inclusionAI/Ring-lite-rl-data huggingface dataset:
In a \(5 \times 5\) table, numbers (not necessarily integers) are placed such that each number is three times smaller than the number in the cell to its right and twice larger than the number in the cell below it. Find the number in the central cell, given that the sum of all the numbers in the table is 341.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{36}{11}}
Let $f(x)$ be a continuous function satisfying $f(x)=1+k\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(t)\sin (x-t)dt\ (k:constant\ number)$

Find the value of $k$ for which $\int_0^{\pi} f(x)dx$ is maximized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$k=\displaystyle\frac{2}{\pi}$}
Example 8 (2004 Jiangsu Summer Camp) Let $S=\frac{1}{\sqrt{1}+\sqrt{2}}+$ $\frac{1}{\sqrt{3}+\sqrt{4}}+\cdots+\frac{1}{\sqrt{2003}+\sqrt{2004}}$, find the integer part of $S$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Let $F$ be a field and consider the field extensions $L = F(xz, yz, xy)$ and $K = F(x, y, z)$. Determine if the degree $[K:L]$ of the extension $K$ over $L$ is finite, and if so, compute its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( \mathbf{A} \) be an \( m \times n \) matrix and \( \mathbf{B} \) be a symmetric \( n \times n \) matrix, where \( m < n \). Given that \( \operatorname{Rank}(\mathbf{A}) = \operatorname{Rank}(\mathbf{B}) = m \), determine the rank of the matrix product \( \mathbf{ABA}^\top \). Is it true that \( \operatorname{Rank}(\mathbf{ABA}^\top) = m \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m}
Given $a > 0$ and $a, b, x \in \mathbb{R}^n$, if for any $x > 0$, whenever $a^T x \leq 1$ we have $b^T x \leq 1$, does it imply that $a \geq b$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \geq b}
Ali and Veli go hunting. The probability that each will successfully hit a duck is 1/2 on any given shot. During the hunt, Ali shoots 12 times, and Veli shoots 13 times. Find the probability that Veli hits more ducks than Ali. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Find the Euler characteristic of the surface defined by \( N = \{(x,y,z) \in \mathbb{R}^3 \mid x^4 + y^4 + z^2 = 1\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the relation $x = Ky$ where $K$ is a fixed scalar, and a function $f$ with its partial derivative $f'(x) = \frac{\partial f}{\partial x}(Ky)$, find the expression for $\frac{\partial f}{\partial y}(x)$ in terms of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{K f'(x)}
Given that \(\mathbf{m}=(\cos \theta) \mathbf{i}+(\sin \theta) \mathbf{j}\) and \(\mathbf{n}=(\sqrt{2}-\sin \theta) \mathbf{i}+(\cos \theta) \mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are the usual unit vectors along the \(x\)-axis and the \(y\)-axis respectively, and \(\theta \in (\pi, 2\pi)\). If the length or magnitude of the vector \(\mathbf{m} + \mathbf{n}\) is given by \(|\mathbf{m} + \mathbf{n}| = \frac{8 \sqrt{2}}{5}\), find the value of \(5 \cos \left(\frac{\theta}{2} + \frac{\pi}{8}\right) + 5\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The sum of the first $15$ positive even integers is also the sum of six consecutive even integers. What is the smallest of these six integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Calculate the intersection multiplicity of the curves defined by $f=y^2-x^3$ and $g=y^3-x^7$ at the point $(0,0)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
What fraction of the area of a regular hexagon of side length 1 is within distance $\frac{1}{2}$ of at least one of the vertices?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi \sqrt{3}}{9}}
Determine the Galois group of the quadratic extension $\mathbb{Q}(\sqrt{p})$, where $p$ is a prime number not congruent to 1 modulo 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/2\mathbb{Z}}
Find the range of values for the real number $a$ such that the inequality $2_x \ln x \geqslant -x^2 + ax - 3$ holds for all $x \in (0, +\infty)$. Express your answer as an interval or a specific numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 4]}
A fair six-sided die has faces numbered 1, 2, 3, 4, 5, 6. The die is rolled four times, resulting in numbers a, b, c, d. Find the probability that one of the numbers in the set {a, a+b, a+b+c, a+b+c+d} equals 4. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{343}{1296}}
A positive integer is called fancy if it can be expressed in the form $$2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{100}}$$ where $a_{1}, a_{2}, \ldots, a_{100}$ are non-negative integers that are not necessarily distinct. Find the smallest positive integer $n$ such that no multiple of $n$ is a fancy number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{101}-1}
Consider the sequence: 1, 1, 2, 1, 2, 3, 1, 2, 3, _, 1, 2, 3. What is the tenth number in the sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In an M/G/1 queue with Poisson arrivals at rate \( \lambda = 1 \) and service time being the sum of two exponentially distributed variables with rates \( \mu_1 = 1 \) and \( \mu_2 = 2 \), calculate the probability that the waiting time \( W_q \) for an average customer is zero, i.e., find \( P(W_q = 0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Provide an example of a polynomial \( f(x) \) with integer coefficients that factors over \( \mathbb{Z}/n\mathbb{Z} \) for some integer \( n \), but has no roots in \( \mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + 1}
Two particles move along the edges of equilateral $\triangle ABC$ in the direction $A\Rightarrow B\Rightarrow C\Rightarrow A,$ starting simultaneously and moving at the same speed. One starts at $A$, and the other starts at the midpoint of $\overline{BC}$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $R$. What is the ratio of the area of $R$ to the area of $\triangle ABC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{16}}
Evaluate the Riemann-Stieltjes integral: \[ \int_{0}^{2}x^2d\left\lfloor 2x\right\rfloor \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{2}}
Evaluate \( \frac{1}{2 \pi i} \oint_{|z|=3} \frac{f'(z)}{f(z)} \, dz \), where \( f(z) = \frac{z^2 (z-i )^3 e^z}{3 (z+2 )^4 (3z - 18 )^5} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the integers $a$ and $b$ such that $a^{4}+(a+b)^{4}+b^{4}$ is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a = 0, b = 0}
In how many ways can 13 identical balls be distributed into 6 different cells such that no cell contains exactly 3 balls?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4092}
Evaluate the complex integral \(\int_0^{2\pi} e^{2ie^{it}}e^{-2it} dt\) using elementary methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4\pi}
In a bank with 1000 customers, the number of withdrawal requests per month follows a Poisson distribution with an average of 60 requests. Each withdrawal amount is exponentially distributed with an average of 700 euros. Calculate the probability that the total amount withdrawn in a given month exceeds 50,000 euros.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.149}
Find the value of \(a\) such that the solution space of the system represented by the augmented matrix
\[\begin{pmatrix} 1 & 4 & -3 & 2 & 0\\ 2 & 7 & -4 & 4 & 0\\ -1 & a & 5 & -2 & 0\\ 3 & 10 & -5 & a^2 + 4a + 1 & 0 \end{pmatrix}\]
is two-dimensional.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
Determine the number of real roots of the function \( g(x) = \left(x^2-2\right)^2 \sinh (x) - \left(4-x^2\right)^2 \sinh (2x) - \left(\frac{x}{3}\right)^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the invariant factors of the finitely generated Abelian group $M$ given by the generators $x, y, z$ and the relations $4x + y + 2z = 0$, $5x + 2y + z = 0$, and $6y - 6z = 0$. Provide your answer as a list of invariant factors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[3, 6]}
Determine the order of the point \( M(0,1) \) on the curve defined by the polynomial \( f(x,y) = (x^2-1)^2 - y^2(3-2y) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $G$ be an abelian group and $M$ be a sequence of $m$ elements from $G$. If $S$ and $F$ are subsets of $G$ with $|S|=n$ and $|F|=n-m$, determine if it is possible to find a sequence of elements from $S$ such that the sum of each element in $M$ with the corresponding element in the sequence is distinct and not in $F$. Answer with 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
Define $ a \circledast b = a + b-2ab $. Calculate the value of
$$A=\left( ...\left(\left(\frac{1}{2014}\circledast \frac{2}{2014}\right)\circledast\frac{3}{2014}\right)...\right)\circledast\frac{2013}{2014}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
If each of the three operation signs, $+$, $\text{--}$, $\times$, is used exactly ONCE in one of the blanks in the expression

$$5\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}4\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}6\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}3$$

then the value of the result could equal




Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
19}
Find the number of integer solutions to the equation \( x + y + z = 100 \) such that \( 3 \leq x \leq y \leq z \leq 60 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{430}
Evaluate the expression:
\[
\left(\frac{\left(z^{2 / p}+z^{2 / q}\right)^{2}-4 z^{2 / p+2 / q}}{\left(z^{1 / p}-z^{1 / q}\right)^{2}+4 z^{1 / p+1 / q}}\right)^{1 / 2}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left| z^{1 / p} - z^{1 / q} \right|}
Let $ABCD$ be a regular tetrahedron of side length $12$. Select points $E,F,G,H$ on $AC,BC,BD,AD$, respectively, such that $AE=BF=BG=AH=3$. Compute the area of quadrilateral $EFGH$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
27
}
Find the limit $\lim_{n\to\infty} \left(\sin \frac{x}{2} \cdot \sin \frac{x}{4} \cdot \sin \frac{x}{8} \cdots \sin \frac{x}{2^n}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \( \int_{\mathbb{R}^1} f(x) \, dx \), where \( f(x) = |(A-x) \cap B| \) and \( A, B \) are measurable subsets of \( \mathbb{R}^1 \). Here, \( |\cdot| \) denotes the measure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|A| \cdot |B|}
Find a value of $n$ such that there are 100 consecutive composite numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101! + 2}
Determine the order of growth of the sum \( \sum_{i=0}^{k-1}n^{(2^{i+1}-1)/2^{i+1}} \) as \( k \to \infty \), where \( n \) is a fixed number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(k)}
Find the minimum value of \( \tan^2 x + \frac{\tan^2 y}{4} + \frac{\tan^2 z}{9} \) given that \( \tan x + 2\tan y + 3\tan z = 40 \) and \( x, y, z \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{800}{49}}
Consider the sequence defined by \( a_1 = e^i \) and \( a_{n+1} = a_n + e^{a_n} \) for all \( n \geq 1 \). Determine if the imaginary part of the sequence \( \Im(\lim_{n \to \infty} a_n) \) converges, and if so, find the value of this limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\pi}
Compute the integral \( \frac{2}{\pi}\int_{-\pi}^\pi\frac{\sin\left(\frac{9x}{2}\right)}{\sin\left(\frac{x}{2}\right)}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
For a fixed positive integer $m$, determine if the inequality \( H_m = \liminf_{n \to \infty} (p_{n + m} - p_n) > p_{m + 2} \) holds for sufficiently large $n$. Answer "yes" or "no".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
Compute the integral \( \int_{\mathbb{R}} \int_{\mathbb{R}} \chi_S(x,y)\, d c(x)\, d \lambda(x) \) where \( S = \{ (t,t) \in \mathbb{R}^2, t \in [0,1]\} \), \( c \) is the counting measure, and \( \lambda \) is the Lebesgue measure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the integral \( \int_{S} (x + 2y - z) \, dV \), where \( S \) is the tetrahedron in \( \mathbb{R}^3 \) with vertices \((0,0,0), (1,2,3), (0,1,2), (-1, 1, 1)\). Use a suitable linear transformation as a change of variables to simplify the integration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
In chess, a king threatens another king if, and only if, they are on neighboring squares, whether horizontally, vertically, or diagonally . Find the greatest amount of kings that can be placed on a  $12 \times 12$  board such that each king threatens just another king. Here, we are not considering part colors, that is, consider that the king are all, say, white, and that kings of the same color can threaten each other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Given the polynomial \( \prod_{k=1}^{99} (x+k) = \sum_{k=0}^{99} a_k \cdot x^k \), and the expression \( M= \frac{\sum_{k=1}^{99} k\cdot a_k}{100!}= \sum_{i=1}^{99} \frac{1}{r_i} \) where \( r_i < r_j \) for all \( i < j \), find the value of \( S = \left\lfloor \sum_{k=1}^{99} \frac{1}{\sqrt{r_k-1}} \right\rfloor \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Evaluate the contour integral \( I = \frac{1}{2\pi i} \int_{\gamma} z^7 \cos \left(\frac{1}{z^2}\right) dz \). The function \( \cos \left(\frac{1}{z^2}\right) \) is not analytic at \( z=0 \) and does not have a pole there. Determine if the integral is zero based on the Cauchy Integral Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{24}}
There are 7 seats in a row, and 4 people are to be seated such that exactly two empty seats are adjacent, and person A and person B cannot sit next to each other. How many different seating arrangements are there? (Answer with a number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{336}
3. In how many ways can you partition the set $\{1,2, \ldots, 12\}$ into six mutually disjoint two-element sets in such a way that the two elements in any set are coprime?

(Martin Panák)

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
Let \( G \) be a finite abelian group such that for any \( x \in G \), \( x + x = 0 \), i.e., \( G = \mathbb{Z}_{2}^{\oplus k} \) for some \( k \in \mathbb{N} \). Consider a symmetrical bilinear form \((\cdot, \cdot): G \times G \to \mathbb{Z}_{2}\). Given the conditions:
\[(a, m) = 0,\]
\[(a, p) = 1,\]
\[(b, m) = 1,\]
\[(b, p) = 0,\]
determine if it is true that \((a, b) = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the improper integral \( \int_0^\infty \frac{\sin x(1-\cos x)}{x^2}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Determine whether the number 7 is prime, irreducible, or neither in the ring \( \mathbb{Z}_{21} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{prime}
Bart wrote the number 1015 as a sum of numbers using only the digit 7. He used a 7 a total of 10 times, including using the number 77 three times. Now he wants to write the number 2023 as a sum of numbers using only the digit 7, using a 7 a total of 19 times. How many times will the number 77 occur in the sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given the set \( M = \left\{ x \ | \ 5 - |2x - 3| \in \mathbf{N}^{*} \right\} \), the number of all non-empty proper subsets of \( M \) is?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{510}
Evaluate the double integral \( \int\int_R \sqrt{xy - y^2} \, dx \, dy \), where \( R \) is the triangular region with vertices at \((0,0)\), \((10,1)\), and \((1,1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Use Stokes' theorem to compute the line integral \(\oint_{\gamma} \mathbf{u} \cdot \,\mathrm{d}\mathbf{r}\), where \(\gamma\) is the boundary of the triangle in the \(xy\)-plane with vertices at \((0,0,0)\), \((1,0,0)\), and \((0,1,0)\), traced out clockwise. The vector field is \(\mathbf{u} = (x \cos(x) \sin(x) + y, y - z, x + z^3)\). Assume the triangle is oriented with its normal vector \(\mathbf{N} = (0, 0, 1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Solve the equation: $$\sqrt{\frac{\sqrt{x!}\times \sqrt{x!!}\times \sqrt{x!!!}}{\sqrt{\left( x-2 \right)!+x}}}=12$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $f(x)=x^{2021}+15x^{2020}+8x+9$ have roots $a_i$ where $i=1,2,\cdots , 2021$. Let $p(x)$ be a polynomial of the sam degree such that $p \left(a_i + \frac{1}{a_i}+1 \right)=0$ for every $1\leq i \leq 2021$. If $\frac{3p(0)}{4p(1)}=\frac{m}{n}$ where $m,n \in \mathbb{Z}$, $n>0$ and $\gcd(m,n)=1$. Then find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{104}
Determine whether a coffee mug with a tube running through its middle is topologically equivalent to a 2-fold torus, a 3-fold torus, or another shape.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2-fold torus}
Suppose a right circular cylinder has a radius of $r = 5$ units and a height of $h = 12$ units. If the diagonal across the circular top and the height measures $13$ units, what is the radius of the cylinder?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let the function \( f(x) = \sin^4 \left( \frac{kx}{10} \right) + \cos^4 \left( \frac{kx}{10} \right) \), where \( k \) is a positive integer. If for any real number \( a \), the set \(\{ f(x) \mid a < x < a+1 \} = \{ f(x) \mid x \in \mathbf{R} \}\), then find the minimum value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Calculate the expected value \( \mathbb{E}\big|X_P-X_Q\big|^2 \) for points \( P=(X_P,Y_P) \) and \( Q=(X_Q,Y_Q) \) chosen independently from the square with vertices \((-1,0), (0,-1), (1,0), (0,1)\) using geometric probability.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
8. There are two inlet pipes, A and C, and two outlet pipes, B and D, in a reservoir. To fill the reservoir, it takes 3 hours for pipe A alone and 5 hours for pipe C alone. To drain the reservoir, it takes 4 hours for pipe B alone and 6 hours for pipe D alone. Currently, the reservoir is $\frac{1}{6}$ full. If the pipes are opened in the order of A, B, C, D, and each pipe is opened for one hour per day in this cycle, how long will it take for the water to start overflowing from the reservoir?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\frac{3}{4}}
Determine the number of zeros of the function \( f(z) = e^{2z} - 2ze^z - 1 \) within the annulus defined by \( 1 < |z| < 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the smallest positive integer \( n \) such that \( x^2 + 7 \) can be expressed as the sum of squares of \( n \) polynomials with rational coefficients, \( f_1(x)^2 + f_2(x)^2 + \cdots + f_n(x)^2 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Solve the recurrence relation for the time complexity: \[ T(n) = T(\sqrt{n}) + n^2 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^2)}
Bob's Rice ID number consists of six digits, each ranging from $1$ to $9$. Each digit can be repeated any number of times. The ID number satisfies the following divisibility conditions:

- The number formed by the first two digits is divisible by $2$.
- The number formed by the first three digits is divisible by $3$.
- The number formed by the first four digits is divisible by $4$.
- The number formed by the first five digits is divisible by $5$.
- The entire six-digit ID number is divisible by $6$.

One example of such an ID number is $123252$. Determine how many different possibilities exist for Bob's ID number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{324}
Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 7), (4, 4), (8, 1)}
Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively. Suppose $AD, BE, CF$ are concurrent at $P$. If $\frac{PF}{PC} = \frac{2}{3}$, $\frac{PE}{PB} = \frac{2}{7}$ and $\frac{PD}{PA} = \frac{m}{n}$, where $m, n$ are positive integers with $\gcd(m, n) = 1$, find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Given that $a_{1}, a_{2}, b_{1}, b_{2}, \cdots, b_{k}$ are vectors in the plane that are pairwise non-parallel, with $\left|a_{1}-a_{2}\right|=1$, and $\left|a_{i}-b_{j}\right| \in\{1,2,3\} (i=1,2; j=1,2, \cdots, k)$, determine the maximum value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
a train of length 250 m crosses a bridge of length 150 m in 32 seconds . what is the speed of train ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.5}
$ \frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+...+\sqrt{10+\sqrt{99}}}{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+...+\sqrt{10-\sqrt{99}}}=? $
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \sqrt{2}}
Find the number of integers $n$ such that $2000 \leq n \leq 2010$ and $7$ divides $\left\lfloor \frac{\sqrt{2}}{4}(1+\sqrt{2})^n \right\rfloor$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the spectrum of the operator \( T \) on the Banach space \( \ell^1 \), where \( \|x_n\|=\sum_{n=1}^\infty |x_n| \) and \( T\{x_n\}=\{x_2,x_1, x_4, x_3, x_6, x_5, x_8,x_7, \ldots \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{-1, 1\}}