inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the number of pairs of integers $x, y$ with different parities such that $\frac{1}{x}+\frac{1}{y} = \frac{1}{2520}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Determine the cardinality of the set of all functions that can be constructed as iterations of pointwise convergence starting from continuous functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Determine the conditions under which the linear operator $T$ on $\mathbb{R}^4$, represented by the matrix \( Q = \begin{bmatrix} 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 \\ 0 & b & 0 & 0\\0 & 0 & c & 0 \end{bmatrix} \) relative to the standard basis, is diagonalizable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a = b = c = 0}
Let \( x \) be a real number. How many of the first 1000 positive integers can be expressed in the form \( [2x] + [4x] + [6x] + [8x] \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600}
Find the least natural number $n$, such that the following inequality holds:
\[
\sqrt{\frac{n-2011}{2012}} - \sqrt{\frac{n-2012}{2011}} < \sqrt[3]{\frac{n-2013}{2011}} - \sqrt[3]{\frac{n-2011}{2013}}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4024}
Determine the cardinality of the set $M$, where $M$ consists of all subsets of the natural numbers $\mathbb{N}$ such that both the subset and its complement are infinite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
设$S = \{ 1 , 2 , 3 , 4 \} , n$项数列$a _ { 1 } , a _ { 2 } , \cdots , a _ { n }$有以下性质：对于$S$的任何一个非空子集$B$，在该数列中有相邻的$|B|$项恰好组成集合$B$.求$n$的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Consider a sequence of independent and identically distributed random variables drawn from either a uniform or normal distribution. Calculate the probability that any point in the sequence, excluding the first and last points, is a local extremum (either a local maximum or minimum).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Consider a sequence  $\{a_n\}$  of integers, satisfying  $a_1=1, a_2=2$  and  $a_{n+1}$  is the largest prime divisor of  $a_1+a_2+\ldots+a_n$ . Find  $a_{100}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
He then went to see the oranges being harvested. He found out that they harvest 82 sacks of ripe oranges and some sacks of unripe oranges per day. If they have 2050 sacks of ripe oranges after 25 days of harvest, how many sacks of unripe oranges do they harvest per day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
fantasticbobob is proctoring a room for the SiSiEyMB with $841$ seats arranged in $29$ rows and $29$ columns. The contestants sit down, take part 1 of the contest, go outside for a break, and come back to take part 2 of the contest. fantasticbobob sits among the contestants during part 1, also goes outside during the break, but when he returns, he finds that his seat has been taken. Furthermore, each of the $840$ contestants now sit in a chair horizontally or vertically adjacent to their original chair. How many seats could fantasticbobob have started in?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{421}
In a certain school cafeteria, there are only four main dishes available for breakfast: steamed buns, baozi (stuffed buns), noodles, and fried rice with eggs. There are 5 students going to have breakfast, each choosing only one of these dishes, and each type of dish is chosen by at least one student. It is known that there are not enough baozi for more than one person to eat, and Student A, due to stomach issues, will not choose fried rice with eggs. The number of different meal choice combinations for these 5 students is ______. (Answer with a number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
Given non-negative real numbers \( a, b, c, x, y, z \) that satisfy \( a+b+c = x+y+z = 1 \), find the maximum value of \( \left(a-x^{2}\right)\left(b-y^{2}\right)\left(c-z^{2}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{16}}
Evaluate the integral:

$$\int_0^\infty \frac{\tan^{-1}\left(\frac{x^2-x^{1/2}}{1+x^{5/2}}\right)}{1+x^2}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the global maximum value of the function \( f(x) = \frac{J_2(x)}{x^2} \), where \( J_2(x) \) is a Bessel function of the first kind of order 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8}}
Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\angle AEB, \angle BFC, \angle CGD, \angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98
}
Given three 3D vectors $W$, $T_1$, and $T_2$ that lie on the same plane, and the equation $W = T_1 \cos^2(\theta) + T_2 \sin^2(\theta)$, find the angle between $W$ and $T_1$. Express your answer in terms of $\theta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arctan(\tan^2 \theta)}
Find a continuous and absolutely integrable function \( f(x) \) that satisfies the integral equation \( \int_{-\infty}^{\infty} e^{-|t-x|}f(x) \, dx = e^{-|t|} + |t|e^{-|t|} \) using the Convolution Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-|x|}}
What is the smallest number which when diminished by 24, is divisible by 5, 10, 15 and a certain number and the result is 84?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2544}
A snowball with a temperature of $0^{\circ} \mathrm{C}$ is launched at a speed $v$ towards a wall. Upon impact, $k=0.02\%$ of the entire snowball melts. Determine what percentage of the snowball will melt if it is launched towards the wall at a speed of $\frac{v}{2}$? The specific heat of fusion of snow is $\lambda = 330$ kJ/kg. Assume that all the energy released upon impact is used for melting.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.005\%}
The sum of several (not necessarily different) real numbers from $[0,1]$ doesn't exceed $S$.
Find the maximum value of $S$ such that it is always possible to partition these numbers into two groups with sums $A\le 8$ and $B\le 4$.

[i](I. Gorodnin)[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11.2}
If the graph of the function $f(x) = \sin \omega x$ is shifted to the right by $\frac{\pi}{6}$ units to obtain the graph of $f(x) = \sin(\omega x - \frac{4}{3}\pi)$, then the minimum value of $|\omega|$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the value of $\lfloor 1 \rfloor + \lfloor 1.7 \rfloor + \lfloor 2.4 \rfloor + \lfloor 3.1 \rfloor + \cdots + \lfloor 99 \rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6987}
A dealer purchases 15 articles of type A for Rs. 25 each and 20 articles of type B for Rs. 35 each. He sells 12 articles of type A for Rs. 33 each and 18 articles of type B for Rs. 45 each. Find the overall profit or loss percentage he made from these transactions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.19\%}
Given points D and E on sides AB and AC of ∠CAB respectively, the angle bisectors of ∠BDE and ∠CED intersect at F. If ∠BAC = 38°, find the measure of ∠DFE. Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71^\circ}
A certain middle school assigns numbers to each student, where the last digit indicates the gender of the student: 1 for male and 2 for female. If 028432 represents "a female student who is number 43 in class 8 and enrolled in the year 2002," then the number for a male student who is number 23 in class 6 and enrolled in the year 2008 is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{086231}
Determine the values of $m$ for which the integral \( \int_0^{\pi/2} \frac{1-\cos(x)}{x^m} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 3)}
If \(\cos^5 \theta - \sin^5 \theta < 7 (\sin^3 \theta - \cos^3 \theta)\), \(\theta \in [0, 2\pi)\), then what is the range of values for \(\theta\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{\pi}{4}, \frac{5 \pi}{4}\right)}
Evaluate the surface integral $\iint_S f(x,y,z)dS$ where $f(x,y,z) = x^2+y^2 + z^2 $ and $S$ is the part of the cone $z = \sqrt{x^2+y^2} $ above the circle $x^2+y^2 \leq 2x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\pi\sqrt2}
Consider a finite group $S$ with identity element $1$. Let $G$ and $H$ be subgroups of $S$ with orders $m$ and $n$, respectively. If the least common multiple of $m$ and $n$ is $k$, determine the maximum possible order of $G \cap H$. \( \text{Express your answer as a number.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{mn}{k}}
What is the range of values for \( m \) such that the equation \( m\left(x^{2}+y^{2}+2 y+1\right) = (x - 2y + 3)^{2} \) represents an ellipse?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(5,+\infty)}
A train is 800 meters long and is running at a speed of 115 km/hour. In what time will it pass a bridge of 375 meters length?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36.78}
Four princesses each guessed a two-digit number, and Ivan guessed a four-digit number. After they wrote their numbers in a row in some order, they got the sequence 132040530321. Find Ivan's number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5303}
In a setup with 5 blue nails in one column and 3 red nails in another column, you can attach a string between any red nail and any blue nail. Determine the number of ways to attach strings such that every nail has at least one string attached to it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16081}
Find the highest lower bound for the probability of getting exactly one head when tossing $d$ pairwise independent coins, each with a probability of $1/d$ of landing heads.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{d}}
Calculate the center of gravity for a solid cone with a circular base of radius 1 and height 3, where the top of the cone is located at the origin and the base is in the xy-plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 0, 3/4)}
Calculate the expected number of coin tosses needed to obtain the sequence H-T-H-T-T-H with a fair coin, where each toss is independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Find the polar coordinates (ρ, θ) of the center of the circle given by the polar equation ρ = -2cosθ. Express your answer in the format (ρ, θ), where θ is in radians.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, π)}
Determine the maximum value of \( \sum_{i=1}^{k} a_{i} \) for integers \( a_{1}, a_{2}, \ldots, a_{k} \) greater than 1, such that \( (a_{1}!) (a_{2}!) \cdots (a_{k}!) \mid 2017! \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5024}
In the "Three Kingdoms Kill" game, there is 1 'Monarch', 3 'Loyalists', 5 'Rebels', and 1 'Traitor', with a total of 10 people playing the game. The Monarch and the Loyalists are on the same team, the Rebels are on another team, and the Traitor is on his own team. If two brothers participate in the same round of the "Three Kingdoms Kill" game, what is the probability that the two brothers are on the same team? (Express the result as a simplified fraction)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{16}{45}}
Compute the integral \(\displaystyle \int_{\gamma} \frac{z-1}{z(z-i)(z-3i)}dz\) where \(\gamma\) is the curve defined by \(|z-i| = \frac{1}{2}\). Decompose the integrand into partial fractions to aid in the computation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi(1 + i)}
A dart is thrown at a square dartboard of side length 2 so that it hits completely randomly. What is the probability that it hits closer to the center than any corner, but within a distance of 1 of a corner?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi - 2}{4}}
Consider numbers of the form  $1a1$ , where  $a$  is a digit. How many pairs of such numbers are there such that their sum is also a palindrome?

*Note: A palindrome is a number which reads the same from left to right and from right to left. Examples:  $353$ ,  $91719$ .*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Determine the cardinality of the set of all convergent sequences in \(\mathbb{Q}\) that approach 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Determine the relationship among the three numbers $2011^{0.99}$, $0.99^{2011}$, and $\log_{0.99}2011$. Express your answer as an ordering of the three functions in the format $a < b < c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log_{0.99}2011 < 0.99^{2011} < 2011^{0.99}}
In triangle $ABC$, points $D$ and $E$ lie on the interior of segments $AB$ and $AC$, respectively, such that $AD = 1$, $DB = 2$, $BC = 4$, $CE = 2$, and $EA = 3$. Let $DE$ intersect $BC$ at $F$. Determine the length of $CF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a game played by two players, A and B, they take turns filling six boxes with digits 1-9, forming a six-digit number. If this six-digit number is divisible by an integer N, B wins; otherwise, A wins. Find the value of N less than 15 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}
Determine the asymptotic behavior, in terms of the length L, for the number of topologically different knots that can be formed using a perfectly flexible, non-self-intersecting rope of length L and radius 1, with the ends glued together after knotting.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\Theta(L)}}
In a $9 \times 9$ grid, 29 squares are colored black. If $m$ represents the number of rows that contain at least 5 black squares, and $n$ represents the number of columns that contain at least 5 black squares, determine the maximum value of $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In the polynomial quotient ring \(B = \mathbb{Z}_2[x]/(x^3+x^2+x+1)\), find a polynomial \([g(x)]\) such that \([x^2+1] \cdot [g(x)] = [0]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[x + 1]}
27.7. (New York, 76). The sequence ( $i_{1} ; i_{2} ; \ldots ; i_{n}$ ) is formed from the first $n>3$ natural numbers, arranged in a random order. What is the probability that for all $k=1,2, \ldots, n$ the inequality $i_{k} \geqslant k-3$ holds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4^{n-3}\cdot3!}{n!}}
Evaluate the limit: $$\lim_{x\to0}\frac{\int_{0}^{x^2} t^{t-1} dt}{x^4}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given the polynomial $f(x) = x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64$, calculate the value of $v_4$ when $x = 2$ using Horner's method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Determine the values of $a \in \mathbb{R}$ for which the function $f(x) = x^{100} - ax^{97} + 5$ has multiple local minimum values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
On a two-dimensional coordinate plane, the line a = x^2 - x^3 touches the x-axis in 2 places. What is the first term of the equation representing the line?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-x^3}
How many permutations \( \pi \) of \( \{ 1,2,\ldots ,7\} \) are there such that \( \pi \left( k\right)  \leq  {2k} \) for \( k = 1,\ldots ,7 \) ? A permutation \( \pi \) of a set \( S \) is a
 function from \( S \) to itself such that if \( a \neq  b \) ,then \( \pi \left( a\right)  \neq  \pi \left( b\right) \) . For example, \( \pi \left( x\right)  = x \) and \( \pi \left( x\right)  = 8 - x \) are permutations
 of \( \{ 1,2,\ldots ,7\} \) but \( \pi \left( x\right)  = 1 \) is not.
 2019 CCA Math Bonanza Individual Round #7
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
576
}
Given $\sin \alpha + \cos \alpha = - \frac{\sqrt{5}}{2}$, and $\frac{5\pi}{4} < \alpha < \frac{3\pi}{2}$, calculate the value of $\cos \alpha - \sin \alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{2}}
Sharik and Matroskin ski on a circular track, half of which is an uphill slope and the other half is a downhill slope. Their speeds are identical on the uphill slope and are four times less than their speeds on the downhill slope. The minimum distance Sharik falls behind Matroskin is 4 km, and the maximum distance is 13 km. Find the length of the track.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24 \text{ km}}
Given a triangular prism \( S-ABC \) with a base that is an equilateral triangle. The projection of point \( A \) on the side \( S-BC \), denoted as \( H \), is the orthocenter of \( \triangle SBC \). The dihedral angle between planes \( H-AB \) and \( BC \) is \( 30^\circ \). Additionally, \( SA = 2\sqrt{3} \). What is the volume of the triangular prism \( S-ABC \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{4} \sqrt{3}}
Suppose you and I each have a box of 600 marbles. In my box, 4 of the marbles are black, while 3 of your marbles are black. We each draw 300 marbles with replacement from our own boxes. Using Poisson approximation, calculate the probability that you and I draw the same number of black marbles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.216}
3. The mathematical committee consists of $2 n$ members, $n \geqslant 3$. It is known that each member of the committee is in a dispute with exactly one other member of the committee (this relationship is symmetric). In how many ways can the committee be divided into three subcommittees: one for creating problems, one for evaluating problems, and one for organizing the competition, such that each subcommittee has at least two members and no two members of the committee who are in a dispute are in the same subcommittee?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6^{n}-3\cdot2^{n}(2n+1)}
Let \( f(x) = \frac{1}{(1+x)(1+x^2)(1+x^4)} \). Find the coefficient of the term \( x^{50} \) in \( (f(x))^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
For any positive integer  $x$ , let  $f(x)=x^x$ . Suppose that  $n$  is a positive integer such that there exists a positive integer  $m$  with  $m  \neq 1$  such that  $f(f(f(m)))=m^{m^{n+2020}}$ . Compute the smallest possible value of  $n$ .

*Proposed by Luke Robitaille*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13611}
A driver departed from point A to point B, which are 100 km apart. When the navigator indicated that 30 minutes of travel remained, the driver reduced speed by 10 km/h for the first time. When the navigator indicated that 20 minutes of travel remained, the driver reduced speed by another 10 km/h. (The navigator determines the remaining time and distance based on the current speed.) Determine the initial speed of the car, given that the car traveled 5 minutes longer at the reduced speed than it would have at the twice-reduced speed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100 \text{ km/h}}
Let $E$ be a point inside the rhombus $ABCD$ such that $|AE| = |EB|$, $m(\widehat{EAB}) = 12^\circ$, and $m(\widehat{DAE}) = 72^\circ$. What is the value of $m(\widehat{CDE})$ in degrees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is monotonically bounded if $1 \leq a_i \leq i$ for all $1 \leq i \leq 2^{16}$. We say that a term $a_k$ in the sequence with $2 \leq k \leq 2^{16}-1$ is a mountain if $a_k$ is greater than both $a_{k-1}$ and $a_{k+1}$. Evan writes out all possible monotonically bounded sequences. Let $N$ be the total number of mountain terms over all such sequences he writes. Find the remainder when $N$ is divided by $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49153}
An object travels 90 feet in 3 seconds. What is the object's approximate speed in miles per hour if 1 mile is equal to some number of feet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20.45}
Evaluate the surface integral \( \iint_S \nabla \times \vec{v} \cdot \vec{N} \, dS \) where \( \vec{v} = (xz, -y, x^2y) \) and \( S \) consists of the five faces of the cube \([0,2] \times [0,2] \times [0,2]\) that are not on the xy-plane. The vector \( \vec{N} \) points outward from the surface. Use Stokes' theorem to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find $A^2_{}$, where $A^{}_{}$ is the sum of the absolute values of all roots of the following equation: $x = \sqrt{19} + \frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{{\sqrt{19}+\frac{91}{x}}}}}}}}}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$383$}
In triangle $ABC$, we have $AB=36$, $BC=48$, and $CA=60$. The incircle of $\triangle ABC$ is centered at $I$ and touches $AB$, $AC$, and $BC$ at points $M$, $N$, and $D$, respectively. Ray $AI$ meets $BC$ at $K$. The radical axis of the circumcircles of triangles $\triangle MAN$ and $\triangle KID$ intersects lines $AB$ and $AC$ at points $L_1$ and $L_2$, respectively. If $L_1L_2 = x$, compute $x^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{720}
Find the number of prime factors in the factorization of $12!$, where $12! = 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
已知数列 $\left\{x_{n}\right\}$ 满足:
$x_{1}=a, x_{2}=b, x_{n+2}=3 x_{n+1}+2 x_{n}$.
若存在整数 $k \geqslant 3$, 使得 $x_{k}=2019$, 则这样的有序正整数对 $(a, b)$ 的个数为 $\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{370}
In an $11 \times 11$ square, the central cell is colored black. Maxim found the largest rectangular grid area that is entirely within the square and does not contain the black cell. How many cells are in that rectangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}