inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the condition under which the expression \(\sqrt{A \pm \sqrt{\bar{B}}}\) can be represented as a binomial \(\sqrt{x} \pm \sqrt{y}\), where \(x\) and \(y\) are rational numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A^2 - B \text{ must be a perfect square}}
Determine the sum of the coefficients of the terms with even powers of $x$ in the expansion of $\frac{(1+x)^{2n}}{(1-x)^{n}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2n-1}}
Given that real numbers \( a \) and \( b \) are such that the equation \( a x^{3} - x^{2} + b x - 1 = 0 \) has three positive real roots, find the minimum value of \( P = \frac{5 a^{2} - 3 a b + 2}{a^{2}(b - a)} \) for all such \( a \) and \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12 \sqrt{3}}
John makes a contribution to a charity fund at school, and the average contribution size reaches $75 per person. There were 1 other contributions made before John's, and the size of his donation is $100. By what percentage did the average contribution size increase after John's donation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16.67\%}
Find the number of four-element subsets of $\{1,2,3,4,\cdots, 20\}$ with the property that two distinct elements of a subset have a sum of $16$, and two distinct elements of a subset have a sum of $24$. For example, $\{3,5,13,19\}$ and $\{6,10,20,18\}$ are two such subsets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
Given an ellipse $C$ with foci at $F_{1}(-2\sqrt{2},0)$ and $F_{2}(2\sqrt{2},0)$, and a major axis length of $6$. Let the line $y=x+2$ intersect the ellipse $C$ at points $A$ and $B$. Find the coordinates of the midpoint of the line segment $AB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(-\frac{9}{5}, \frac{1}{5}\right)}
Six soccer teams are competing in a tournament in Waterloo. Every team is to play three games, each against a different team. (Note that not every pair of teams plays a game together.) Ignoring the order and times of the games, calculate the number of different possible schedules.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
The letter T is formed by placing a $2\:\text{inch}\!\times\!6\:\text{inch}$ rectangle vertically and a $3\:\text{inch}\!\times\!2\:\text{inch}$ rectangle horizontally on top of the vertical rectangle at its middle, as shown. What is the perimeter of this T, in inches?
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Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Suppose $a_0, a_1, \ldots, a_{2018}$ are integers such that \[ (x^2 - 3x + 1)^{1009} = \sum_{k=0}^{2018} a_k x^k \] for all real numbers $x$. Compute the remainder when $a_0^2 + a_1^2 + \cdots + a_{2018}^2$ is divided by $2017$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the sum $\sum_{r=0}^{2n}(-1)^r(r+1)\binom{2n}{r}\cdot\binom{2n}{r}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the minimum of the function $f(x_1,x_2) = (x_1 + x_2 + 3)^2 + (3x_1 + x_2 - 1)^2 + (2x_1 + x_2 + 1)^2$. We've been instructed to solve this by writing $f(x_1,x_2)$ as $||Ax - b||^2$, and $x = [x_1,x_2]^T$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
6. Given that $n, k$ are positive integers, $n>k$. Given real numbers $a_{1}, a_{2}, \cdots, a_{n} \in(k-1, k)$. Let positive real numbers $x_{1}, x_{2}$, $\cdots, x_{n}$ satisfy that for any $k$-element subset $I$ of $\{1,2, \cdots, n\}$, we have $\sum_{i \in I} x_{i} \leqslant \sum_{i \in I} a_{i}$. Find the maximum value of $x_{1} x_{2} \cdots x_{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_{1}a_{2}\cdotsa_{n}}
Real numbers \(a\) and \(b\) are chosen with \(1 < a < b\) such that no triangle with positive area has side lengths \(1, a, b\) or \(\frac{1}{b}, \frac{1}{a}, 1\). The original answer is in the form \(\frac{k+\sqrt{n}}{m}\). Find the value of \(k + m + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Shirley sold 54 boxes of Trefoils, 36 boxes of Samoas, and 48 boxes of Thin Mints within 3 hours. Each case she needs to deliver contains 4 boxes of Trefoils, 3 boxes of Samoas, and 5 boxes of Thin Mints. How many cases does Shirley need to deliver to fulfill the orders?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Find a function $h(y)$ that satisfies the integral equation \[ y = \int_{0}^{\infty} h\left(\frac{y}{x}\right) dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta(y - 1)}
Find the number of subgroups of index 3 in the group \( P = \Bbb Z_4 \times \Bbb Z_6 \times \Bbb Z_9 \). Here, \( \Bbb Z_m \) denotes the cyclic group of order \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
For a random sample $x_1, x_2, \ldots, x_9$ drawn from a Poisson distribution with mean $\theta = 0.4$, the probability density function is given by $f(x;\theta) = \frac{\theta^x e^{-\theta}}{x!}$. A hypothesis test is conducted where the null hypothesis is accepted if $\frac{1}{3} \leq \overline{x} \leq \frac{2}{3}$, and rejected otherwise. Calculate the type I error rate for this test.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.376}
Determine the number of ways to paint $n$ sectors of a disk using $n$ different colors such that no two adjacent sectors have the same color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n-1)!}
Evaluate the limit: $$\lim_{\epsilon \to 0^+} \left(\int_{2}^{2+\epsilon}+\int_{3-\epsilon}^{3}\right)\frac{\ln (x-1)}{(x-2)^2} dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the limit \( \lim_{n \to \infty} \frac{2n!}{n!} \) using Stirling's approximation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Estimate the number of terms required in the Taylor expansion of \( \ln(1+x) \) to approximate \( \ln\left(\frac{31}{16}\right) \) to four decimal places. The Taylor series is given by:

\[ \ln(1+x) = \sum_{n=1}^\infty (-1)^{n+1} \frac{x^n}{n} \]

Given \( x=\frac{15}{16} \), use the alternating series error bound to determine the smallest integer \( N \) such that:

\[ \left|\ln\left(1+\frac{15}{16}\right) - \sum_{n=1}^N (-1)^{n+1} \frac{\left(\frac{15}{16}\right)^n}{n}\right| < 0.0001 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Calculate \( \mathbb{E}\left[\sinh \left(B_{t}\right)\right] \) where \( B_t \) is a Brownian motion, without using Itô's formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a single-elimination tournament with $2^n$ players, where each player has an equal 50% chance of winning against any other player, calculate the probability that the second-best player wins the tournament.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2^n}}
Determine the number of passes in a monkey saddle, given that a pass moves from a hollow, through the critical point, into the adjacent hollow. A monkey saddle has three hollows adjacent to the critical point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the number of positive integers $a \le 250$ for which the set $\{a+1, a+2, \dots, a+1000\}$ contains

$\bullet$ Exactly $333$ multiples of $3$,
$\bullet$ Exactly $142$ multiples of $7$, and
$\bullet$ Exactly $91$ multiples of $11$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
21
}
75. Arrange 10 small balls labeled with numbers 1 to 10 in a row, such that the sum of the numbers on every 3 adjacent balls is a multiple of 3. The total number of arrangements is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1728}
There are 2023 distinct points on a plane, which are coloured in white or red, such that for each white point, there are exactly two red points whose distance is 2023 to that white point. Find the minimum number of red points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45
}
Let the function $f(x)=(2x^2-ax+b)\ln(x-1)$, where $a,b\in\mathbb{R}$, and $f(x) \geq 0$ holds true when $x > 1$. The range of values for $a$ is \_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \leq 6}
A box initially contains 3 large donuts and 7 small donuts. Each hour, a donut is randomly selected: small donuts are consumed, while large ones are halved, with one half eaten and the other half returned as a small donut. What is the expected number of hours until only small donuts remain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{113}{12}}
If for any real number $x \in (0, \frac{1}{2}]$, the inequality $2^{-2x}-\log_{a}x < 0$ always holds, then the range of the real number $a$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4} < a < 1}
Let $f$ be a function from the set $\{1,2,3, \dots ,10 \}$ to $\mathbb{R}$ such that \[ \left( \sum_{i=1}^{10}\frac{|f(i)|}{2^i}\right)^2 = \left( \sum_{i=1}^{10} |f(i)|^2 \right) \left(\sum_{i=1}^{10} \frac{1}{4^i} \right). \] Let $S$ be the set of all functions that satisfy this equality. Determine the cardinality of the set $S$. \(\text{Express your answer as a number.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024}
Determine the parameter $b$ for a Bernoulli random variable $Y$ such that the sum $X + Y$ of two Bernoulli random variables $X$ and $Y$ is also a Bernoulli random variable with parameter $p + b$. Given that $X$ is a Bernoulli random variable with parameter $p$, find the value of $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - p}
Consider the sequence defined by \(x_1 = 1\) and \(x_n = \sup\{x \in [0, x_{n-1}): \sin(1/x) = 0\}\) for \(n \geq 2\). Determine the value of \(\limsup x_n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
What is the maximum number of elementary matrices required to express any invertible 2x2 matrix as a product?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the limit: $$ \lim_{x\rightarrow 0} \frac{\sin x^2}{ \ln ( \cos x^2 \cos x + \sin x^2 \sin x)} $$ without using L'Hôpital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Let \( f \) be a nonnegative measurable function on \( \mathbb{R}^d \) such that \( m(\{x \mid f(x) > \lambda\}) = \frac{1}{1+\lambda^2} \). For which values of \( p \) is \( f \in L^p(\mathbb{R}^d) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2)}
For \(x \to 0^+\), find an asymptotic equivalent of the series \(\sum_{n=1}^{+\infty} f_n(x)\), where \(f_n(x) = n^x \exp(-nx)\). Assume \(f(x) = \sum_{n=1}^{+\infty} f_n(x)\) is defined on \((0, \infty)\) and is convex. Use the hint \(n \ln(n) \sim n\) and explore methods to find asymptotic equivalents of series of functions without using the \(\Gamma\) function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{x}}
Calculate the multiplicity of the intersection of the two plane curves \(F=2X_0^2X_2-4X_0X_1^2+X_0X_1X_2+X_1^2X_2\) and \(G=4X_0^2X_2-4X_0X_1^2+X_0X_1X_2-X_1^2X_2\) at the point \((1:0:0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the maximum value of \( M \) such that for all real numbers \( a, b, c \), there exists a point \( x_0 \) in the interval \([0, 4]\) where \(|x_0^3 + ax_0^2 + bx_0 + c| \geq M\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral \( \iint\limits_R \min\{\cos x, \cos y\} \, dx \, dy \) over the region \( R = \{0 \le x \le \frac{\pi}{2},\ 0 \le y \le \frac{\pi}{2}\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi - 2}
Find the norm of the operator \( T_f: (C([a,b],\mathbb{C}), \lVert \cdot \rVert_1) \to \mathbb{C} \) defined by \( T_f(g) = \int_a^b f(x)g(x) \, dx \) for a given function \( f \in C([a,b],\mathbb{C}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\|f\|_\infty}
Evaluate the surface integral \( \iint_{S} (\mathbf{F} \cdot \mathbf{n}) \, dS \) of the vector field \( \mathbf{F}(x, y, z) = (-x\mathbf{i} - 3x\,y^2\mathbf{j} + z\mathbf{k}) \) over the surface \( S \) defined by the region \( V \), where \( V \) is the solid enclosed by the paraboloid \( z = 2xy \) and the cylinder \( x^2 + y^2 = 1 \), with \( 0 \le z \le 2xy \). Given that \( \nabla\cdot\mathbf{F} = -6xy \), compute the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{4}}
A knight is placed on the bottom left corner of a $3 \times 3$ chessboard. Determine the number of ways to move the knight to the upper right corner in exactly 12 moves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{992}
Determine the number of ways to partition a set of \( n \) elements into \( k+1 \) non-empty, pairwise disjoint subsets, where the subsets are not labeled.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S(n, k+1)}
Determine the number of functions \( f: \mathbb{Z} \to \mathbb{Z} \) that satisfy the equation \( f(h+k) + f(hk) = f(h)f(k) + 1 \) for all integers \( h \) and \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the general solution to the differential equation: \[ t y y'' - t y'^2 + 2y y' = 0 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = C_1 e^{C_2/t}}
Evaluate the limit: \[ L = \lim_{x \to 0} \frac{1-\cos(\sin x)+\ln(\cos x)}{x^4}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{7}{24}}
Determine the condition on the parameter \(a\) such that the solution \(u(x, t)\) of the partial differential equation \(\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - au\), with boundary conditions \(u(0, t) = u(\pi, t) = 0\) for \(t \geq 0\) and initial condition \(u(x, 0) = (\sin x)^3\), converges to zero uniformly as \(t \to \infty\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > -1}
In a series of coin tosses, each with an equal probability of heads or tails, calculate the expected number of tosses required to observe a sequence of 80 heads within a window of 120 consecutive tosses.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{81} - 2}
Let $G$ be the group of symmetries of an equilateral triangular lattice $L$. Find the index in $G$ of the subgroup $T \cap G$, where $T$ is the group of translations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given the pair of straight lines represented by the equations \(x^2 - 2pxy - y^2 = 0\) and \(x^2 - 2qxy - y^2 = 0\), where each pair bisects the angle between the other pair, find the value of \(pq\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
How many terms of the Maclaurin expansion of \((1+x)^{82}\) are needed to approximate \(1.1^{82}\) to an accuracy of \(10^{-6}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Evaluate the integral \( \int_{0}^{\pi}\ln(1-2\pi\cos(x)+\pi^2)\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi \ln \pi}
In an infinite sequence of tosses of a fair coin, what is the expected number of tosses required for the pattern H T T H to appear?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let \( X = \{ x = (x_1, x_2, \ldots) : x_i \in \mathbb{R}, x_i = 0 \text{ for almost all } i \in \mathbb{N} \} \) with the norm \( \|x\| = \sup_{i \in \mathbb{N}} |x_i| \). Let \( S: X \to X \) be a linear operator defined by \( Sx = (x_1 + x_2 + x_3 + \ldots, x_2 + x_3 + x_4 + \ldots, x_3 + x_4 + \ldots, \ldots) \). Compute the operator norm \( \|S\| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let \( A \) be a subset of \(\{1, 2, 3, \ldots, 2019\}\) having the property that the difference between any two of its elements is not a prime number. What is the largest possible number of elements in \( A \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{505}
How many of the fi\frst $1000$ positive integers can be written as the sum of fi\fnitely many distinct numbers from the sequence $3^0$, $3^1$, $3^2$ ,$...$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105
}
Through points \(A(0 ; 14)\) and \(B(0 ; 4)\), two parallel lines are drawn. The first line, passing through point \(A\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(K\) and \(L\). The second line, passing through point \(B\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(M\) and \(N\).

What is the value of \(\frac{A L - A K}{B N - B M}?\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.5}
In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Additionally, at least one head of each dragon always tells the truth. One day, 530 dragons sat at a round table, and each of them said:

- 1st head: "To my left is a green dragon."
- 2nd head: "To my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."

What is the maximum number of red dragons that could have been at the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{176}
A gives 24 apples to B and C, and each of the three people has at least two apples. Find the number of distinct ways to distribute the apples. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{190}
Uri buys three burgers and two sodas for $\$5.10$, and Gen buys two burgers and three sodas for $\$5.40$. How many cents does a soda cost if a side dish costs $\$0.30$ and Uri buys one side dish?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
In the expansion of $(1-x)^n = a + a_1x + a_2x^2 + a_3x^3 + \ldots + a_nx^n$, if $2\binom{n}{2} - a_{n-5} = 0$, then find the value of the natural number $n$. Express your answer as a natural number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Two circles are tangent to each other internally at point \( A \). A chord \( BC \) of the larger circle is tangent to the smaller circle at point \( D \). The line \( AD \) intersects the larger circle again at point \( M \). Find \( MB \), given that \( MA = a \) and \( MD = b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{a \cdot b}}
The line $y=kx$ intersects the circle $(x-2)^2+y^2=4$ at points O and A. If $|OA|=2\sqrt{3}$, then the value of the real number $k$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm\frac{\sqrt{3}}{3}}
Cars A and B simultaneously depart from locations $A$ and $B$, and travel uniformly towards each other. When car A has traveled 12 km past the midpoint of $A$ and $B$, the two cars meet. If car A departs 10 minutes later than car B, they meet exactly at the midpoint of $A$ and $B$. When car A reaches location $B$, car B is still 20 km away from location $A$. What is the distance between locations $A$ and $B$ in kilometers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Given the function $y=f(x)$ ($x \in \mathbb{R}$) that satisfies $f(x+2)=f(x)$, and for $x \in (-1,1]$, $f(x)=|x|$, find the number of intersection points between the graph of the function $y=f(x)$ and the graph of the function $y=\log_{4}|x|$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $a, b, c, d$ be four positive integers such that $a > b > c > d$. Given that $ab + bc + ca + d^2 \mid (a + b)(b + c)(c + a)$. Find the minimal value of $\Omega(ab + bc + ca + d^2)$. Here $\Omega(n)$ denotes the number of prime factors $n$ has. For example, $\Omega(12) = 3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A man lost 10% by selling oranges at the rate of 12 a rupee. He changed the rate to 7.5 a rupee to gain a certain percentage. What was the percentage of his gain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44\%}
An equiangular hexagon has side lengths $1,1, a, 1,1, a$ in that order. Given that there exists a circle that intersects the hexagon at 12 distinct points, we have $M<a<N$ for some real numbers $M$ and $N$. Determine the minimum possible value of the ratio $\frac{N}{M}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 \sqrt{3}+3}{2}}
Let $C$ be the part of the intersection curve of the paraboloid $z = x^2 + y^2$ and the plane $2x + 2y - z + 2 = 0$ that starts at $(2, 0, 4)$ and ends at $(0, 2, 4)$. Calculate $\int_C (x^2 + y^2) \, dl$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4π}
On the coordinate plane, consider squares where all vertices have non-negative integer coordinates, and the center is at point \((60, 45)\). Find the number of such squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2070}
From the numbers 1, 2, 3, 4, 5, two numbers are randomly selected to be the base and the true number (antilogarithm) of a logarithm, respectively. The total number of different logarithmic values that can be obtained is ___.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
The graph of the power function $f(x)$ passes through the point $(\sqrt{3}, 3)$. If the function $g(x) = f(x) + 1$ has a range of $[1, 5]$ on the interval $[m, 2]$, then the range of the real number $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-2, 0]}
In triangle \(ABC\) with area 1, let \(A_1\), \(B_1\), and \(C_1\) be the midpoints of sides \(BC\), \(CA\), and \(AB\) respectively. Points \(K\), \(L\), and \(M\) are located on segments \(AB_1\), \(CA_1\), and \(BC_1\) respectively. Determine the minimum area of the intersection of triangles \(KLM\) and \(A_1B_1C_1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8}}
Define the sequences \(x_0, x_1, x_2, \ldots\) and \(y_0, y_1, y_2, \ldots\) such that \(x_0 = 1\), \(y_0 = 2021\), and for all nonnegative integers \(n\), we have \(x_{n+1} = \sqrt{x_n y_n}\) and \(y_{n+1} = \frac{x_n + y_n}{2}\). There is some constant \(X\) such that as \(n\) grows large, \(x_n - X\) and \(y_n - X\) both approach \(0\). Estimate \(X\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{353}
Let $f(n) = 9n^5- 5n^3 - 4n$. Find the greatest common divisor of $f(17), f(18),... ,f(2009)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}