse Laplace transform of \( F(s) = \frac{1}{s(\cosh{\sqrt{2s}}-1)} \). If an analytic expression is not possible, provide an asymptotic expansion for \( f(t) = \mathcal{L}^{-1}[F](t) \) as \( t \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t - \dfrac{1}{6}}
Given points $P = (1: -2: 3)$, $Q = (2: 2: -1)$, and $R = (3: 0: 2)$ on a line $g$ in $\\mathbb{P}^{2}\mathbb{R}$, find a coordinate for $g$ such that $(P, Q, R)$ has coordinates $(0, 1, ∞)$. Determine the number of possible coordinates for $g$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \(g\) be a continuously differentiable function on \(\mathbb{R}^3\). Consider the closed ball \(B = \{(x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \leq 1\}\). Since \(g\) is continuous on \(B\), it attains its maximum value at some point \(u \in B\), with \(g(u) < \infty\). If \(u\) lies on the boundary of the ball (\(x^2 + y^2 + z^2 = 1\)), there exists an \(a \in \mathbb{R}\) such that \(\nabla g(u) = au\). Determine the sign of \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{positive}
Let \(k\) be an algebraically closed field, and let \(q(x) = \frac{r(x)}{s(x)}\) be a nonzero rational function in \(k(x)\), where \(r\) and \(s\) are coprime polynomials and \(d = \deg r \ge \deg s\). Determine whether the degree of the field extension \([k(x):k(q(x))]\) is equal to \(d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d}
Find the splitting field and its degree over \( \mathbb{Q} \) for the polynomial \( f(x) = x^4 + 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $f_n$ be Riemann integrable on $[a,b]$ for all $n$, and let $f, g \in C[a, b]$ be continuous functions. Suppose $f_n \rightarrow f$ in $L^2$ and $f_n \rightarrow g$ pointwise. Is it necessarily true that $f = g$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f = g}
Let $D \to A$ be a finite type algebra with $D$ a domain. Suppose $V \subset \operatorname{Spec}A$ is open and dense. Determine if $\dim V = \dim A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dim V = \dim A}
Let $g$ be a non-negative function defined on the interval $[0, \pi]$. Given that \( \int_0^x \cos g'(t) \, dt = \int_0^x g(t) \, dt \) for \(0 \leq x \leq \pi\) and \(g(0) = 0\), determine the possible values of \(g\left(\frac{\pi}{2}\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( D \) be a point inside the acute triangle \( \triangle ABC \). Given that \( \angle ADB = \angle ACB + 90^\circ \) and \( AC \cdot BD = AD \cdot BC \), find the value of \( \frac{AB \cdot CD}{AC \cdot BD} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Let \( \{a_n\}_{n\geq1} \) be an absolutely convergent sequence. Determine whether the sequence \( K_n = \frac{1}{\ln(n^2+1)}\sum_{k=1}^{n}a_k\frac{3k^3-2k}{7-k^3}\sin k \) is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Exercise 4. Consider an $8 \times 8$ chessboard whose squares are alternately colored white and black. An infernal rook is a piece that can attack squares of its own color on its row, as well as squares of the opposite color on its column. What is the maximum number of infernal rooks that can be placed on the chessboard such that no two infernal rooks can ever attack each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
if an object travels 200 feet in 2 seconds , what is the object ' s approximate speed in miles per hour ? ( note : 1 mile = 5280 feet )
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68.18}
A cross, consisting of two identical large squares and two identical small squares, is placed inside an even larger square. Calculate the side length of the largest square in centimeters if the area of the cross is $810 \mathrm{~cm}^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36 \text{ cm}}
When Lauren was born on January 1, 1990, her grandparents put $\$1000$ in a savings account in her name. The account earned $7.5\%$ annual interest compounded quarterly every three months. To the nearest dollar, how much money was in her account when she turned two?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1160}
Let $f:\mathbb{R}^n\to \mathbb{R}^m$ be a linear and positively homogeneous function of degree zero. If $f$ vanishes on a plane, does this imply that $f$ is zero everywhere or it has a discontinuity at a direction parallel to that plane? Answer with 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
Let $G$ be a group, and let $H$ and $K$ be distinct subgroups of $G$. If $H \cup K$ is also a subgroup of $G$, does it necessarily follow that either $H \subset K$ or $K \subset H$? Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
Let $X$ be a Banach space, and let $E$ be a dense subspace of $X$. Suppose $A: X \to X$ is a bounded operator that maps $E$ into itself, and the spectral radius of $A$ restricted to $E$ is a finite $r > 0$. Is the spectral radius of $A$ on $X$ also equal to $r$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r}
Let $X$ be a Banach space and $X^*$ its dual space. Suppose $u \in X$ satisfies $\langle f, u \rangle_{X^* \times X} = 0$ for all $f \in X^*$. Is it necessarily true that $\|u\|_X = 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $K=\mathbb{R}(X)$ be a rational function field, and let $F= \mathbb{R}(X^4 -\frac{1}{X^4 })$. Find the Galois closure $L$ of $K$ over $F$ and determine the degree $[L:F]$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Determine the number of possible Jordan canonical forms and similarity classes for a $5 \times 5$ matrix with the minimal polynomial $p(x) = (x-2)(x^2+1)^2$ over $\mathbb{R}$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Every cell of $8\times8$ chessboard contains either $1$ or $-1$. It is known that there are at least four rows such that the sum of numbers inside the cells of those rows is positive. At most how many columns are there such that the sum of numbers inside the cells of those columns is less than $-3$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6
}
Evaluate the surface integral \( \iint_S x\;dy \times dz + y \; dz \times dx + z \; dx \times dy \) over the part of the plane \( x+y+z=2 \) in the first octant, where the normal vector \( n \) satisfies \( n \cdot (0,1,0) \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let  $P(x)$  be a polynomial with integer coefficients and roots  $1997$  and  $2010$ . Suppose further that  $|P(2005)|<10$ . Determine what integer values  $P(2005)$  can get.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a financial market with a bond $A$, a stock $S$, and call options $f$ on the stock, consider two times $0$ and $T$ with two possible outcomes for $S$ at time $T$. The bond and stock values are $A(0)=100$, $A(T)=120$, $S(0)=50$, $S^u(T)=70$, $S^d(T)=30$. Construct a portfolio consisting of shares of $S$ and call options $f$ with strike $K=50$ and expiration $T$ such that the portfolio value at time $T$ is $120$ regardless of the stock's movement. What is the price of the call option?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.5}
Find all non-decreasing functions $f: \mathbb{Z}^+ \to \mathbb{Z}^+$ satisfying the equation:

$$f\left(\sum_{i=1}^{2014} a_{i}^{2014}\right) = \frac{1}{2014^{2013}}\sum_{i=1}^{2014} \left(f(a_{i})\right)^{2014}$$

for all $a_i \in \mathbb{Z}^+$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(n) = 2014n}
How many of the numbers
\[ 
a_1\cdot 5^1+a_2\cdot 5^2+a_3\cdot 5^3+a_4\cdot 5^4+a_5\cdot 5^5+a_6\cdot 5^6
\]
are negative if  $a_1,a_2,a_3,a_4,a_5,a_6 \in \{-1,0,1 \}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{364}
An aluminum part and a copper part have the same volume. The density of aluminum is $\rho_{A} = 2700 \, \text{kg/m}^3$, and the density of copper is $\rho_{M} = 8900 \, \text{kg/m}^3$. Find the mass of the aluminum part, given that the masses of the parts differ by $\Delta m = 60 \, \text{g}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26.13}
Two robots are programmed to communicate numbers using different bases. The first robot states: “I communicate in base 10,
 which interestingly is a perfect square. You communicate in base 16, which is not a perfect square." The second robot states: "I
 find it more interesting that the sum of our bases is the factorial of an integer." The second robot is referring to the factorial of
 which integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4
}
Let $ \left(a_{n} \right)_{n = 1}^{\infty }$ be a sequence on real numbers such that $ a{}_{n + 1} = a_{n} a_{n + 2}$ for every $ n\ge 1$. The number of elements in the set $ \left\{a_{n} : n\ge 1\right\}$ cannot be
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5
}
In the triangle $ABC$, $\angle ABC = \angle ACB = 78^o$. On the sides $AB$ and $AC$, respectively, the points $D$ and $E$ are chosen such that $\angle BCD = 24^o$, $\angle  CBE = 51^o$. Find the measure of angle $\angle  BED$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$12^\circ$

}
Determine the number of ring homomorphisms from \( \mathbb{Z}[x,y] \) to \( \frac{\mathbb{F}_2[x]}{\langle x^3+x^2+x+1 \rangle} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Determine if the derivative of the function \( F(x) = \int_0^x |\cos(1/u)| \, du \) exists at \( x = 0 \) and find \( F'(0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{\pi}}
Evaluate the integral \( \int_0^\infty\int_0^\infty e^{-sxy}\cos(x)\,dx\,dy \) for \( s>0 \) using the double integration method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2s}}
Two ants, one starting at $(-1,1)$ and the other at $(1,1)$, walk to the right along the parabola $y=x^{2}$ such that their midpoint moves along the line $y=1$ with constant speed 1. When the left ant first hits the line $y=\frac{1}{2}$, what is its speed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \sqrt{3} - 3}
What is the minimum number of moves required to transform a $100 \times 100$ table, initially all white, into a checkerboard pattern by choosing any 99 cells from one row or one column and flipping their colors in each move?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Find the optimal value of \( k_n \) in terms of \( n \) such that for any positive integer \( n \), the inequality \( \left(\sum_{i=1}^{3n} a_{i}\right)^{3} \geqslant k_{n}\left(\sum_{i=1}^{n} a_{i} a_{n+i} a_{2n+i}\right) \) holds for all sequences \( 0 \leqslant a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{3n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27n^2}
Determine whether the image of the function $f(z) = 2z^{14}\cos^2z$ is $\mathbb{C}$ or $\mathbb{C} \setminus \{a\}$ for some $a \in \mathbb{C}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{C}}
Determine the number of differentiable functions \( y:(-\infty, \infty) \rightarrow [0, \infty) \) that satisfy the differential equation \( y' = 2 \sqrt{|y|} \) with the initial condition \( y(0)=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
If the graph of the function $f(x)=2\sin (2x+\varphi )$ $(0 < \varphi < \frac{\pi}{2})$ passes through the point $(0,\sqrt{3})$, then the decreasing interval of the function $f(x)$ on $[0,\pi ]$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[ \frac{\pi}{12}, \frac{7\pi}{12}\right]}
$f(x)$ is a nonconstant polynomial. Given that $f(f(x)) + f(x) = f(x)^2$, compute $f(3)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $I$ be an ideal in the polynomial ring $R = k[x,y]$. Consider the algebraic closure $\overline{k}$ of $k$ and the ring $S = \overline{k}[x,y]$. Determine whether $IS \cap R = I$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{IS \cap R = I}
For which values of \( a \) are the flows of \( \overrightarrow{x}' = A \overrightarrow{x} \) and \( \overrightarrow{y}' = B \overrightarrow{y} \) topologically conjugate, given that \( A \) is a hyperbolic matrix such that all solutions of \( \overrightarrow{x}' = A \overrightarrow{x} \) tend to the origin as \( t \to \infty \), and \( B = \begin{bmatrix}a-3 & 5 \\ -2 & a+3\end{bmatrix} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a < 0}
If \(L\) is a language and the language \(\tilde{L} = \{x^k \mid x \in L, k \in \mathbb{N}\}\) is regular, does it necessarily follow that \(L\) is regular? Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
If \(2^7\cos^5x \cdot \sin^3x = a\sin8x - b\sin 6x + c\sin 4x + d\sin 2x\) where \(x\) is real, find the value of \(a^4 + b^4 + c^4 + d^4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1329}
Hoping to join Chad and Jordan in the Exeter Space Station, there are $2014$ prospective astronauts of various nationalities. It is given that $1006$ of the astronaut applicants are American and that there are a total of $64$ countries represented among the applicants. The applicants are to group into $1007$ pairs with no pair consisting of two applicants of the same nationality. Over all possible distributions of nationalities, what is the maximum number of possible ways to make the $1007$ pairs of applicants? Express your answer in the form $a \cdot b!$, where $a$ and $b$ are positive integers and $a$ is not divisible by $b + 1$.

Note: The expression $k!$ denotes the product $k \cdot (k - 1) \cdot ... \cdot  2 \cdot  1$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$499968 \cdot 1006!$}
In a math competition, all teams must consist of between $12$ and $15$ members, inclusive. Mr. Beluhov has $n > 0$ students and he realizes that he cannot form teams so that each of his students is on exactly one team. Find the sum of all possible values of $n$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
480
}
For $i=1,2,\ldots,7$, Zadam Heng chooses a positive integer $m_i$ at random such that each positive integer $k$ is chosen with probability $\frac{2^i-1}{2^{ik}}$. If $m_1+2m_2+\ldots+7m_7\neq35$, Zadam keeps rechoosing the $m_i$ until this equality holds. Given that he eventually stops, what is the probability that $m_4=1$ when Zadam stops?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{5}}
在●○●○○●○○○●○○○○●○○○○○中，空心圈出现的频率是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.75}
Determine the number of different bases that can be formed from the set of all vectors in \(\mathbb{C}^3\) where each coordinate is either 0 or 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
There are 4 spheres in space with radii 2, 2, 3, and 3, respectively. Each sphere is externally tangent to the other 3 spheres. Additionally, there is a small sphere that is externally tangent to all 4 of these spheres. Find the radius of the small sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["6/11"]}
Given that point P is on the curve $y=\frac{1}{2}{{e}^{x}}$ and point Q is on the curve $y=x$, determine the minimum value of the distance $|PQ|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\sqrt{2}}{2}(1 - \\ln 2)"]}
Paint three faces of an $8 \times 8$ cube red and three faces blue (requiring that any three faces sharing a common vertex cannot all be the same color), then cut it into 512 $1 \times 1 \times 1$ small cubes. Among these 512 small cubes, there are $\qquad$ small cubes that have both red and blue faces.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Let \( f(x) = \sqrt{x + \sqrt{0 + \sqrt{x + \sqrt{0 + \sqrt{x + \dots}}}}} \,.\) If \( f(a) = 4 \), find \( f'(a) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{31}}
For real numbers $$ a $$ and $$ b $$, define $$ a \$b = (a - b)^2 $$. What is $$ (x - y)^2 \$(y - x)^2 $$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0

}
已知$\vartriangle ABC$的内角$A$,$B$,$C$的对边分别为$a$,$b$,$c$,且${{\left( a+b+c \right)}^{2}}+{{\left( a+b-c \right)2}}=4{{c}^{2}}+2ab$,$B=30{}^\circ $,$a=2$,则$\vartriangle ABC$的面积为______.
______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt{3}$；}
Calculate the Euler characteristic of the surface given by the presentation \( \langle a,b,c \mid abcb^{-1}a^{-1}b^{-1}\rangle \) and determine the type of surface it represents.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let  $\triangle ABC$  be a right-angled triangle and  $BC > AC$ .  $M$  is a point on  $BC$  such that  $BM = AC$  and  $N$  is a point on  $AC$  such that  $AN = CM$ . Find the angle between  $BN$  and  $AM$ .

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Calculate the last two digits of the sum $f(17) + f(18) + f(19) + f(20)$, where $f(x) = x^{x^{x^x}}$. Given that $f(20)$ ends with $00$, determine the last two digits of $f(17), f(18),$ and $f(19)$ using modular arithmetic, and find the last two digits of the entire sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Given two independent random variables $X$ and $Y$ with moment-generating functions (mgfs) $M_{X}(t) = (1-t)^{-1}$ for $t<1$ and $M_{Y}(t) = (1-2t)^{-1}$ for $t<0.5$, respectively. Let $U = \frac{1}{2}(X+Y)$ and $V = \frac{1}{2}(X-Y)$. Determine the covariance, $\text{Cov}(U,V)$. Use the relationships $U+V = X$ and $U-V = Y$ if they simplify the calculation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{3}{4}}
Alissa used discarded A4-size paper to make notepads for her friends. She would fold 8 pieces of A4-size paper 4 times then cut along the edges to form smaller note papers. She would then stack these smaller note papers and staple them together. How long would 1 notepad last if someone wrote 15 notes per day on both sides of the note papers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
There is a billiard table in shape of rectangle $2 \times 1$, with pockets at its corners and at midpoints of its two largest sizes. Find the minimal number of balls one has to place on the table interior so that any pocket is on a straight line with some two balls. (Assume that pockets and balls are points).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The minimal number of balls required is 4.

}
[ Ordinary fractions ]  [ Inclusion-exclusion principle ]

How many irreducible fractions with numerator 2015 are there, which are less than $1 / 2015$ and greater than $1 / 2016$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1440}
Given the function $f(x)=\cos ^{2}\omega x+2\sin \omega x\cos \omega x-\sin ^{2}\omega x$ ($\omega\ \gt 0$), which has a maximum value and no minimum value in the interval $\left(\frac{π}{{12}},\frac{π}{3}\right)$, find the range of values for $\omega$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{3}{8},\frac{3}{2}\right)}
A palindrome number is a positive integer that reads the same forward and backward. For example, $1221$ and $8$ are palindrome numbers whereas $69$ and $157$ are not.  $A$ and $B$ are $4$-digit palindrome numbers.  $C$ is a $3$-digit palindrome number.  Given that $A-B=C$, what is the value of $C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
4. At a rectangular table, eight chairs are placed, four on one side and four opposite them on the other side. In how many ways can eight friends be seated at this table so that Ana and Bane sit opposite each other, and Vesna and Goran sit next to each other? (It is known that all friends have distinct names.)


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1152}
G7 Let $A B C$ be an isosceles triangle with $A C=B C$. The point $D$ lies on the side $A B$ such that the semicircle with diameter $[B D]$ and center $O$ is tangent to the side $A C$ in the point $P$ and intersects the side $B C$ at the point $Q$. The radius $O P$ intersects the chord $D Q$ at the point $E$ such that $5 \cdot P E=3 \cdot D E$. Find the ratio $\frac{A B}{B C}$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6}{5}}
Consider a sequence of subdivisions \(\{x_i^n\}_{i=0}^{m_n-1}\) of the interval \([0,1]\) such that \(\max_{i=1,...,m_n-1}\Delta x_i^n\to 0\). Determine whether the following limit always holds: \[ \lim_{n\to \infty }\sum_{i=0}^{m_n-1}o(\Delta x_i^n)=0. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the limit of the sequence: \[ \lim_{{n}\to{\infty}} (n+1)!(e-1-{1\over2}-\cdots-{1\over n!}) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the digits $x, y, z$ that satisfy the equation $\overline{x 5} \cdot \overline{3 y z}=7850$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x = 2, y = 1, z = 4}
Find the maximum integer $k \ge 7$ such that there exists an integer $n$ for which the digit sums of the squares of $n+1, n+2, \ldots, n+k$ are all perfect squares. What is the largest such $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $a, b, c$ be positive integers and $p$ be a prime number. Assume that
$$a^n(b + c) + b^n(a + c) + c^n(a + b) \equiv 8 \pmod{p}$$
for each nonnegative integer $n$. Let $m$ be the remainder when $a^p + b^p + c^p$ is divided by $p$, and $k$ the remainder when $m^p$ is divided by $p^4$. Find the maximum possible value of $k$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
399
}
Let $n \ge 3$ be a fixed integer. The numbers $1,2,3, \cdots , n$ are written on a board. In every move one chooses two numbers and replaces them by their arithmetic mean. This is done until only a single number remains on the board.

Determine the least integer that can be reached at the end by an appropriate sequence of moves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2
}