inclusionAI/Ring-lite-rl-data huggingface dataset:
A game is played by two players, A and B. They first agree on an integer N, and then A starts. They take turns filling one of the six boxes with any of the digits 1, 2, 3, 4, 5, 6, 7, 8, or 9, where each box can hold only one digit and digits can repeat, forming a six-digit number. If this six-digit number is divisible by N, B wins; otherwise, A wins. When N is less than 15, what is the value of N that guarantees a winning strategy for B? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Evaluate the Riemann-Stieltjes integral \( \int_2^7 t^2 \, d\alpha(t) \) where \( \alpha(t) = -t + k \) for \( k \le t < k+1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{70}{3}}
Solve the equation \( \sin x = 1 - x \) for \( x \) without using graphical methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.511}
Find the distribution of the ratio $Y/X$ where $X$ and $Y$ are random variables uniformly distributed over the region $A = \{(x,y) \in \mathbb{R^2}: 0 \leq y \leq x \leq 1\}$ with joint density $d\mu_{(X,Y)}(x,y) = 2\mathbf{1}_A(x,y)dxdy$. Provide the distribution of $Y/X$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{U(0,1)}
A room is shaped like an 'L'. One part is a rectangle that is 23 feet long and 15 feet wide, and a square with a side of 8 feet is attached at one end of the rectangle, extending its width. Calculate the ratio of the total length of the room to its perimeter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
对每个正整数 $n$ ，定义集合
$P_{n}=\left\{n^{k} \mid k=0,1, \cdots\right\}$ 。
对于正整数 $a 、 b 、 c$, 若存在某个正整数 $m$ ，使得 $a-1 、 a b-12 、 a b c-2015$ 这三个数
（不必两两不等）均属于集合 $P_{m}$ ，则称正整数组 $(a, b, c)$ 为"幸运的". 求所有幸运的正整数组的个数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Find the greatest common divisor (GCD) of the complex numbers \(5-3i\) and \(7+i\) in the ring of Gaussian integers \(\mathbb{Z}[i]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + i}
Alexis noticed her brother Joshua measuring something on the floor. He was designing a sandcastle and needed to measure the base of a spiral staircase. The castle walls form a rectangle, and the center of the staircase is: 
- 9 inches from one corner, 
- 15 inches from another corner, 
- 16 inches from a third corner, 
- and some whole number of inches from the furthest corner. 

Find the distance from the center of the staircase to the furthest corner of the rectangular castle, assuming all four distances are measured on the same plane (the ground).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the value of $x$ that satisfies the equation $2^{x^2-3}=x^{-1/3}$ using a numerical approximation method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.660}
Embed $\mathbb{R}^2$ in the projective plane $\mathbb{RP}^2$ using the map $(x, y) \mapsto [1, x, y]$. Determine the point of intersection in $\mathbb{RP}^2$ of the projective lines corresponding to the parallel lines $y = mx$ and $y = mx + c$ in $\mathbb{R}^2$, where $c \neq 0$. Provide your answer as a point in homogeneous coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 1, m]}
Evaluate the double integral \( F(\Lambda) = \int_0^{\infty}\int_0^{\infty}\left(\frac{e^{-\Lambda x}-e^{-\Lambda y}}{x-y}\right)^2 \,dx\,dy \), where \( \Lambda \) is a positive real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \ln 2}
A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90\,^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{49}{512}$}
Consider a group $H$ with elements $x$ and $y$ such that $|x|=|y|=4$, $x^2=y^2$, $xy=x^{-1}y$, $x\neq y$, and $H=\langle x,y\rangle$. Determine the order of the group $H$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the dimension of the vector space $\Bbb Q[x,y]/J$ over $\Bbb Q$, where $J=\langle(x-1)^3,(x-1)(y+2),(y+2)^2\rangle$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Integrate the function \( f(x,y,z) = 14xz \) over the region in the first octant \((x, y, z \ge 0)\) that lies above the parabolic cylinder \( z = y^2 \) and below the paraboloid \( z = 8 - 2x^2 - y^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1024}{3}}
Let $V$ be a finite-dimensional vector space over the field $\mathbb{F}_q$ with $q$ elements, and let $U$ be a subspace of $V$. Determine the number of subspaces $W$ of $V$ such that $W \cap U = \{0\}$ and $V = W \oplus U$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{q^{k(n - k)}}
Given the joint mass density function \( f_{X,Y}(x,y) = \frac{{e}^{-\left(y + \frac{x}{y}\right)}}{y} \) for \( x > 0 \) and \( y > 0 \), calculate the covariance \( \text{Cov}[X,Y] = E[XY] - E[X]E[Y] \). It is known that \( Y \) follows an exponential distribution with rate parameter \( \lambda = 1 \), hence \( E[Y] = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find a non-constant polynomial \( p(x) \) in \( \mathbb{Z}_4[x] \) such that \( p(x) \) is a unit and has a degree strictly between 2 and 6.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 2x^3}
Consider three mutually interpenetrating solids: two right circular pyramids and one right circular prism. The base of each pyramid and both bases of the prism are sealed by flat circular surfaces. Determine the maximum number of bounded volumes that can be formed, considering only the surfaces of the solids as boundaries and counting only volumes that are not further subdivided.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In a positive non-constant geometric progression, the arithmetic mean of the second, seventh, and ninth terms is equal to some term of this progression. What is the smallest possible index of this term?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the vectors \(\mathbf{v}_1 = (1, 2, 0, 2)\), \(\mathbf{v}_2 = (1, 1, 1, 0)\), and \(\mathbf{v}_3 = (2, 0, 1, 3)\), find a fourth vector \(\mathbf{v}_4\) such that the set \(\{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4 \}\) forms a basis for \(\mathbb{R}^4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 0, 0, 1)}
The volume of a cube in cubic meters and its surface area in square meters is numerically equal to four-thirds of the sum of the lengths of its edges in meters. What is the total volume in cubic meters of twenty-seven such cubes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216}
Determine the cardinality of the set \( \bigcup_n \mathcal{P} (B-A_n) \), where \( n \in \mathbb{N} - \{0,1\} \), \( A_n=[0,1/n] \), and \( B=\{1/k | k \in \mathbb{N} - \{0\}\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
Determine the dimension of the subset of $2 \times 2$ matrices $(c_1, c_2)$ that generate a 2-dimensional Lie algebra, given by the condition $[c_1, c_2] = k_1 c_1 + k_2 c_2$, where $[a,b]$ denotes the commutator $ab - ba$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the number of unique ways to color a triangular array of 10 balls, arranged in rows of 1, 2, 3, and 4 balls respectively, using 2 colors. Consider colorings that are equivalent under rotation to be the same.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{352}
Given a grid where each cell contains a number representing the height of a mountain, moving from a mountain of height \( a \) to a mountain of height \( b \) takes \( (b-a)^{2} \) time. Starting from the mountain of height 1, determine the minimum amount of time required to reach the mountain of height 49, moving only up, down, left, or right.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Consider an analytic function with a Laurent series expansion that converges on an annulus. Determine the maximum possible radius of convergence for a power series expansion about a point within the annulus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{R - r}{2}}
Let $u_1=(1,1,1)$, $u_2=(2,0,0)$, and $u_3=(0,3,0)$ be a basis of $\mathbb{R}^3$. Calculate the coordinates of $w=2e^{*2}+3e^{*3}$ in the dual basis $\{u^{*1}, u^{*2}, u^{*3}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(5, 0, 6)}
Determine the Lipschitz constant for a solution $x(t)$ to the differential equation $y'(t) = \sin(3\pi(y(t)+2t))$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\pi}
Points $P, Q, R, S, T, U, V,$ and $W$ lie, in that order, on line $\overline{PW}$, dividing it into seven equal segments, each of length 1. Point $X$ is not on line $PW$. Points $Y$ and $Z$ lie on line segments $\overline{XR}$ and $\overline{XW}$ respectively. The line segments $\overline{YQ}, \overline{ZT},$ and $\overline{PX}$ are parallel. Determine the ratio $\frac{YQ}{ZT}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{6}}
A large bucket is filled with water. You want to divide this water into 7 smaller buckets of the same shape and size. If you divide it by 900 grams (g), 200 grams (g) remain. How many kilograms (kg) of water is in the large bucket?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.5}
For positive integers $n$, let $c_{n}$ be the smallest positive integer for which $n^{c_{n}}-1$ is divisible by 210, if such a positive integer exists, and $c_{n}=0$ otherwise. What is $c_{1}+c_{2}+\cdots+c_{210}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{329}
Determine the radius of convergence of the power series \( \sum_{n=0}^\infty b_n x^n \), where the coefficients \( b_n \) are defined by the recurrence relation:

\[ \begin{aligned} b_{n+2} &= \frac{(n+2)(n+1) b_{n+1} - b_n}{(n+2)(n+1)}, \quad n\ge 1 \\
 b_2 &= -b_0/2 \end{aligned} \]

with arbitrary initial values \( b_0 \) and \( b_1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the sum of all values of $a$ that satisfy the equation: \[ \int_{0}^{\pi/2} (\sin x + a \cos x)^3 \, dx - \frac{4a}{\pi-2} \int_{0}^{\pi/2} x \cos x \, dx = 2. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{3}{2}}
Determine the maximum number of points in general position in a finite projective plane of order $n$, where $n$ is an odd integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + 1}
Find the units digit of the sum:
\[2^{2^2} + 4^{4^4} + 6^{6^6} + \cdots + 2016^{2016^{2016}} + 2018^{2018^{2018}}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the integral \( \oint_{|z| = 3} \frac{\mathrm{d}z}{\sqrt{z^2 - 1}} \) using a branch of the function that is analytic outside the interval \([-1, 1]\) and positive on \((1, \infty)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
Determine the fundamental group of the space \( \mathbb{C}^2 \setminus \Delta \), where \( \Delta = \{(z,z) : z \in \mathbb{C}\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Find the intersection point of the functions $x^{0.999}$ and $\frac{x}{\log(x)}$ for $0 < p < 1$. Assume $p = 0.999$. Determine the value of $x$ where these two functions are equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.721}
Determine the smallest integer \( N \) such that the difference \( |s - s_N| \) is less than \( 10^{-6} \), where \( s = \sum_{n=1}^\infty \frac{2}{2+e^n} \) and \( s_N = \sum_{n=1}^N \frac{2}{2+e^n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In a finite arithmetic progression, the sum of the absolute values of its terms is 100. If each term is increased by 1 or by 2, the sum of the absolute values of the terms remains 100 in both cases. Determine the possible values of the expression \( n^2 d \), where \( n \) is the number of terms and \( d \) is the common difference of the progression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
Consider an infinite sequence of independent and fair coin tosses. Let $H_i$ denote the event that the $i$th coin lands heads (where $H_i = 1$ for heads and $0$ for tails). Compute the probability of the event:

$$\mathbb{P}\left(\bigcap_{i=1}^{\log_2(n)} H_{n+i} \text{ occurs infinitely often}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the limit: $$ \lim_{n \to \infty }\left[\frac{\rm f\left(1 + 1/n\right)}{\rm f\left(1\right)}\right]^{1/n} $$ where ${\rm f}$ is differentiable at $1$ and ${\rm f}\left(1\right) > 0$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r}M$, where $r$ and $M$ are positive integers and $M$ is not divisible by $3$. What is $r$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5
}
Given a real number $a$, for each $n \ge 0$, let $a_n$ be the distance from $a$ to the nearest rational number of the form $\frac{m}{2^n}$, where $m$ is an integer. Determine the maximum possible value of $\sum\limits_{n=0}^{\infty} a_n$. \(\text{Express your answer as a single number.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Find the sum of the indexes of the singular points other than zero of the vector field:
\[ z\overline{z}^2 + z^4 + 2\overline{z}^4 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the smallest constant $M$, such that for any positive real numbers $x$, $y$, $z$, and $w$,
\[\sqrt{\frac{x}{y + z + w}} + \sqrt{\frac{y}{x + z + w}} + \sqrt{\frac{z}{x + y + w}} + \sqrt{\frac{w}{x + y + z}} < M.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{\sqrt{3}}}
A regular sphere has a surface area given by \( A = 4\pi r^2 \). Determine the distance from the bottom of a hemisphere such that the surface area above and below this point is equal, excluding the sliced area.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{r}{2}}
In \(\triangle ABC\), \(AB = 2\), \(BC = \sqrt{3}\), and \(\angle ABC = 150^\circ\). \(P\) is a point on the plane such that \(\angle APB = 45^\circ\) and \(\angle BPC = 120^\circ\). Find \(BP\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{\sqrt{5}}}
Evaluate the improper integral \[ \int_0^{\infty} \frac{2 \sin(x) \cos^2(x)}{x e^{x \sqrt{3}}} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Find the remainder when $40!$ is divided by $5^{10}$. Express your answer as $40! \mod 5^{10}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7812500}
Find the solutions to the complex equation \( \frac{1}{4}\ln(z)=i\pi+8k\pi \), where \( k \) is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{32k\pi}}
Minimize the function \( c = 5x^2 - 80x + y^2 - 32y \) subject to the constraints \( x, y \geq 0 \) and \( x + y \geq 20 \) using the Karush-Kuhn-Tucker (KKT) conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-576}
In a game, a sequence of numbers is generated as follows: the first number is chosen randomly from the interval (0, 1), and each subsequent number is chosen randomly from the interval between 0 and the previous number. The game ends when a number less than a target value \( y \) (where \( 0 < y < 1 \)) is generated. What is the expected length of the sequence in terms of \( y \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \ln y}
In a sequence of $n$ integers, the sum of any three consecutive numbers is positive, and the sum of any five consecutive numbers is negative. Determine the maximum possible value of $n$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given that $x^3[f(x+1)-f(x)]=1$ and $f$ is continuous, determine $\lim_{x\rightarrow \infty}f(x)$. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the cardinality of the set \( \{ (x,y) \in \mathbb{R}^2 \ | \ y > x > 0 , x^x = y^y \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Calculate the area enclosed by the polar curve \( r = 1 + \sin\theta \) and outside the curve \( r = 2 \sin\theta \) using double integrals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
An equilateral triangle \(ABC\) is placed on the floor. Three nails are driven into the floor, one near each side of the triangle, preventing it from rotating without being lifted. The first nail divides side \(AB\) in the ratio \(1:3\) from vertex \(A\), and the second nail divides side \(BC\) in the ratio \(2:1\) from vertex \(B\). Determine the ratio in which the third nail divides side \(AC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:3}
If the minute hand is moved back by 5 minutes, the number of radians it has turned is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\pi}{6}"]}
Compute the homotopy group \( \pi_3(S^2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Determine the order of convergence of the series \( S_N(x) = \sum_{n=1}^{N}\frac{\cos(x n)}{n} \) to its limit \( -\log\left(2\sin\left(\frac{x}{2}\right)\right) \) as \( N \to \infty \). Find a function \( g(N) \) such that:
\[ S_N(x) = -\log\left(2\sin\left(\frac{x}{2}\right)\right) + \mathcal{O}(g(N)), \]
or equivalently,
\[ \left|S_N(x) + \log\left(2\sin\left(\frac{x}{2}\right)\right)\right| \leq C g(N), \]
for some constant \( C > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{N}}
Evaluate the integral \( \int_{0}^{1} \frac{x^{t}-1}{\ln x} \, dx \) for \( \text{Re}(t) > -1 \) without using Feynman's technique.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln(t + 1)}
Given the expressions a = 2ln 1.01, b = ln 1.02, and c = sqrt(1.04) - 1, determine the correct ordering of a, b, and c. Express your answer as a sequence of variables using standard comparison operators (>, <, =).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > c > b}
You have a $9 \times 9$ board with white squares. A tile can be moved from one square to another neighbor (tiles that share one side). If we paint some squares of black, we say that such coloration is good if there is a white square where we can place a chip that moving through white squares can return to the initial square having passed through at least $3$ boxes, without passing the same square twice.
Find the highest possible value of $k$ such that any form of painting $k$ squares of black are a good coloring.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21
}
Given a polynomial $F\in \mathbb{Z}\left [ X_{1},X_{2},X_{3},X_{4},X_{5} \right ]$ defined as $F=(X_{1}+X_{2}+X_{3})^{2}X_{4}X_{5}+X_{1}X_{2}X_{3}(X_{4}+X_{5})$, and a set $A = \{ a_1,a_2,a_3,a_4,a_5 \} \subseteq \mathbb{R}$ with $|A| = 5$, find the maximum number of elements of the set:
$$\{F(a_{\sigma(1)},a_{\sigma(2)},a_{\sigma(3)},a_{\sigma(4)},a_{\sigma(5)}) \} \mid \sigma \in S_5\}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
从 1,2,\cdots,2024 中任取两个数 a,b(a\leq b)，则 3^a+7^b 的值中，个位数字为 8 的数有__________个.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{384307}
There are many two-digit multiples of 7, but only two of the multiples have a digit sum of 10. The sum of these two multiples of 7 is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$119$}
Let \( g(x) = \int_0^x f(y)(y-x)dy \) be a function that is three times continuously differentiable. Determine the greatest integer \( n \) such that the function \( f \) must be \( n \) times continuously differentiable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If there are two points on the parabola \( y = ax^2 - 1 \) that are symmetric with respect to the line \( x + y = 0 \), find the range of values for \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > \frac{3}{4}}
Georgia is working on a math test with 75 problems on it. After 20 minutes, she has completed 10 problems. After another 20 minutes, she has completed twice as many problems. She has some time left to complete the rest of the test and has 45 problems left to solve. How many minutes does she have left to complete the test?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Evaluate the limit using Taylor expansions: $$ \lim_{z \to 0} \frac{(1-\cos z)^2}{(e^z-1-z)\sin^2z} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Let $f:\mathbb{N}\mapsto\mathbb{R}$ be the function\[f(n)=\sum_{k=1}^\infty\dfrac{1}{\operatorname{lcm}(k,n)^2}.\]It is well-known that $f(1)=\tfrac{\pi^2}6$. What is the smallest positive integer $m$ such that $m\cdot f(10)$ is the square of a rational multiple of $\pi$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
42}
In the given pattern of numbers, each row \( n \) contains the number \( n \) multiple times. Determine \( a(n) \), the number of times the number \( n \) appears in the \( n \)-th row of the pattern.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Calculate \(\left\lfloor \sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\ldots+\sqrt[2009]{\frac{2009}{2008}} \right\rfloor\), where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2008}