inclusionAI/Ring-lite-rl-data huggingface dataset:
Identify the Banach space that is isometrically isomorphic to the quotient space $c/Y$, where $c$ is the space of convergent sequences and $Y$ is the subspace of sequences converging to zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Two $4 \times 4$ squares are randomly placed on an $8 \times 8$ chessboard so that their sides lie along the grid lines of the board. What is the probability that the two squares overlap?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{529/625}
a and b are two multiples of a certain number, and q is the set of consecutive integers between a and b, inclusive. q contains 11 multiples of that number. There are 21 multiples of 7 in q. What is the number that a and b are multiples of?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In one of the hotels on the wellness planet Oxys, there are $2019$ saunas. The managers have decided to accommodate $k$ couples for the upcoming long weekend. The following information is known about the guests:

1. If two women know each other, then their husbands also know each other, and vice versa.
2. Each sauna can be used by either men only or women only (there is no limit on the number of people using a sauna at once, as long as they are of a single gender).
3. Each woman is only willing to share a sauna with women whom she knows.
4. Each man is only willing to share a sauna with men whom he does not know.

What is the greatest possible $k$ for which we can guarantee, without knowing the exact relationships between the couples, that all the guests can use the saunas simultaneously while respecting the restrictions above?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2018}
How many sequences of real numbers $a_1, a_2, \ldots, a_9$ satisfy 
\[ |a_1 - 1| = |a_2 - a_1| = \cdots = |a_9 - a_8| = |1 - a_9| = 1? \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
Example 1 Given 20 numbers $1,2, \cdots, 20$, two people, A and B, take turns placing a “+” or “-” sign in front of these numbers (the order of placement is not limited). After 20 steps, the absolute value $S$ of the resulting algebraic sum is calculated. A wants to make $S$ as small as possible, while B wants to make $S$ as large as possible. What is the largest $S$ that B can guarantee?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
给定整数 $n \geqslant 3$ 。求最小的正整数 $k$, 使得存在一个 $k$ 元集合 $A$ 和 $n$ 个两两不同的实数 $x_{1}, x_{2}, \cdots, x_{n}$, 满足 $x_{1}+x_{2}, x_{2}+x_{3}, \cdots$, $x_{n-1}+x_{n}, x_{n}+x_{1}$ 均属于 $A$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the integral \( \int_T f \) over the region \( T \) in \( \mathbb{R}^3 \) with vertices \((1,0,1), (0,2,1), (2,1,0), (1,-1,2)\), where \( f(x,y,z) = y - 2x + z \). Use a linear transformation to simplify the bounds of integration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{6}}
Let ${f'}\left(x\right)$ be the derivative of the function $f\left(x\right)$, and ${f'}\left(x\right) \lt f\left(x\right)\left(x\in R\right)$, $f\left(1\right)=e$, then the solution set of the inequality $f\left(\ln x\right) \gt x$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,e)}
A cup is filled with a saline solution of $15\%$ concentration. There are three iron balls of different sizes: large, medium, and small, with their volumes in the ratio 10:5:3. Initially, the small ball is submerged in the cup of saline causing $10\%$ of the saline solution to overflow. Then the small ball is removed. Next, the medium ball is submerged in the cup and then removed. This is followed by submerging and removing the large ball. Finally, pure water is added to the cup until it is full again. What is the concentration of the saline solution in the cup at this point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10\%}
Compute the largest squarefree positive integer \( n \) such that \( \mathbb{Q}(\sqrt{-n})\cap \overline{\mathbb{Z}} \) is a unique factorization domain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{163}
In scalene $\triangle ABC$, $I$ is the incenter, $I_a$ is the $A$-excenter, $D$ is the midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$, and $M$ is the midpoint of side $BC$. Extend ray $IM$ past $M$ to point $P$ such that $IM = MP$. Let $Q$ be the intersection of $DP$ and $MI_a$, and $R$ be the point on the line $MI_a$ such that $AR\parallel DP$. Given that $\frac{AI_a}{AI}=9$, the ratio $\frac{QM} {RI_a}$ can be expressed in the form $\frac{m}{n}$ for two relatively prime positive integers $m,n$. Compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
11
}
Evaluate the integral \( \int_{\gamma} f(z) \, dz \), where \( f(z) = \frac{1}{[(z-\frac{1}{2}-i)(z-1-\frac{3i}{2})(z-1-\frac{i}{2})(z-\frac{3}{2}-i)]} \) and \( \gamma \) is the polygon with vertices \( 0, 2, 2+2i, 2i, 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $\mathbf{F}$ be a vector field defined on $\mathbb{R}^2 \setminus\{(0,0)\}$ by $$\mathbf{F}(x,y) = \frac{y}{x^2+y^2}i - \frac{x}{x^2+y^2}j.$$ Let $\gamma, \alpha: [0,1] \to \mathbb{R}^2$ be defined by $$\gamma(t) = (8\cos 2\pi t, 17\sin 2\pi t)$$ and $$\alpha(t) = (26\cos 2\pi t, -10\sin 2\pi t).$$ If $$3\int_{\alpha} \mathbf{F \cdot dr} - 4 \int_{\gamma} \mathbf{F \cdot dr} = 2m\pi,$$ find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
8. Let \( A B C \) be a triangle with sides \( A B = 6 \), \( B C = 10 \), and \( C A = 8 \). Let \( M \) and \( N \) be the midpoints of \( B A \) and \( B C \), respectively. Choose the point \( Y \) on ray \( C M \) so that the circumcircle of triangle \( A M Y \) is tangent to \( A N \). Find the area of triangle \( N A Y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{600}{73}}
Let $p$ and $q$ be prime numbers such that $(p-1)^{q-1}-1$ is a positive integer that divides $(2q)^{2p}-1$. Compute the sum of all possible values of $pq$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
Determine the minimum number of real roots of the equation \((x^2+px+3q)(x^2+rx+q)(-x^2+sx+2q)=0\) for real numbers \(p, q, r, s\), without expanding the expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What percentage error do we make if we approximate the side of a regular heptagon by taking half of the chord corresponding to the $120^\circ$ central angle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2\%}
In a round-robin chess tournament with 12 players, each player creates 12 lists. The first list contains only themselves, and the $(k+1)$-th list contains the players from the $k$-th list and those who were defeated by any of the players in the $k$-th list. It is known that the 12th list of each player is different from their 11th list. Determine the number of draws that occurred in the tournament.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Let  $x, y,z$  satisfy the following inequalities  $\begin{cases} | x + 2y - 3z| \le 6 
| x - 2y + 3z|  \le  6  
| x - 2y - 3z|  \le 6 
| x + 2y + 3z|  \le  6 \end{cases}$ Determine the greatest value of  $M = |x| + |y| + |z|$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
On a two-dimensional coordinate plane, the line a = x^2 - x^3 touches the x-axis in 2 places. What is the second term of the equation representing the line?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{"-x^3"}
已知平面直角坐标系中的点集L=\{(x,y)\ |\ x,y\in\{1,2,\cdots,100\}\}.设 A 是由平面上若干个凸多边形组成的集合，满足：A 中每个凸多边形的所有顶点均属于 L，并且 L 中的每个点都恰好是 A 中一个凸多边形的顶点.求 A 中所有凸多边形的面积之和的最小可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1667}
Find the smallest radius \( r \) such that there is at least one local maximum point of the function \( f \) inside or on the boundary of a circle of radius \( r \) centered at \( (x_0, f(x_0)) \), given the conditions \( f(x_0) \ge f(x_0+h) \) and \( f(x_0) \ge f(x_0-h) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{h}
Determine the minimum number of colors required to paint all the cells of a 6 by 6 square grid such that each row, column, and diagonal of the square has all cells with different colors. A diagonal is defined as all rows of at least two cells going diagonally from one edge of the square to the other at an angle of $45^\circ$ or $135^\circ$ to the horizontal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Calculate the probability \( \mathbb{P}(Z_1 > 0, Z_1 + Z_2 < 0) \) where \( Z_1 \) and \( Z_2 \) are independent standard normal random variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8}}
A bicycle is traveling along a path with a 5-degree incline. The front wheel has a perimeter of 30 cm, and the back wheel has a perimeter of 20 cm. The surface is such that the front wheel experiences a slip ratio of 5%, while the back wheel experiences a slip ratio of 7%. If the front wheel revolves 240 times, how many revolutions will the back wheel take to cover the same distance along the incline?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{368}
Determine the number of points within the interval \([0, 6\pi]\) where the function \( f(x) \) is continuous, given that \( f(x) = \tan x \) for rational \( x \) and \( f(x) = x^2 + 1 \) for irrational \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Natural numbers \( x_{1}, x_{2}, \ldots, x_{13} \) are such that \( \frac{1}{x_{1}} + \frac{1}{x_{2}} + \ldots + \frac{1}{x_{13}} = 2 \). What is the minimum value of the sum of these numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
In a plane, point C is located 300 meters due east of point B. A tower, not on the line connecting B and C, is observed from points B and C with vertical angles of 45 degrees and 60 degrees, respectively. From point D, which is 500 meters west of B, the tower is observed with a vertical angle of 30 degrees. Calculate the height of the tower.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300\sqrt{5}}
Select 4 students to participate in a relay race from a group of 8 students, which includes 6 boys and 2 girls, using stratified sampling by gender. If girls are not to be the last runner, find the number of different arrangements. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{720}
A mixture of $30$ liters of paint is $25\%$ red tint, $30\%$ yellow tint and $45\%$ water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Let $\gamma : [0,1] \to \mathbb{C}$ be the contour given by $\gamma(t) = t + it$. Suppose $f : \mathbb{C} \to \mathbb{C}$ is a complex differentiable function which satisfies $|f(z)| \le 3$ for all $z$ such that $|z| \le 100$. Determine the maximum possible value of \(\int_\gamma f(z)\,dz\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2}}
At a ball, there are \(n\) married couples. In each couple, the husband and wife are of the exact same height, but no two couples share the same height. When the waltz begins, all attendees randomly pair up: each man dances with a randomly chosen woman. Find the expected value of the random variable \(X\), which represents the number of men who are shorter than their partners.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n-1}{2}}
Let  $ d_1,d_2 \ldots, d_r$  be the positive divisors of  $ n$  $ 1\equal{}d_1<d_2< \ldots <d_r\equal{}n$ 
If  $ (d_7)^2 \plus{} (d_{15})^2\equal{} (d_{16})^2$  find all posible values of  $ d_{17}$ 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ d_{17} = 28 }
How many positive integers less than or equal to $150$ have exactly three distinct prime factors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Evaluate the limit \( \lim_{n \to \infty} n^2 \sum_{k=1}^n \frac{1}{(n^2+k^2)^2} \) using the concept of Riemann sums.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine all positive real numbers \(a\) for which the series \( \sum_{A \in \mathcal{F}} \frac{1}{\sum_{k \in A} a^{k}} \) is convergent, where \(\mathcal{F}\) is the family of all nonempty finite subsets of \(\mathbb{N} \cup \{0\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 2}
In how many ways can you select an even number of green shirts from 50 drawers, each containing a blue shirt, a red shirt, a green shirt, and a yellow shirt, considering that the order of non-green shirts matters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{99} + 2^{49}}
As shown in the diagram, the houses of Xiaohong, Xiaolan, and Xiaolu are located along a circular road where one can only travel in a counterclockwise direction. It takes Xiaohong 30 meters to walk to Xiaolan's house, Xiaolu 26 meters to walk to Xiaohong's house, and Xiaolan 28 meters to walk to Xiaolu's house. Calculate the distance it takes Xiaolan to walk to Xiaohong's house in the reverse direction. Express your answer in meters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If the area $S= \frac {1}{4}(b^{2}+c^{2}-a^{2})$, then $\angle A=$ _____ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {\pi}{4}}
The  $\emph{Stooge sort}$  is a particularly inefficient recursive sorting algorithm defined as follows: given an array  $A$  of size  $n$ , we swap the first and last elements if they are out of order; we then (if  $n\ge3$ ) Stooge sort the first  $\lceil\tfrac{2n}3\rceil$  elements, then the last  $\lceil\tfrac{2n}3\rceil$ , then the first  $\lceil\tfrac{2n}3\rceil$  elements again. Given that this runs in  $O(n^\alpha)$ , where  $\alpha$  is minimal, find the value of  $(243/32)^\alpha$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
In how many ways can you divide the set of eight numbers \( \{ 2,3,\cdots ,9\} \) into 4 pairs such that no pair of numbers has \( \gcd \)
 equal to 2 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36
}
Given a matrix $A \in \mathbb{R}^{m \times n}$ with full rank and $m \le n$, solve the optimization problem:
\[ \mathop{\arg\max}\limits_{\substack{v \in \mathbb{R}^m \\ \| v \| = 1}} v^T A A^T v \]
using the singular value decomposition (SVD) of $A$. Determine the vector $v$ that maximizes the expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{u_1}
Evaluate the expression $a/b$, where $a = \sum_{r=1}^8 \tan^2\left(\frac{r\pi}{17}\right)$ and $b = \prod_{r=1}^8 \tan^2\left(\frac{r\pi}{17}\right)$. Determine the value of $a/b$ without using graphing calculators.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Agent 1 and Agent 2 each have two tasks to complete. The time it takes Agent 1 to complete Task 1 and Task 2 follows an exponential distribution with means of 6 and 4, respectively. Similarly, the time it takes Agent 2 to complete Task 1 and Task 2 follows an exponential distribution with means of 3 and 1, respectively. Task 2 for each agent can only be started after Task 1 is completed. Assuming all task completion times are independent, calculate the probability that Agent 1 completes Task 2 before Agent 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{5}}
In a conference of 50 participants, each person shakes hands with everyone else, except for the committee members and the participants who have a red badge. The committee consists of 10 participants, and there are 5 participants with red badges. Determine the maximum number of handshakes that can occur without any circular handshakes involving the committee members or sequences that involve participants with red badges shaking hands with each other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{595}
Let the minimal polynomial of a matrix \( A \) be \( f(x)=(x-c_1)^{n_1}\cdots(x-c_k)^{n_k} \). Determine the minimal polynomial of the matrix \( \begin{pmatrix}A & 2I \\0 & A \\\end{pmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x - c_1)^{n_1 + 1} \cdots (x - c_k)^{n_k + 1}}
Let $g: H \to D(0,1)$ be a holomorphic function, where $H$ is the right half-plane minus the y-axis. Given that $g(1) = 0$ and $|g(z)| < 1$ for all $z \in H$, determine an upper bound for $|g'(1)|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
If a random variable X follows a t-distribution with n degrees of freedom, what is the probability distribution of the random variable X^2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F(1, n)}
有长为 $2^{n}(n=0,1, \cdots, 1009)$ 的线段各三条，则由这 3030 条线段能构成不全等的三角形的个数为 （用数字作答）。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{510555}
Let $G = \mathbb{Z} \oplus \mathbb{Z}$ and $H = \langle (2,2) \rangle$. Determine the order of the factor group $G/H$. \( \text{Note: } H = \{ (x,y) \mid x,y \in 2\mathbb{Z} \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Simplify the expression \( \frac{( b \times (a \times b)) \cdot (b \times (a\times b))}{(a\times b) \cdot (a\times b)} \), where \(a\) and \(b\) are 3-dimensional vectors, and express it without a denominator.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b \cdot b}
The championship finals of the Chinese Volleyball Super League adopts a seven-game four-win system, which means that if one team wins four games first, that team will be the overall champion, and the competition will end. Now team A and team B are in the finals. Since the two teams have similar strengths, the probability of each team winning each game is $\frac{1}{2}$. According to past data, the ticket revenue for the first game is 5 million yuan, and the ticket revenue for each subsequent game increases by 100,000 yuan compared to the previous game. What is the probability that the total ticket revenue for the finals is exactly 45 million yuan?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{16}}
The angle between the line $x+ \sqrt {3}y+2=0$ and the line $x+1=0$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
In triangle $ABC$, where $\angle A = 100^\circ$ and $\angle B = \angle C = 40^\circ$, the side $AB$ is extended to a point $D$ such that $AD = BC$. Determine the measure of $\angle BCD$ without using trigonometry.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10^\circ}
A train passes a platform in 25 seconds. The same train passes a man standing on the platform in 20 seconds. The speed of the train is some km/hr. The length of the platform is 75.006 meters. What is the speed of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54.00432}
Evaluate the contour integral \( \int_\gamma \frac{\mathrm{Im} (z)}{z - \alpha} dz \), where \( \alpha \) is a complex number such that \( 0 < |\alpha| < 2 \) and \( \gamma \) is the circle oriented in the positive sense, centered at the origin with radius 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \alpha}
Let $p = 101$ and let $S$ be the set of $p$-tuples $(a_1, a_2, \dots, a_p) \in \mathbb{Z}^p$ of integers. Let $N$ denote the number of functions $f: S \to \{0, 1, \dots, p-1\}$ such that:
- $f(a + b) + f(a - b) \equiv 2\big(f(a) + f(b)\big) \pmod{p}$ for all $a, b \in S$.
- $f(a) = f(b)$ whenever all components of $a-b$ are divisible by $p$.

Compute the number of positive integer divisors of $N$. (Here addition and subtraction in $\mathbb{Z}^p$ are done component-wise.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5152}
Let  $MATH$  be a square with  $MA = 1$ . Point  $B$  lies on  $AT$  such that  $\angle MBT = 3.5 \angle BMT$ . What is the area of  $\vartriangle BMT$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}-1}{2}}
Given that \(x\) satisfies \(\log _{5x} (2x) = \log _{625x} (8x)\), find the value of \(\log _{2} x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\ln 5}{2 \ln 2 - 3 \ln 5}}
Given the Hilbert space \( l^2 = \{ (x_1, x_2, x_3, \dots) : x_n \in \mathbb{R}, \, \sum_{n=1}^{\infty} x_n^2 < \infty \} \) with the norm \( \lvert \lvert (x_1, x_2, x_3, \dots) \rvert \rvert_2 = \left( \sum_{n=1}^{\infty} x_n^2 \right)^{\frac{1}{2}} \), consider the subspace \( M \subset l^2 \) defined by \( \sum_{n=1}^{\infty} \frac{x_n}{4^n} = 0 \). The orthogonal complement of \( M \) is denoted as \( M^{\perp} \). Given the vector \( v = \left(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \right) \in l^2 \), its orthogonal projection onto \( M^{\perp} \) is \( \alpha \left(\sum_{n=1}^{\infty} \frac{1}{n4^n}\right) \left(\frac{1}{4}, \frac{1}{4^2}, \frac{1}{4^3}, \dots \right) \) for some \( \alpha \in \mathbb{R} \). Find the value of \( \alpha \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Using $600$ cards, where $200$ of them have the number $5$, $200$ have the number $2$, and the other $200$ have the number $1$, a student wants to create groups of cards such that the sum of the card numbers in each group is $9$. What is the maximum number of groups that the student can create?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Find the value of the constant \( k \) such that the cubic equation \( x^3 - 12x + k = 0 \) has distinct roots in the interval \([0, 2]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
For each positive integer $n$, let $f(n) = \sum_{k = 1}^{100} \lfloor \log_{10} (kn) \rfloor$. Find the largest value of $n$ for which $f(n) \le 300$. <b>Note:</b> $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$109$}
Inside a grid rectangle with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut out along the cell borders (the hole does not contain boundary cells). If you cut this figure along all the horizontal grid lines, you get 20 strips that are 1 cell wide. How many strips will you get if you cut it along all the vertical grid lines instead? (A $1 \times 1$ square is also considered a strip!)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}