inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine all integers $n$ for which the polynomial $P(x) = 3x^3-nx-n-2$ can be written as the product of two non-constant polynomials with integer coeffcients.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$n \in \{-2, 26, 38, 130\}$}
Bianca has a rectangle whose length and width are distinct primes less than 100.  Let  $P$  be the perimeter of her rectangle, and let  $A$  be the area of her rectangle.  What is the least possible value of  $\frac{P^2}{A}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{82944}{5183}}
Find the second derivative of the function \( f(x) = \sec(3x)\sqrt{324\cos^2(3x) + 396 + 121\sec^2(3x)} \). Then, evaluate the second derivative at \( x = \frac{\pi}{12} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1584}
In 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 values are: $A(0)=100$, $A(T)=120$, $S(0)=50$, $S^u(T)=70$, $S^d(T)=30$. Construct a portfolio $\\phi$ of shares of $S$ and call options $f$ with strike $K=50$ and expiration $T$ such that $V(T)=120$ regardless of the stock's movement. Determine the price of the call option at time $0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.5}
Let $\Phi_1$ and $\Phi_2$ be cumulative probability distribution functions with domain $[L, \infty)$, $L\geq 0$, both distributions having the same expectation $\mu$ and the same second moment. If the variance of the distribution behind $\Phi_2$ is higher than that of $\Phi_1$, determine whether the following inequality holds:

$$G'-G \geq 0$$

where:

$$G=1-\frac{1}{\mu}\int_L^\infty \left(1- \Phi_1(x)\right)^2 \, \mathrm{d} x$$

$$G'=1-\frac{1}{\mu}\int_L^\infty \left(1- \frac{1}{2}\left(\Phi_1(x)+\Phi_2(x)\right)\right)^2 \, \mathrm{d} x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{G' - G \geq 0}
The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Moreover, the angles in each of these two triangles also form an arithemetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240
}
What is the area of the triangle formed by the lines $y=5$, $y=1+x$, and $7=1-x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Determine whether the language \( L_G = \{ H \mid G \text{ contains a copy of } H \} \) for some fixed graph \( G \) is in \( P \) or is \( NPC \). Assume \( P \neq NP \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P}
Calculate the work done by a gas in the cycle $1-2-3-4-5-6-1$, given that $p_{0}=10^{5} \Pi a$ and $V_{0}=1$. Note that states 1 and 3 coincide, as do states 4 and 6.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100000}
Let $V$ be a real $d$-dimensional vector space with $1<k<d$. Given a basis $\{v_i\}$ for $V$ and an induced basis for $\bigwedge^k V$ consisting of $v_{i_1} \wedge \dots \wedge v_{i_k}$, suppose there exists another basis $\{w_i\}$ such that each $v_{i_1} \wedge \dots \wedge v_{i_k}$ can be expressed as $\lambda_{i_1,\ldots,i_k}w_{j_1} \wedge \dots \wedge w_{j_k}$, where $\lambda_{i_1,\ldots,i_k} \in \mathbb R$ are non-zero scalars. Determine whether there exists a permutation $\sigma \in S_d$ such that $\text{span}(v_i)=\text{span}(w_{\sigma(i)})$. Provide your answer as 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
Let $z_1,$ $z_2,$ $z_3,$ and $z_4$ be the four distinct complex solutions of the equation
\[
  z^4 - 6z^2 + 8z + 1 = -4(z^3 - z + 2)i.
\]Find the sum of the six pairwise distances between $z_1,$ $z_2,$ $z_3,$ and $z_4$ in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\sqrt{3}+6}
Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\text{Dan}$}
Determine whether the solution to the recurrence relation \( T(n) = T\left(\frac{1}{8n}\right) + 8 \) is infinite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Is the largest singular value of a matrix $A$, denoted as $\sigma_1$, the solution to the optimization problem \( \max_{\|x\|\leq1,\|y\|\leq1} \quad x^TAy \) where $A \in \mathbb{R}^{m \times n}$, $x \in \mathbb{R}^{m \times 1}$, and $y \in \mathbb{R}^{n \times 1}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sigma_1}
Let $ABC$ be at triangle with incircle $\Gamma$. Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$ be three circles inside $\triangle ABC$ each of which is tangent to $\Gamma$ and two sides of the triangle and their radii are $1,4,9$. Find the radius of $\Gamma$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Is the number\[
\left( 1 + \frac12 \right) \left( 1 + \frac14 \right) \left( 1 + \frac16 \right)\cdots\left( 1 + \frac{1}{2018} \right)
\]greater than, less than, or equal to $50$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The number is less than 50.}
Find the Maclaurin series expansion for the function \( f(z) \) that satisfies \( f(z)e^{f(z)}= z \). Determine the radius of convergence of this series using the Lagrange inversion formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{e}}
Determine whether the function \( f(z) = \mathfrak{Re}(z) \mathfrak{Im}(z) \) is pointwise differentiable at \( z=0 \) using the limit definition: 
\[ \lim_{z\to 0} \frac{f(z) - f(0)}{z}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $C \subset \mathbb{R}$ be an infinite and perfect set. Suppose there exists a family of compact sets $\{Q_k\}_{k\in \mathbb{N}}$ such that $Q_{k+1} \subseteq Q_k$ and $Q_k \cap C$ is infinite for all $k$. Given that $Q := \bigcap_{k\in \mathbb{N}} Q_k \neq \emptyset$, is it true that $Q \subseteq C$ or at least $Q \cap C \neq \emptyset$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Q \cap C \neq \emptyset}
Suppose $A, R, S$, and $T$ all denote distinct digits from $1$ to $9$. If $\sqrt{STARS} = SAT$, what are $A, R, S$, and $T$?


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

$A = 3, T = 9, S = 1$ and $R = 2$}
Liquid $X$ does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film $0.1$cm thick. A rectangular box measuring $6$cm by $3$cm by $12$cm is filled with liquid $X$. Its contents are poured onto a large body of water. What will be the radius, in centimeters, of the resulting circular film?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt{\dfrac{2160}{\pi}}$

}
Let $x \in \mathbb{R}^+$ and define the operation $x \diamond 1 = x$ and $x \diamond (n+1) = x^{x \diamond n}$ for $n \in \mathbb{N}$. For example, $2 \diamond 3 = 2^{2^2} = 16$. Given a unique positive real number $t$ such that $t^t = 10^{10^{10}}$, it is known that $10^{10} > t$. Let $k$ be the unique positive real number such that $k \diamond 50 = 10 \diamond 51$, and $s$ be the unique positive real number such that $s \diamond 51 = 10 \diamond 52$. Determine if $k > s$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k > s}
26th BMO 1990 Problem 2 The fractional part { x } of a real number is defined as x - [x]. Find a positive real x such that { x } + { 1/x } = 1 (*). Is there a rational x satisfying (*)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5}+3}{2}}
Define a common chord between two intersecting circles to be the line segment connecting their two intersection points. Let $\omega_1,\omega_2,\omega_3$ be three circles of radii $3, 5,$ and $7$, respectively. Suppose they are arranged in such a way that the common chord of $\omega_1$ and $\omega_2$ is a diameter of $\omega_1$, the common chord of $\omega_1$ and $\omega_3$ is a diameter of $\omega_1$, and the common chord of $\omega_2$ and $\omega_3$ is a diameter of $\omega_2$. Compute the square of the area of the triangle formed by the centers of the three circles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
You have a 4-digit whole number that is a perfect square.
 Another number is built adding 1 to the unit’s digit,
 subtracting 1 from the ten's digit,adding 1 to the hundred's
 digit and subtracting 1 from the ones digit of one thousanc
 If the number you get is also a perfect square, find the
 original number. It's unique?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3025
}
p5. Hao has $32$ ounces of pure lemonade in his juice box. On the first day, he drinks an ounce and then refills the juice box with an ounce of Hater-ade and mixes the mixture thoroughly. On the second day, he drinks $2$ oz of the juice and then refills the juice box with $2$ oz of Hater-ade. On the third day, he drinks $3$ oz and then refills the juice box with $3$ oz of Hater-ade, and so on. Hao stops when the juice box has no more lemonade in it. How many ounces of Hater-ade did Hao drink?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{496}
Determine the points at which the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \sin x \) if \( x \) is rational, and \( f(x) = 0 \) otherwise, is continuous.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k\pi}
Consider a power series \(\sum_{k=0}^\infty a_k x^k\) with a radius of convergence of 1. Assume that \(\sum_{k=0}^\infty a_k = \infty\). Determine whether \(\lim_{x\to 1^-} \sum_{k=0}^\infty a_k x^k = \infty\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
The polynomial $f(x)=x^3-4\sqrt{3}x^2+13x-2\sqrt{3}$ has three real roots, $a$, $b$, and $c$. Find
$$\max\{a+b-c,a-b+c,-a+b+c\}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3} + 2\sqrt{2}}
Determine if the only solution to the differential equation \( \frac{dy}{dx}=8y^2x^3 \) with the initial condition \( y(2)=0 \) is the constant solution \( y=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = 0}
Consider an infinite double random array $\{X_{ij}:i\geq 1, j\geq 1\}$ where all elements are independent, identically distributed (i.i.d.) with mean zero and unit variance. Define $Y_{i,n}=\frac{1}{n}\sum_{j=1}^nX_{ij}$. By the strong law of large numbers, we know that for any $i\geq 1$, $Y_{i,n}\to 0$ a.s. and $Y_{i,n}^2\to 0$ a.s. as $n\to\infty$. Does this imply that $\frac{1}{n}\sum_{i=1}^nY_{i,n}^2\to 0$ a.s.?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A traveler token is placed on the central tile of a 19-tile hexagonal grid. Each turn, the traveler moves to an adjacent tile based on a six-sided die roll, with all directions equally probable. The game ends when the traveler exits the board. Calculate the expected number of turns until the game concludes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{55}{9}}
Suppose $X(k)$ is a $k \times m$ matrix with elements $x_{ij}(k)$. If \(\lim_{k \to \infty} \frac{X'(k)X(k)}{k}=R,\) where $R$ is positive semi-definite, determine if the following holds: \(\max_{i=1, \ldots, k} x_i'(X'(k)X(k))^{-1}x_i =0, \text{ as } k \to \infty.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $x,y$ be complex numbers such that $\dfrac{x^2+y^2}{x+y}=4$ and $\dfrac{x^4+y^4}{x^3+y^3}=2$. Find all possible values of $\dfrac{x^6+y^6}{x^5+y^5}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$10 \pm 2\sqrt{17}$}
All twelve points on the circle are at equal distances. The only marked point inside is the center of the circle. Determine which part of the whole circle in the picture is filled in.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{1}{2}$}
Determine the cardinality of the set formed by the union of all Cartesian products of the natural numbers with itself, i.e., the set of all sequences of natural numbers of any finite length. Is the cardinality of this set aleph-one or higher?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
Determine whether the set of computable real numbers has a positive Hausdorff dimension.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine whether the limit \( \lim_{x \to 0} x \cdot (\log(x))^\alpha = 0 \) holds for every \( \alpha > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let F be a countable field and B an infinite set. Let \((F^B)_0\) be the set of all functions with finite support from F to B. Determine whether \(|(F^B)_0| = |B|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|(F^B)_0| = |B|}
Find all solutions $a, b, c, d, e$ if it is known that they represent distinct
digits and satisfy the following:
$\begin{array}{c}& a & b & c & d \\+ & a & c & a & c \\ \hline c & d & e & b & c \\\end{array}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(5,6,1,0,7)}
Suppose that for real $x,y,z,t$ the following equalities hold:$\{x+y+z\}=\{y+z+t\}=\{z+t+x\}=\{t+x+y\}=1/4$.
Find all possible values of $\{x+y+z+t\}$.(Here$\{x\}=x-[x]$)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$0, \frac{1}{3} , \frac{2}{3} $
}
Jerry begins his English homework at $07:39$ a.m. At $07:44$ a.m., he has finished $2.5\%$ of his homework. Subsequently, for every five minutes that pass, he completes three times as much homework as he did in the previous five minute interval. If Jerry finishes his homework at $AB : CD$ a.m., what is $A + B + C + D$? For example, if he finishes at $03:14$ a.m., $A + B + C + D = 0 + 3 + 1 + 4$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Determine whether the first-order predicate formula \((A \land B) \Rightarrow C\) is valid, where:

- \(A = \forall a (\neg P(a,a))\)
- \(B = \forall a \forall b \forall c (P(a,b)\wedge P(b,c)\Rightarrow P(a,c))\)
- \(C = \forall a \forall b (P(a,b)\Rightarrow \neg P(b,a))\)

A formula is valid if there is no truth assignment that makes it false.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Valid}
Determine whether the integral \( \int_0^2 \frac{5\,dx}{\sqrt{2x-x^2}} \) converges or diverges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\pi}
Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for which $f(0)=2010.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{163}
Let  $O$  be the centre of the square  $ABCD$ . Let  $P,Q,R$  be respectively on the segments  $OA,OB,OC$  such that  $OP=3,OQ=5,OR=4$ . Suppose  $S$  is on  $OD$  such that  $X=AB\cap PQ,Y=BC\cap QR$  and  $Z=CD\cap RS$  are collinear. Find  $OS$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{60}{23}}
一个有盖的长方体形状的文具盒的长、宽、高分别是12cm,4cm,3cm,那么它最多能放13 cm长的铅笔.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13 \text{ cm}}
Determine whether the Krull dimension of any commutative semilocal Hilbert ring is equal to zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the asymptotic value of \(\underset{c \in \text{Coloring}}{\min} \text{ score}(c) = \underset{c \in \text{Coloring}}{\min} \left(\underset{\begin{subarray}{c} B \subset \text{Colors}, \\ |B|\leq n  \end{subarray}}{\max} V_c(B)\right)\). Is it \(\Theta(n^2)\) or \(o(n^2)\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{o(n^2)}
Evaluate the integral \(\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos(\theta)}\,d\theta.\) Use the substitution \(\theta=2\arctan t\) to express the integral in terms of \(t\) and find its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
King Midas spent $\frac{100}{x}\%$ of his gold deposit yesterday. He is set to earn gold today. What percentage
of the amount of gold King Midas currently has would he need to earn today to end up with as much
gold as he started?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{100}{x - 1}\%}
Determine the values of 's' for which the matrix A is positive semi-definite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1]}
Let \( f(x), g(x), \) and \( h(x) \) be linear functions. Define \( j(x) = \max\{f(x), g(x), h(x)\} \) and \( k(x) = \min\{f(x), g(x), h(x)\} \). Given the graph of \( y = j(x) \) for \(-3.5 \le x \le 3.5\), determine the value of \( \ell^2 \), where \( \ell \) is the length of the graph of \( y = k(x) \) over the same interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
Is 45 vertices the current record for the smallest triangle-free 6-chromatic graph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Last year a bicycle cost $160 and a cycling helmet $40. This year the cost of the bicycle increased by $5\%$, and the cost of the helmet increased by $10\%$. The percent increase in the combined cost of the bicycle and the helmet is:
$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\%\quad}
From the given, find the ratio of the number of triangles to the number of circles inside the rectangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:2}
Find the solutions to
\[\frac{13x - x^2}{x + 1} \left( x + \frac{13 - x}{x + 1} \right) = 42.\]Enter all the solutions, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 6, 3 + \sqrt{2}, 3 - \sqrt{2}}
No math tournament exam is complete without a self referencing question. What is the product of
the smallest prime factor of the number of words in this problem times the largest prime factor of the
number of words in this problem
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1681}
Equiangular hexagon \( {ABCDEF} \) has side lengths
 \( {AB} = {CD} = {EF} = 1 \) and \( {BC} = {DE} = {FA} = r \) . The
 area of \( \bigtriangleup  {ACE} \) is \( {70}\% \) of the area of the hexagon. What is the
 sum of all possible values of \( r \) ?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6
}
Each of the $5{ }$ sides and the $5{ }$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\dfrac{253}{256}$}
The integers $a$, $b$, $c$ and $d$ are such that $a$ and $b$ are relatively prime, $d\leq 2022$ and $a+b+c+d = ac + bd = 0$. Determine the largest possible value of $d$,
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
Let $a, b, c$ be two-digit, three-digit, and four-digit numbers, respectively. Assume that the sum of all digits of number $a+b$, and the sum of all digits of $b + c$ are all equal to $2$. The largest value of $a + b + c$ is

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
10199}
Let $x,y,z$ is positive. Solve:
$\begin{cases}{x\left( {6 - y} \right) = 9}\\
{y\left( {6 - z} \right) = 9}\\
{z\left( {6 - x} \right) = 9}\end{cases}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
x = 3, y = 3, z = 3
}
Suppose $G$ is the Galois group of an irreducible degree 5 polynomial $f \in \mathbb{Q}[x]$ with $|G|=10$. Determine whether $f$ must have precisely 1 or 5 real roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
例4 有1000张编号为000,001，···，999的证件和100个编号为00，01，···，99的盒子．若盒子的号码可以由证件的号码划掉一个数字而得到，则该证件可以放入该盒子中.若选择k个盒子可以装下所有证件，求k的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
It takes 60 grams of paint to paint a cube on all sides. How much paint is needed to paint a "snake" composed of 2016 such cubes? The beginning and end of the snake are shown in the diagram, and the rest of the cubes are indicated by ellipses.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80660}
In the diagram, \( AB \) and \( CD \) intersect at \( E \). If \(\triangle BCE\) is equilateral and \(\triangle ADE\) is a right-angled triangle, what is the value of \( x \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find all values of $z$ such that $z^4 - 4z^2 + 3 = 0$.  Enter all the solutions, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\sqrt{3},-1,1,\sqrt{3}}
Simplify the expression \u221a[3]{5+2\u221a{13}} + \u221a[3]{5-2\u221a{13}} and find its exact value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors. 
Suppose $M$ can be written as  $\Pi_{k=1}^{n}p_k^{a_k}$ where the $p_k$’s are distinct primes and the $a_k$’s are positive integers. 
Find $\Sigma_{k=1}^{n}(p_k + a_k)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{510}
In how many ways can the letters in $\textbf{BEEKEEPER}$ be rearranged so that two or more $\textbf{E}$s do not appear together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A circle of radius $2$ is centered at $O$. Square $OABC$ has side length $1$. Sides $AB$ and $CB$ are extended past $B$ to meet the circle at $D$ and $E$, respectively. What is the area of the shaded region in the figure, which is bounded by $BD$, $BE$, and the minor arc connecting $D$ and $E$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\pi}{3}+1-\\sqrt{3}"]}
In the cube $ABCD-A_1B_1C_1D_1$, it is known that $M$ is the midpoint of $BB_1$. Calculate the cosine value of the dihedral angle between planes $M-CD_1$ and $A$. Express your answer in simplified form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
The sequence $a_0$, $a_1$, $a_2$, $\ldots$ satisfies the recurrence equation:
\[
a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3}
\]
for every integer $n \ge 3$. If $a_{20} = 1$, $a_{25} = 10$, and $a_{30} = 100$, what is the value of $a_{1331}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{181}
若抛物线 $y=a x^2-1$ 上存在关于 $x+y=0$ 成轴对称的两个点, 试求 $a$ 的取值范围.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$a>\frac{3}{4}$ 为所求.}