inclusionAI/Ring-lite-rl-data huggingface dataset:
Let \( A = \{1, 2, 3, \ldots, 9\} \). Determine the number of bijective functions \( f: A \rightarrow A \) such that there exists at least one \( i \in A \) for which \( \left| f(i) - f^{-1}(i) \right| > 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360260}
How to find the sum of the series $\sin(x)-\dfrac{1}{2}\sin(2x)+\dfrac{1}{3}\sin(3x)-\dfrac{1}{4}\sin(4x)+...$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{x}{2}}
Let  $n \ge 3$  be an integer. What is the largest possible number of interior angles greater than  $180^\circ$  in an  $n$ -gon in the plane, given that the  $n$ -gon does not intersect itself and all its sides have the same length?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n-3 }
A cyclist climbs a hill with a length of 400 m with a certain speed. When descending, the speed is two times greater. The cyclist takes 30 minutes to climb and descend the hill. What is the speed of the cyclist when climbing the hill?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
6. Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C: \frac{x^{2}}{4}-\frac{y^{2}}{5}=1$, respectively. Point $P$ is on the right branch of the hyperbola $C$, and the excenter of $\triangle P F_{1} F_{2}$ opposite to $\angle P F_{1} F_{2}$ is $I$. The line $P I$ intersects the $x$-axis at point $Q$. Then
$$
\frac{|P Q|}{|P I|}+\frac{\left|F_{1} Q\right|}{\left|F_{1} P\right|}=
$$
$\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
How can we count the number of homomorphisms from $S_5$ to $A_6$ with a kernel of $A_5$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $G = \Bbb Z \times \Bbb Z$ with group law given by addition. Let $H$ be the subgroup generated by $(2,3)$, so $H$ consists of all elements of the form $(2a,3a)$ for some $a \in \Bbb Z$. To which group is $G/H$ isomorphic?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Consider the sequence \( \{a_{n}\} \) defined by \( a_{1} = 1 \), \( a_{2} = 2 \), \( a_{3} = 3 \), \( a_{4} = 4 \), \( a_{5} = 5 \), and for \( n \geq 5 \), \( a_{n+1} = a_{1} a_{2} \cdots a_{n} - 1 \). Determine the number of positive integers \( m \) such that \( a_{1}a_{2}\cdots a_{m} = a_{1}^{2} + a_{2}^{2} + \cdots + a_{m}^{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the spectrum of the operator $B: D \subset Y \rightarrow Y$, where $D=\{g \in \mathbb{L}^2(\mathbb{R}) : \int |yg(y)|^2 < \infty \}$, $Y=\mathbb{L}^2(\mathbb{R})$, and $Bg(y):=yg(y)$. Provide your answer in terms of the spectrum of $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Evaluate the limit: $$\lim_{N\to \infty} \frac{2}{N} \sum_{j=0}^{N-1} \left(\frac{3}{2}\left[\frac{2j}{N} + 1\right]\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given matrices $A$, $B$, and $C$ with sizes $m \times m$, $n \times n$, and $n \times m$, respectively, where $\det(A) = 2$ and $\det(B) = 3$, find the determinant of the block matrix:
\[ \det \begin{pmatrix} 0 & A \\ B & C \end{pmatrix} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^{mn} \cdot 6}
If $X$ is a random variable that follows a Poisson distribution with parameter $u$, what is the variance of $X^2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4u^3 + 6u^2 + u}
Evaluate the expression \( \log_{2}\left(\prod_{a=1}^{2015}\prod_{b=1}^{2015}\left(1+e^{\frac{2\pi iab}{2015}}\right)\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13725}
Determine the signature of the symmetric bilinear form on \( \mathbb{R}^n \) defined by \( e_i \cdot e_j = 1 - \delta_{ij} \), where \( e_1 \cdot e_1 = e_2 \cdot e_2 = e_3 \cdot e_3 = \cdots = 0 \) and \( e_1 \cdot e_2 = e_1 \cdot e_3 = e_2 \cdot e_3 = \cdots = 1 \). The signature is the number of vectors in an orthogonal basis squaring to \(+1, -1, 0\). What is the signature for \( n > 1 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, n-1, 0)}
Compute the derivative of the function \( \frac{d(x^TWW^Tx)}{dW} \), where \( x \) is an \( n \times 1 \) vector and \( W \) is an \( n \times m \) matrix, using the fact that \( \frac{d(x^TWx)}{dW} = x^Tx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2xx^TW}
Evaluate the limit: $$\lim_{n\to\infty} \frac{1+\frac{n}{1!}+\cdots+\frac{n^n}{n!}}{e^n}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Solve for \(x\) in the equation: \[ 7000 = \sum_{n=1}^{5} \frac{1000}{(1 + x)^n} + \sum_{n=6}^{10} \frac{2000}{(1 + x)^n}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.14}
Find the Fredholm index of the Toeplitz operator $T_{e^{2ix}}$ on the Hilbert space $L_2^+(S^1)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Determine the cardinality of the set of all real-valued functions that have a countable number of discontinuities.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
3. Through a point lying inside a triangle, three lines parallel to its sides are drawn, which divide the triangle into six parts: three triangles and three quadrilaterals. The areas of all three inner triangles are equal. Determine the range in which the ratio of the area of each inner triangle to the area of the original triangle can lie.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{1}{9}"]}
Consider the Collatz Conjecture, where for a given non-zero natural number, if it is even, you divide it by 2, and if it is odd, you multiply it by 3 and add 1. This process is repeated, and the number of transformations needed for a number to first reach 1 is called its path length. Calculate the sum of all natural numbers that have a path length of 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1572}
Find the expected value of \( Z = 2Y_1 + Y_2 \), where \( Y_1 = \min(X_1, X_2) \) and \( Y_2 = \max(X_1, X_2) \). Each \( X_i \) is an independent exponential random variable with parameter \( \theta = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{4}}
Calculate the Hilbert polynomial of the twisted cubic $C$ in $\mathbb{P}^3$, defined by the ideal $V(XZ-Y^2, YW-Z^2, XW-YZ)$. Denote this polynomial as $P_C(n)$. What is $P_C(n)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3n + 1}
Find the kernel of the ring homomorphism \( \phi: \mathbb{Z}[a, b, c, d] \to \mathbb{Z}[p, q, r, s] \) defined by \( \phi(a) = pq, \phi(b) = pr, \phi(c) = qs, \phi(d) = rs \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(ad - bc)}
How many integers between 30,000 and 100,000 have a sum of digits that is 15 or less?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5852}
Find the residue of the function \( \frac{1}{e^{e^{\frac{1}{z}}}} \) at \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{e}}
Find the limit of the power series \( \lim_{{x\rightarrow +\infty}} \sum_{k=1}^\infty (-1)^k \frac{x^k}{k^{k-m}} \), where \( m>1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find a non-constant polynomial in \( \mathbb{Z}/8\mathbb{Z}[x] \) that is a unit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 4x}
Let $\mathcal{H} = l^2(\mathbb{N})$. For $N \geq 1$, define the linear functional $L: \mathcal{H} \rightarrow \mathbb{F}$ by $L(\{\alpha_n\}) = \alpha_N$. Find the vector $h_0$ in $\mathcal{H}$ such that $L(h) = \langle h, h_0 \rangle$ for every $h$ in $\mathcal{H}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e_N}
Given that $\Phi(x)$ is the fundamental solution matrix of the system $\dot{x}=Ax$, where $A$ is a constant matrix, and $\Phi(0)=E$ (the identity matrix), find an expression for $\Phi^{-1}(x)$. Provide your answer in terms of $\Phi(x)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Phi(-x)}
Evaluate the limit: $$\lim_{n\to\infty}\frac{\displaystyle\sum_{k=1}^{n}\frac{2n}{k^2+kn+n^2}-\int_0^1\frac{1}{x^2-x+1}dx}{\displaystyle\sum_{k=1}^{n}\frac{n}{k^2+kn+n^2}-\int_0^1\frac{1}{x^2+x+1}dx}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Angela has $128$ coins. $127$ of them have the same weight, but the one remaining coin is heavier than the others. Angela has a balance that she can use to compare the weight of two collections of coins against each other (that is, the balance will not tell Angela the weight of a collection of coins, but it will say which of two collections is heavier). What is the minumum number of weighings Angela must perform to guarantee she can determine which coin is heavier?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the set of values for $n$ such that the integral \( \int_0^\infty \frac{\sin(x+x^2)}{x^n} \, \mathrm{d}x \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 2)}
A plane has 144 seats, and it is known that 15% of ticket holders do not show up for the flight, with each person's decision being independent. How many tickets can you sell to be 99% sure that not more than 144 people will show up for the flight?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{157}
Identify the space for which the set \( \{1, \sin(x), \cos(x), \sin(2x), \cos(2x), \ldots\} \) forms a basis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{L^2([-\pi, \pi])}
Find the residue of the function \( f(z) = \frac{e^z\sin(z)}{z(1-\cos(z))} \) at \( z = 0 \) using its Laurent series expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a triangulation of the $k$-simplex \( \sigma_k = \{(x_1, \ldots, x_{k+1}) : x_1 + \cdots + x_{k+1} = 1, x_i \ge 0 \} \), determine the number of $k$-simplices that contain a given non-boundary $(k-1)$-simplex.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider the positive integers less than or equal to one billion, i.e., $1, 2, 3, \dots, 10^{9}-1, 10^{9}$. What fraction of these integers can be expressed in the form $x^2 + y^3 + z^5$, where $x, y$, and $z$ are positive integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \(X_1, \ldots, X_n\) be iid Bernoulli\((p)\) random variables with \(0 < p < 1\). Given a sample where \(x_1 = \ldots = x_n = 1\), and assuming the prior distribution of \(p\) is Uniform\((0,1)\), consider a Bayesian test for \(H_0: p \geq 0.9\) versus \(H_1: p < 0.9\). Find the necessary and sufficient condition on \(n\) such that \(H_0\) is rejected.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \leq 5}
Determine the maximum number of rooks that can be placed on an \( n \times n \) chessboard such that each rook is threatened by at most \( 2k \) other rooks.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(k + 1)}
What is the maximum number of smaller balls that can be placed inside a larger ball without intersecting each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Determine the number of subgroups of index 4 in the group \((\mathbb{Z}/4\mathbb{Z})^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Calculate \( F''(\pi) \) for the function \( F(x) = \int_x^{\sin x}\left( \int_0^{\sin t}\sqrt{1+u^4} \, du \right) dt \) using the fundamental theorem of calculus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a pirate coin distribution game, determine the minimum number of coins the third youngest pirate must be offered to vote in favor of a proposal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the range of values for $r$ such that the series \(-\sum\limits_{n=1}^{\infty} \frac{1}{n^{r-1}}\sin(nx)\) does not converge uniformly for \(x \in \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 2]}
Frank mistakenly believes that the number $1011$ is prime and for some integer $x$ writes down $(x+1)^{1011}\equiv x^{1011}+1\pmod{1011}$. However, it turns out that for Frank's choice of $x$, this statement is actually true. If $x$ is positive and less than $1011$, what is the sum of the possible values of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3030}
The sum of the three largest natural divisors of a natural number \( N \) is 10 times the sum of its three smallest natural divisors. Find all possible values of \( N \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Determine the limit superior of the sequence \(|\sin n|^n\) as \(n\) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider the set \( A = \{1, 2, \cdots, 6\} \). A mapping \( f: A \rightarrow A \) is a bijection. Define \( f^{(2)}(x) = f(f(x)) \) and \( f^{(n+1)}(x) = f(f^{(n)}(x)) \) for \( n \geq 1 \). Determine the number of bijections \( f \) such that \( f^{(12)}(x) = x \) for all \( x \in A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{576}
Determine the values of the constant $c$ for which the series \( S=\sum_{n=1}^{\infty}\frac{n^{cn+d}}{(cn+d)^n} \) converges, given that $c \neq 1$ and $c$ and $d$ are positive constants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 < c < 1}
A function $f$ is defined by \(f(z)=(4+i)z^2+a z+b\) for all complex numbers $z$, where $a$ and $b$ are complex numbers. Given that $f(1)$ and $f(i)$ are both purely real, find the minimum value of $|a|+|b|$. \(i=\sqrt{-1}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Let $L$ be the line of intersection of the planes $ax+by+cz=1$ and $a'x+b'y+c'z=1$. Given that $$(a-a')^2+(b-b')^2+(c-c')^2=16$$ and $$(ab'-a'b)^2+(bc'-b'c)^2+(ac'-a'c)^2=9,$$ find the perpendicular distance of $L$ from the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Determine the greatest possible value of \( \frac{P}{20!} \), where \( P \) is the product of the numbers in a list of 20 numbers. For each positive integer \( n \) from 1 to 20, the \( n \)th number in the list is either \( \frac{1}{2}n \) or \( 2n \). The mean of the numbers in the list is exactly 12.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Compute \( \int_{S^1} i^* \omega \) where \( \omega \in \Omega^1(\mathbb{R}^2) \) is a 1-form such that \( d\omega = dx \wedge dy \). Let \( i: S^1 \to \mathbb{R}^2 \) be the inclusion map and \( S^1 \) be the unit circle. Use Stokes' theorem to evaluate the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Find all positive integers \( n \) such that \( n \mid a^{25} - a \) for all positive integers \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2730}
Given a solution $u \in H^2(\mathbb{R}^3)$ to the Hartree equation with an odd real-valued potential $V$, find the time derivative of the modified Hartree energy.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given independent random variables \( X_1 \sim N(\mu = 2, \sigma = 2) \), \( X_2 \sim N(\mu = 1, \sigma = 4) \), and \( X_3 \sim N(\mu = -4, \sigma = 3) \), consider \( Y = (X_1 + 2X_2 + X_3)^2 \). If \( E(Y) = 53 \), determine the probability \( P(Y > E(Y)) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.3173}
In how many ways can 15 different books be distributed to 15 children such that each child receives exactly one book, and then the books are collected and redistributed so that no child receives the same book they initially received?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15! \times !15}
Let the function \( f(x) = 1 - |1 - 2x| \) and \( g(x) = x^2 - 2x + 1 \) for \( x \in [0,1] \). Define the function \( F(x) \) as follows:

\[ 
F(x) = 
\begin{cases} 
f(x) & \text{if } f(x) \geq g(x), \\
g(x) & \text{if } f(x) < g(x). 
\end{cases}
 \]

Determine the number of real roots of the equation \( F(x) \cdot 2^x = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $ABC$ be a triangle with incenter $I$ such that $AB=20$ and $AC=19$. Point $P \neq A$ lies on line $AB$ and point $Q \neq A$ lies on line $AC$. Suppose that $IA=IP=IQ$ and that line $PQ$ passes through the midpoint of side $BC$. Suppose that $BC=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3902
}
Let $\{N(t): t \geq 0\}$ be a Poisson process with rate $1$, and let $T_1 < T_2 < \cdots$ denote the arrival times. Determine the probability density function of the random variable $Y = \frac{T_1}{T_3}$. Provide your answer in terms of $y$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(1 - y)}
Find the smallest integer $n$ such that the group of integers modulo $n$ has a proper subgroup of order $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2a}
Given the function \( f(x) = \frac{\sqrt{2-e^{2x}}\sqrt[4]{2-e^{4x}}\cdot\ldots\cdot\sqrt[50]{2-e^{50x}}}{(2-e^x)\sqrt[3]{2-e^{3x}}\cdot\ldots\cdot\sqrt[99]{2-e^{99x}}} \), find the derivative \( f'(0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Given the complex matrix \( z = \left| \begin{array}{ccc} 3 + 2i & 1 & i \\ 2 & 3 - 2i & 1 + i \\ 1 - i & -i & 3 \end{array} \right| \), and the condition \( |z + \overline{z}| = k|z| \), find the value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of triples of natural numbers \((a, b, c)\) that satisfy the following system of equations:

\[ 
\left\{\begin{array}{l}
\gcd(a, b, c) = 33, \\
\operatorname{lcm}(a, b, c) = 3^{19} \cdot 11^{15}.
\end{array}\right.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9072}
Find the limit of \( \sqrt[n^2]{a_n} \) given that \( a_2>0,\;a_3>0 ,\; a_{n+2}a_n=(1+1/n)^n a_{n+1}^2, n\ge2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
In a chess marathon with 2000 players, each pair of players plays exactly one game against each other. For every pair of players, at least one of them plays no more than 30 games during the marathon. What is the maximum number of games that can be played in this marathon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{59100}
If \( \lim_{x\to 0}\frac{1}{x^3}\left(\frac{1}{\sqrt{1+x}}-\frac{1+ax}{1+bx}\right)=l \), find the value of \( \frac{1}{a}-\frac{2}{l}+\frac{3}{b} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
For any set  $A = \{x_1, x_2, x_3, x_4, x_5\}$  of five distinct positive integers denote by  $S_A$  the sum of its elements, and denote by  $T_A$  the number of triples  $(i, j, k)$  with  $1 \le i < j < k \le 5$  for which  $x_i + x_j + x_k$  divides  $S_A$ .
Find the largest possible value of  $T_A$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let  $p\ge 3$  be a prime number. Each side of a triangle is divided into  $p$  equal parts, and we draw a line from each division point to the opposite vertex. Find the maximum number of regions, every two of them disjoint, that are formed inside the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3p^2 - 3p + 1}
设整数 $n>4,(x+2 \sqrt{y}-1)^{n}$ 的展开式中 $x^{n-4}$ 与 $x y$ 两项的系数相等, 则 $n$ 的值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
5. (25 points) Six resistors with resistances $R_{1}=1$ Ohm, $R_{2}=2$ Ohm, $R_{3}=3$ Ohm, $R_{4}=4$ Ohm, $R_{5}=5$ Ohm, and $R_{6}=6$ Ohm are connected in series and form a loop. A source of constant voltage is connected to the resulting circuit such that the resistance between its terminals is maximized. The voltage of the source is $U=12 \mathrm{~V}$. Find the power $P_{3}$ dissipated on the resistor $R_{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.32}
The square of a three-digit number ends with three identical digits different from zero. Write the smallest such three-digit number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{462}
Given the line $l$: $x-2y+2=0$ passes through the left focus $F\_1$ and one vertex $B$ of an ellipse. Determine the eccentricity of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{2\\sqrt{5}}{5}"]}
If the shopper saves $\textdollar{3.75}$ when buying a coat on sale and the final price paid for the coat was $\textdollar{50}$, determine the percentage that the shopper saved.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7\%}
Find the number of ordered pairs of integers $(a,b) \in \{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a quadratic residue modulo $x^2+1$ and $35$. This means there exists a polynomial $f(x)$ with integer coefficients such that either of the following equivalent conditions holds:

- There exist polynomials $P(x)$, $Q(x)$ with integer coefficients such that $f(x)^2 - (ax+b) = (x^2+1)P(x) + 35Q(x)$.
- The remainder when the polynomial $f(x)^2 - (ax+b)$ is divided by the polynomial $x^2+1$ is a polynomial with integer coefficients, all divisible by $35$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225}
A student is rearranging all his stamps into a new album. If he pastes 20 stamps per page, the album will be insufficient. If he pastes 23 stamps per page, at least one page will remain empty. If he receives an additional identical album, each page containing 21 stamps, he will have a total of 500 stamps. How many pages are in the album?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In a certain university, there are 5 indicators to be allocated to 3 senior classes, with each class receiving at least one indicator. How many different allocation methods are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given that the power function $f(x) = (m^2 - m - 1)x^{1-m}$ is symmetric about the y-axis, find the value of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
There is a greatest number which, when dividing 1642 and another number, leaves remainders 6 and 4 respectively. The greatest number is 4. What is the second number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}