inclusionAI/Ring-lite-rl-data huggingface dataset:
For a positive constant \( a \), consider the curve \( C: y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}}) \). Let \( l(t) \) be the length of the segment of \( C \) where \( a \leq y \leq t \), and let \( S(t) \) be the area bounded by the line \( y=t \) (where \( a<t \)) and the curve \( C \). Determine \( \lim_{t\to\infty} \frac{S(t)}{l(t)\ln t} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a}
Find the 1000th decimal place of \( \sqrt{\underbrace{111\ldots111}_{1998 \text{ times}}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In how many ways can the letters in BEEKEEPER be rearranged so that two or more Es do not appear together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
24
}
Find the number of permutations \(a_1, a_2, \ldots, a_{100}\) of the numbers \(1, 2, \ldots, 100\) such that:
\[|a_1 - 1| = |a_2 - 2| = \ldots = |a_{100} - 100|\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In triangle \(ABC\), let \(\omega\) be the incircle tangent to \(BC, CA,\) and \(AB\) at points \(D, E,\) and \(F\) respectively. A point \(P\) on \(EF\) is such that \(PD\) is perpendicular to \(BC\). The line \(AP\) intersects \(BC\) at \(Q\). Circles \(\omega_1\) and \(\omega_2\) pass through \(B\) and \(C\) respectively, both tangent to \(AQ\) at \(Q\). Circle \(\omega_1\) meets \(AB\) again at \(X\), and circle \(\omega_2\) meets \(AC\) again at \(Y\). The line \(XY\) intersects \(BC\) at \(Z\). Given \(AB=15\), \(BC=14\), and \(CA=13\), find \(\lfloor XZ \cdot YZ \rfloor\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
Given a list of non-intersecting rectangles $W$, where each rectangle $w$ is represented as $(x, y, \text{width}, \text{height})$ and the list is sorted by $y$-coordinates, determine the maximum factor by which an input rectangle $R$ can grow while maintaining its aspect ratio without intersecting any rectangles in $W$. For example, if $W=\{(10, 2, 2, 8), (5, 5, 3, 3), (7, 7, 2, 2)\}$ and $R=(1, 1, 1, 1)$, find the maximum growth factor for $R$. The aspect ratio of $R$ is $\frac{w}{h}=1$. Determine the maximum factor by which $R$ can grow before it intersects with any rectangle in $W$. Output the maximum growth factor.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the condition number of the matrix product $X \cdot Y$ with respect to the Frobenius norm, where $X \in \mathbb{R}^{M \times N}$ and $Y \in \mathbb{R}^{N \times P}$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given a random vector \(X = (Y, X, Z)\) that follows a Gaussian distribution with mean \(\mu = (1, 2, 4)^T\) and covariance matrix \(\Sigma = \begin{pmatrix} 2 & 3 & 1\\ 3 & 5 & 2\\ 1 & 2 & 6 \end{pmatrix}\), calculate the conditional variance \(\operatorname{Var}(Y | X, Z)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{26}}
Determine the number of homomorphisms from the dihedral group \(D_5\) to the symmetric group \(S_3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the minimum number of unit squares that must be removed from a \(202 \times 202\) grid so that it can be tiled with T-tetrominoes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the big-O bound for the expression: $$\sum^{ N}_{k=0}{2^k \log_2\left(\frac{2^N}{2^k}\right)}$$ where $N = \log M$. Provide your answer in big-O notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(M)}
Find the intersection multiplicity of the curves given by the equations \( f(x,y) = x^5 + x^4 + y^2 \) and \( g(x,y) = x^6 - x^5 + y^2 \) at the point \( P = (0,0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
You have a standard six-sided die and you roll it repeatedly, writing down the numbers that come up. You win when two of your rolled numbers add up to 7. How many turns do you expect to play before you win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{21}{5}}
Determine the form of a matrix $A$ in $\mathbb{R}^3$ that commutes with every other matrix $X$ in the same space, i.e., $AX = XA$ for all matrices $X$. What is the form of matrix $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda I}
What is the value of $\aleph_1^{\aleph_0}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Calculate the surface area of the solid formed by rotating the astroid given by the equation \(x^{2/3} + y^{2/3} = 1\) around the \(y\)-axis, for \(y\) in the interval \([0, 1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{6\pi}{5}}
$a_1,a_2, \cdots , a_{2003}$ are integers such that $|a_1| = 1$ and $|a_{i+1}|=|a_i+1|$ $(1\leq i \leq 2002)$. What is the minimal value of $|a_1+a_2+\cdots + a_{2003}|$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
34
}
Determine the minimum number of black points, $m$, required such that every unit circle centered at a black point contains exactly 10 white points. Assume $m \ge n$ and that the maximum intersection between any two subsets of size 10 is 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
Let \(P_1, P_2, P_3, P_4, P_5\) be five distinct planes in \(\mathbb{R}^3\) such that the intersection of any two distinct planes is a line, the intersection of any three distinct planes is a point, and the intersection of any four distinct planes is empty. Let \(X = \bigcup_{i=1}^5 P_i\) with the topology as a subspace of \(\mathbb{R}^3\). Determine the Euler characteristic of \(X\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In a right triangle \(ABC\) with the right angle at \(C\), the inscribed circle touches the sides \(AB\), \(BC\), and \(CA\) at points \(C_1\), \(A_1\), and \(B_1\) respectively. The altitudes of triangle \(A_1B_1C_1\) intersect at point \(D\). Given that the legs of triangle \(ABC\) are 3 and 4, find the distance between points \(C\) and \(D\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$, respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}$. The area of the union of all triangles $B_{n-1}C_nB_n$ for $n\geq1$ can be expressed as $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{961}
In a game with 2022 nuts on the table, two players, Alice and Bob, take turns eating nuts. On each turn, a player rolls a regular six-sided die and eats the number of nuts shown on the top face. The player who cannot make a move loses the game. Determine the starting number of nuts, \(a\), for which Bob can force a win.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
Calculate the line integral of the scalar field \( x+y \) over the curve \( L \), where \( L \) is the right loop of the polar equation \( r^2 = 2\cos(2\theta) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Calculate the fundamental group \(\pi_1(Y)\) of the space \(Y\), where \(Y\) is the quotient space obtained from \(X = S^2 \times [0,1]\) by identifying each point \(x \in S^2 \times \{1\}\) with its antipodal in \(S^2 \times \{0\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Let $Z_1$ and $Z_2$ be independent identically distributed standard Gaussian random variables with mean 0 and variance 1. Determine the distribution of the random variable \( X = \frac{(Z_1 + Z_2)^2}{(Z_1 - Z_2)^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F(1,1)}
Evaluate the principal value of the improper integral: $$\int_{-1}^{1} \frac{1}{x^{2}}dx$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Calculate the limit: \[ \lim_{n \rightarrow \infty} \left( \frac{n \pi}{4} - \left( \frac{n^2}{n^2+1^2} + \frac{n^2}{n^2+2^2} + \cdots + \frac{n^2}{n^2+n^2} \right) \right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Determine the constant $k$ such that the probability density function \( p(t) = k(1+\sin(20t))e^{-\frac{t}{1.5}} \) is properly normalized and satisfies \( p(t) \leq 1 \) for all \( t > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1802}{2793}}
Determine the order of the group $GL_3(\mathbb{Z_4})$, which consists of all invertible $3 \times 3$ matrices with entries from $\mathbb{Z}_4$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86016}
Find the explicit Cartesian equation for the surface obtained by revolving the curve \( y = \frac{1}{x} \) about the line \( y = x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2xy - z^2 = 2}
Consider a sequence of 10 independent tosses of a biased coin, where the probability of Heads is \( p = \frac{3}{4} \) and the probability of Tails is \( 1 - p \). A reward of one unit is given at time \( k \), for \( k \in \{1, 2, \ldots, 10\} \), if the toss at time \( k \) resulted in Tails and the toss at time \( k-1 \) resulted in Heads. Otherwise, no reward is given at time \( k \). Let \( R \) be the sum of the rewards collected at times \( 1, 2, \ldots, 10 \). Calculate \( E(R^2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{117}{32}}
The amount of a claim that a car insurance company pays out follows an exponential distribution. By imposing a deductible of $d$, the insurance company reduces the expected claim payment by 10%. What is the percentage reduction in the variance of the claim payment?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1\%}
Calculate the probability \( P(N(4) \ge 6 \mid S(4) = 3) \), where \( N(t) \) is the number of events that have occurred between 0 and \( t \), and \( S(n) \) is the arrival time of the \( n \)th event in a Poisson process.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \dfrac{2}{e}}
How many times must a fair coin be tossed to be at least 99% confident that the percentage of heads will fall between 49.5% and 50.5%?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66358}
A casino uses two kinds of dice: a fair die with a probability of \(\frac{1}{6}\) for each number, and a loaded die with a probability of \(\frac{1}{2}\) for rolling a 6 and \(\frac{1}{10}\) for each of the other numbers. The probability of switching from the fair to the loaded die is 0.1, and from the loaded to the fair die is 0.05. Given the observed sequence of numbers \(4, 6, 6, 6\), determine the most likely sequence of dice used, assuming the casino starts with the fair die.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F, L, L, L}
In a tournament with $8$ players, each pair of players played exactly one match. Each match ended either with a win for one player, who received $2$ points while the opponent received $0$ points, or in a draw, where both players received $1$ point each. After all matches were completed, it was found that every player had the same number of points. Determine the minimal total number of draws.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Compute the number of sequences of integers $(a_1,\ldots,a_{200})$ such that the following conditions hold.
· $0\leq a_1<a_2<\cdots<a_{200}\leq 202.$
· There exists a positive integer $N$ with the following property: for every index $i\in\{1,\ldots,200\}$ there exists an index $j\in\{1,\ldots,200\}$ such that $a_i+a_j-N$ is divisible by $203$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20503}
Determine the minimum value of the integral
\[ \mathcal{I}[y] = \int_{-\infty}^{\infty} \left( \frac{1}{2} y'^2 + 1 - \cos y \right) \, dx \]
subject to the boundary conditions \(y(-\infty) = 0\) and \(y(\infty) = 2\pi\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find all solutions for $x$ that satisfy the equation:
\[ \frac{1}{2(x-1)} = \int_0^\infty \frac{\sin (x\arctan y)}{(1+y^2)^{x/2}(e^{y\pi}+1)}\,dy \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a trapezoid \(ABCD\) with \(AD \parallel BC\). It turns out that \(\angle ABD = \angle BCD\). Find the length of segment \(BD\) if \(BC = 36\) and \(AD = 64\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Calculate the probability of generating a dense graph with $n$ vertices from a graphon with a specific set $S$ of positive measure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
已知数列 a_n=n\cos\frac{n\pi}{2},n=1,2,\cdots，函数 f(x)=\df{x^2}{x^2-2x+2}. 则数列 \{f(a_n)\} 的前 2024 项的和 T_{2024} 为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
In the figure, $ABC$ is an obtuse triangle. $BC = 6$ cm, $AB = 5$ cm, and the height $AD$ from $A$ to side $BC$ is 4 cm. If this triangle moves upward along the direction of $DA$ at a speed of 3 cm per second, what is the area swept by the triangle after 2 seconds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66 \text{ cm}^2}
Let $Z \sim N(0,1)$ be a standard normal random variable. Calculate the expected value $E[\max\{Z-1,0\}]$. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0833}
Find the least possible value of \( N \) such that the error in approximating the integral \( \int_0^1 3e^{x^2} \, dx \) using Simpson's Rule is less than or equal to \( 1 \times 10^{-9} \). Use the error bound formula for Simpson's Rule: \( \text{Error}(S_N) \le \frac{K(b-a)^5}{180N^4} \), where \( K \) is the maximum value of the fourth derivative of the integrand on the interval \([a, b]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{244}
In triangle $ABC$, the angles $A$, $B$, and $C$ have tangents that form a geometric sequence: $\tan A$, $\tan B$, $\tan C$. Given that $1 \le \tan A + \tan B + \tan C \le 2015$, determine the number of possible integer values for $\tan B$. (Note: $\tan A$ and $\tan C$ do not need to be integers.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
For a Brownian motion \( B_t \), let \( T_a = \inf\{t \geq 0 \mid B_t = a\} \) be the hitting time of level \( a \). Compute the probability \( P(T_2 < T_{-1} < T_1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the annihilator operator of the function \( f(x) = x(\sin x + 3x - x^{-1}) - \cos x \), where the operator is \( \frac{d}{dx} \equiv D \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{D^3 (D^2 + 1)^2}
In how many ways may thirteen beads be placed on a circular necklace if each bead is either blue or yellow and no two yellow beads may be placed in adjacent positions? (Beads of the same color are considered to be identical, and two arrangements are considered to be the same if and only if each can be obtained from the other by rotation).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
Find the residue at $z = 0$ of the function $(1-e^{-z})^{-n}$, where $n$ is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Two joggers each run at their own constant speed and in opposite directions from one another around an oval track. They meet every 36 seconds. The first jogger completes one lap of the track in a time that, when measured in seconds, is a number (not necessarily an integer) between 80 and 100. The second jogger completes one lap of the track in a time, \(t\) seconds, where \(t\) is a positive integer. The product of the smallest and largest possible integer values of \(t\) is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3705}
Jessica's family is some distance away from New York. They are traveling at the rate of 50 km/h and stop to rest for 30 minutes every 2 hours. It takes them 7 hours to reach New York. How far away is Jessica's family from New York?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Polycarp, the top student, wrote a huge number by writing out all natural numbers from 1 to 500 in sequence: 123...10111213...499500. The underachieving Kolya erased the first 500 digits of this number. What do you think the first digit of the remaining number is?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\ell$ passing through $B$, we color red the foot of the perpendicular from $C$ to $\ell$. The set of red points enclose a bounded region of area $\mathcal{A}$. Find $\lfloor \mathcal{A} \rfloor$, which is the greatest integer not exceeding $\mathcal{A}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{157}
Let $f(x) = \frac{x}{a(x+2)}$, and the equation $x = f(x)$ has a unique solution. Given that $f(x_0) = \frac{1}{1008}$ and $f(x_{n-1}) = x_n$ for $n=1, 2, 3, \ldots$, calculate $x_{2015}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2015}}
Find the minimum possible value of the sum \(\sum_{i=1}^{11}x_i\), where \(x_i\) are positive integers for \(i=1,2,...,11\). The conditions \(x_i+x_{i+1}\geq 100\) and \(|x_i-x_{i+1}|\geq 20\) hold for \(i=1,2,...,10\), and additionally, \(x_{11}+x_{1}\geq 100\) and \(|x_{11}-x_1|\geq 20\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{580}
Each frog needs to eat 30 flies per day to live, but they also cooperate with herons, collectively catching 60 flies per heron present. Each fish needs to eat 8 frogs per day to live. However, the fish and frogs also compete for a common prey: tadpoles. Each fish needs to eat either 8 frogs or 24 tadpoles, and each frog needs to eat either 30 flies or 2 tadpoles per day.

Each gharial needs to eat 15 fish a day to live. Additionally, each heron needs to eat 5 frogs and 3 fish per day to live, and each caiman needs to eat 2 gharials and 2 herons per day to live.

Considering the above interactions and consumption requirements, how many flies get eaten every day in this swamp if it has 9 gharials, 12 herons, 7 caimans, 20 fish, and 50 frogs? Assume that a given ratio of tadpoles and frogs can be determined and use it to calculate the daily total of consumed flies.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2220}
Determine the number of pairs \((x, y)\) of real numbers that satisfy the inequalities \(x^6+y^6 < 1\) and \(x^3+y^3 > 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Among the eight vertices of the cube ABCD-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub>, a plane passes through four of them, and the remaining four vertices are all at equal distances from the plane. Find the number of such planes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given that an agent's task completion times follow an exponential distribution, Agent 1 has Task 1 and Task 2 with mean completion times of 6 and 4 units, respectively. Agent 2's completion times for Task 1 and Task 2 have means of 3 and 1 unit, respectively. Task 2 can only begin after Task 1 is completed by both agents. Assuming all times are independent, calculate the probability that Agent 1 finishes Task 2 before Agent 2 does.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{5}}
Find the sum of \( \frac{\sigma(n) \cdot d(n)}{ \phi (n)} \) over all positive \( n \) that divide \( 60 \).

Note: The function \( d(i) \) outputs the number of divisors of \( i \), \( \sigma (i) \) outputs the sum of the factors of \( i \), and \( \phi (i) \) outputs the number of positive integers less than or equal to \( i \) that are relatively prime to \( i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{350}
A round robin tournament is held with $2016$ participants. Each player plays each other player once, and no games result in ties. We define a pair of players $A$ and $B$ as a \textit{dominant pair} if all other players either defeat both $A$ and $B$ or are defeated by both $A$ and $B$. Find the maximum number of dominant pairs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
For the nodal cubic defined by \(y^2 = x^2(x+1)\), consider the parametrization given by \(x(t) = t^3 - 3t\) and \(y(t) = t^4 - 4t^2\). Find the values of \(t\) where this parametrization fails to be regular.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Ricsi, Dénes, and Attila often play ping-pong against each other, where two of them team up on one side. Dénes and Attila win against Ricsi three times as often as they lose; Dénes wins against Ricsi and Attila as often as he loses; finally, Attila wins against Ricsi and Dénes twice as often as he loses. Recently, they spent an entire afternoon playing six matches, playing each combination twice. What is the probability that Ricsi won at least once, either alone or with a partner?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15}{16}}
Find the limiting value of the ratio of integrals \( \frac{\int_{0}^{
fty}x^{m+1}e^{x}dx}{\int_{0}^{
fty}x^{m}e^{x}dx} \) as the upper limit approaches infinity, for \( m \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Martha needs to paint all four walls in her kitchen, which has a certain length, 16 feet width, and 10 foot high ceilings. Each wall needs one coat of primer and two coats of paint. Martha can paint 40 square feet per hour, and it takes her 42 hours to paint the kitchen. What is the length of the kitchen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the integer values of \( n \) for which the inequality \( n^{\sqrt{n+2}} > (n+1)^{\sqrt{n+1}} \) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 9}
Let \( P(x) \) be a polynomial of degree 4, having extrema at \( x=1 \) and \( x=2 \). Given that \( \lim_{x\to 0}\frac{x^2+P(x)}{x^2}=2 \), find the value of \( P(2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the number of integers $n$ such that $1 \leq n \leq 2017$ and $(n-2)(n-0)(n-1)(n-7)$ is a multiple of $1001$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}