inclusionAI/Ring-lite-rl-data huggingface dataset:
A dilation centered at $2 - 3i$ with a scale factor of $3$ maps which complex number to the point $1 + 2i$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5 - 4i}{3}}
Let $a = 111\ldots111$ (forty ones) and $b = 111\ldots111$ (twelve ones). Determine the greatest common divisor of $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1111}
Example 8 Given $A \subseteq\{1,2, \cdots, 2000\}$, and the absolute difference between any two numbers in $A$ is not equal to 4 or 7, find the maximum value of $|A|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{910}
在透明的球袋中有 1 个黑球和 2 个白球。接下来从袋中摸球，每次摸出一个球，然后放回袋中，并继续往袋中再放入 2 个与摸出球同色的球. 记 $S_{n}$ 为前 n 次中摸出黑球的次数. 已知第 2 次摸出的球是黑球，记 $S_{100}$ 的期望为 E ，则 $[E]=\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Given that $p$ is a sufficient condition for $q$, $s$ is a sufficient condition for $q$, $r$ is a necessary condition for $q$ and also a sufficient condition for $s$, what is the condition of $p$ for $s$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{sufficient}}
Problem 4. A circle is inscribed in trapezoid $ABCD$, touching the lateral side $AD$ at point $K$. Find the area of the trapezoid if $AK=16, DK=4$ and $CD=6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{432}
Given the function $y = \log_a(x - 1) + 8$ $(a > 0$ and $a \neq 1)$, its graph always passes through the fixed point $P(P, P)$ on the graph of the power function $f(x)$. Find the value of $f(3)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
A line $l$ passes through the right focus of the hyperbola $x^2 - y^2 = 1$ and intersects the hyperbola at points $A$ and $B$. If the real number $\lambda$ makes $|AB| = \lambda$ and there are exactly 3 such lines $l$, then $\lambda = \quad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
给定大于 2004 的正整数 $n$ ，将 $1,2, \cdots$, $n^{2}$ 分别填入 $n \times n$ 棋盘（由 $n$ 行 $n$ 列方格构成）的方格中，使每个方格都有 1 个数。如果 1个方格中填的数大于它所在行至少 2004 个方格内所填的数，且大于它所在列至少 2004 个方格内所填的数，则称这个方格为"优格"。求棋盘中"优格"个数的最大值。请给出 n = 2010 时的结果。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12060}
Let $x, y, z$ be positive real numbers such that $x + y + z = 1399$. Find the maximum value of $\lfloor x \rfloor y + \lfloor y \rfloor z + \lfloor z \rfloor x$, where $\lfloor a \rfloor$ is the greatest integer not exceeding $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{652400}
Given a function $f(x)=a\sin x+b\cos x$ where $x\in \mathbb{R}$ and $\frac{b}{a} \in (1,\sqrt{3}]$. If the function $f(x)$ attains its maximum value at $x=x_{0}$, then the range of $\tan 2x_{0}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\sqrt{3}, +\infty)}
Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\theta=\angle PAB=\angle QPC=\angle RQB=\cdots$ will produce a path length of $120$ meters? (Do not assume the zigzag path has exactly four segments as shown; there could be more or fewer.) If the answer is in the form of \(\arccos\frac{k}{m}\), provide the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
(3) 18 football teams are participating in a round-robin tournament, which means that in each round, the 18 teams are divided into 9 groups, and each group's two teams play one match. In the next round, the teams are regrouped to play, for a total of 17 rounds, so that each team plays one match against each of the other 17 teams. After $n$ rounds of the tournament according to any feasible schedule, there always exist 4 teams that have played a total of only 1 match among themselves. Find the maximum possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
【Question 25】  
Among the positive integers whose digit sum is 50 and are multiples of 11, the second smallest number is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{789899}
On a board, nine quadratic trinomials are written: \(x^{2}+a_{1} x+b_{1}, x^{2}+a_{2} x+b_{2}, \ldots, x^{2}+a_{9} x+b_{9}\). It is known that the sequences \(a_{1}, a_{2}, \ldots, a_{9}\) and \(b_{1}, b_{2}, \ldots, b_{9}\) are arithmetic progressions. It turns out that the sum of all nine trinomials has at least one root. What is the maximum number of the original trinomials that may not have any roots?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the difference between 85% of 250 raised to the power of 2, and 3/5 of 175, when they are both divided by 2.3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19587.5}
During the process of selecting test points using the 0.618 method, if the test interval is $[2, 4]$, then the second test point $x_2$ should be chosen at _____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.764}
The distance of a point light source from a sphere is equal to three times the radius of the sphere. How does the illuminated area of the sphere compare to the lateral surface area of the cone of light?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{5}}
Given that $C$ is an interior angle of $\triangle ABC$, and the vectors $\overrightarrow{m}=(2\cos C-1,-2)$, $\overrightarrow{n}=(\cos C,\cos C+1)$. If $\overrightarrow{m}\perp \overrightarrow{n}$, calculate the value of $\angle C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi}{3}}
A jewelry store is restocking its shelves. The necklace stand, which can hold 12 necklaces, currently holds 5 necklaces. The ring display, which can hold 30 rings, currently holds 18 rings. The bracelet display, which can hold 15 bracelets, currently holds 8 bracelets. The store’s supplier charges $4 per necklace, $10 per ring, and $5 per bracelet. However, the supplier offers the following volume discounts:

For necklaces:
- Buy 4 or more, get a 10% discount
- Buy 6 or more, get a 15% discount

For rings:
- Buy 10 or more, get a 5% discount
- Buy 20 or more, get a 10% discount

For bracelets:
- Buy 7 or more, get an 8% discount
- Buy 10 or more, get a 12% discount

Considering the discounts, how much, in dollars, will the store need to pay to fill the displays?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$170}
Michel starts with the string \( H M M T \). An operation consists of either replacing an occurrence of \( H \) with \( H M \), replacing an occurrence of \( M M \) with \( M O M \), or replacing an occurrence of \( T \) with \( M T \). For example, the two strings that can be reached after one operation are \( H M M M T \) and \( H M O M T \). Compute the number of distinct strings Michel can obtain after exactly 10 operations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Evaluate the triple integral $${\int \int \int_{\Omega } xy^{2}z^{2}dxdydz}$$ where $\Omega$ is the region bounded by the surfaces $z=xy$, $y=x$, $x=1$, and $z=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{198}$}
A point in three-space has distances $2,6,7,8,9$ from five of the vertices of a regular octahedron. What is its distance from the sixth vertex?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{21}}
2. Find the probability that a randomly selected five-digit natural number with non-repeating digits, formed from the digits $1,2,3,4,5,6,7,8$, is divisible by 8 without a remainder. (10 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{8}}
How many perfect squares are between 100 and 500?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
On dividing a number by 68, we get 269 as quotient and some remainder. On dividing the same number by 67, the remainder is 1. What is the remainder in the first division?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Figure $ABCD$ is a trapezoid with $AB||DC$, $AB=5$, $BC=3\sqrt{2}$, $\angle BCD=45^\circ$, and $\angle CDA=60^\circ$. The length of $DC$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$8+\sqrt{3}$}
Let $p = 10009$ be a prime number. Determine the number of ordered pairs of integers $(x, y)$ such that $1 \leq x, y \leq p$ and $x^3 - 3xy + y^3 + 1$ is divisible by $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30024}
In $\triangle ABC$, if angles $A$, $B$, and $C$ form an arithmetic sequence, and sides $a=2$, $c=5$, then the area of $\triangle ABC$ is $S_{\triangle ABC}=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\sqrt{3}}{2}}
Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all the numbers between \(\frac{1}{2}\) and \(\frac{2}{3}\). Armed with this information, what number should Carol choose to maximize her chance of winning? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Kanel-Belov A.Y.

The periods of two sequences are 7 and 13. What is the maximum length of the initial segment that can coincide?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Find the maximum value of the expression for \( a, b > 0 \):

$$
\frac{|4a - 10b| + |2(a - b\sqrt{3}) - 5(a\sqrt{3} + b)|}{\sqrt{a^2 + b^2}}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{87}}
From vertex \( B \) of triangle \( ABC \), a perpendicular \( BM \) is dropped to the bisector of angle \( C \). Let \( K \) be the point of tangency of the incircle with side \( BC \).

Find angle \( \angle MKB \) given that \( \angle BAC = \alpha \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^{\circ} - \frac{\alpha}{2}}
In the rectangular coordinate system xOy, it is known that the line $x + \sqrt{2}y - 2\sqrt{2} = 0$ is tangent to the ellipse C: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), and the right focus F($c$, $0$) of the ellipse C is symmetric about the line $y = \frac{c}{b}x$ with the point E on the ellipse C. Find the area of $\triangle$OEF.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In triangle $ABC$, $AC = 13$, $BC = 14$, and $AB=15$. Points $M$ and $D$ lie on $AC$ with $AM=MC$ and $\angle ABD = \angle DBC$. Points $N$ and $E$ lie on $AB$ with $AN=NB$ and $\angle ACE = \angle ECB$. Let $P$ be the point, other than $A$, of intersection of the circumcircles of $\triangle AMN$ and $\triangle ADE$. Ray $AP$ meets $BC$ at $Q$. The ratio $\frac{BQ}{CQ}$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m-n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$218$}
There are $2$ boys for every $3$ girls in Ms. Johnson's math class.  If there are $30$ students in her class, what percent of them are boys?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40\%}
设集合 S=\{1,2,3,\cdots,997,998\}，集合 S 的 k 个 499 元子集 A_1, A_2, \cdots, A_k 满足：对 S 中任一二元子集 B，均存在 i\in\{1,2,\cdots,k\}，使得 B\subseteq A_i. 求 k 的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $m \ge n$ be positive integers. Frieder is given $mn$ posters of Linus with different integer dimensions of $k \times l$ with $1 \ge k \ge m$ and $1 \ge l \ge n$. He must put them all up one by one on his bedroom wall without rotating them. Every time he puts up a poster, he can either put it on an empty spot on the wall or on a spot where it entirely covers a single visible poster and does not overlap any other visible poster. Determine the minimal area of the wall that will be covered by posters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \cdot \frac{n(n+1)}{2}}
During one game, a certain number of people attended a baseball team's games. Forty percent and thirty-four percent of the audiences are supporters of the first and second teams, respectively. There were 3 people who attended the game and did not support either of the teams. How many people attended the game in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Example 4 Given that $m, n$ are integers greater than 7, consider a rectangular array of $m \times n$ points. Color $k$ of these points red, such that the three vertices of any right-angled triangle with its two legs parallel to the sides of the rectangle are not all red points. Find the maximum value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m+n-2}
Given that point $O$ is the centroid of $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $2a\cdot \overrightarrow{OA}+b\cdot \overrightarrow{OB}+ \frac{2\sqrt{3}}{3}c\cdot \overrightarrow{OC}=\overrightarrow{0}$, determine the measure of angle $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
Given $a, b \in \mathbb{R}$, if the inequality $x\ln x - a\ln x \geqslant x + b$ holds for all $x \gt 0$, then the range of $\frac{b}{a}$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -1]}
The left and right foci of a hyperbola are $F_{1}$ and $F_{2}$, respectively. A line passing through $F_{2}$ intersects the right branch of the hyperbola at points $A$ and $B$. If $\triangle F_{1} A B$ is an equilateral triangle, what is the eccentricity of the hyperbola?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Find the smallest natural number \( n \) for which the number \( A = n^3 + 12n^2 + 15n + 180 \) is divisible by 23.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
We have 1985 sets. Each of the sets has 45 elements, the union of any two sets has exactly 89 elements. How many elements has the union of all these 1985 sets? Please provide the final integer value of the total number of elements in the union.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{87341}
How many unordered triples $A, B, C$ of distinct lattice points in $0 \leq x, y \leq 4$ have the property that $2[ABC]$ is an integer divisible by $5$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
The real number sequence \(\{a_n\}\) satisfies
\[1=a_0 \leqslant a_1 \leqslant a_2 \leqslant \cdots,\]

The sequence \(\{b_n\}\) satisfies
\[b_n = \sum_{k=1}^{n}\left(1-\frac{a_{k-1}}{a_k}\right) \frac{1}{\sqrt{a_k}}, \quad (n=1,2, \cdots).\]

Find the largest real number \(C\) such that for any real number \(x \in [0, C)\), there exists a sequence \(\{a_n\}\) satisfying the condition (1), such that the corresponding sequence \(\{b_n\}\) has infinitely many terms greater than \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A cylinder with a volume of 21 is inscribed in a cone. The plane of the upper base of this cylinder truncates the original cone, forming a frustum with a volume of 91. Find the volume of the original cone.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{94.5}
## Task Condition

Find the derivative.

$y=\arccos \frac{x^{2}-4}{\sqrt{x^{4}+16}}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{2\sqrt{2}(4+x^{2})}{x^{4}+16}}
13.

A box of hawthorns has more than one hundred grains. When counted in groups of 3, there is 1 extra; when counted in groups of 4, there are 2 extra; when counted in groups of 5, there are 3 extra; and when counted in groups of 6, there are 4 extra. The box of hawthorns has at most ( ) grains.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{178}
A line passing through the focus of the parabola $y^2=2px$ ($p > 0$) intersects the parabola at points $P(x_1, y_1)$ and $Q(x_2, y_2)$. If $x_1 + x_2 = 3p$, then the length of $PQ$ is $\boxed{\_\_\_\_\_\_\_\_}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4p}
$已知函数f(x)=x-a(1+\ln x).$
$若函数f(x)有两个零点x_1，x_2，求a的取值范围.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,+\infty)}
Given an ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and its eccentricity as $e$. If there exists a point $P$ on the ellipse such that $\dfrac{|PF_1|}{|PF_2|}=e$, then the range of the eccentricity $e$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\sqrt{2} -1,1)}
Given the line $4x+3y+2m=0$ intersects with the circle $C:\left(x+3\right)^{2}+\left(y-1\right)^{2}=1$, one possible value of the integer $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the circle $\left(x-a\right)^{2}+y^{2}=1\left(a\geqslant 0\right)$ and the circle $x^{2}+\left(y-2\right)^{2}=25$, find the range of the real number $a$ such that the two circles have common points. Express your answer in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2\sqrt{3}, 4\sqrt{2}]}
How many zeros are at the end of the product \( s(1) \cdot s(2) \cdot \ldots \cdot s(100) \), where \( s(n) \) denotes the sum of the digits of the natural number \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Find the arithmetic mean of the reciprocals of the first four prime numbers, including the number 7 instead of 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{493}{1848}}
Let $a, b, c, d, e,$ and $f$ be real numbers. Define the polynomials: 
\[ P(x) = 2x^4 - 26x^3 + ax^2 + bx + c \] 
and 
\[ Q(x) = 5x^4 - 80x^3 + dx^2 + ex + f. \] 
Let $S$ be the set of all complex numbers which are a root of either $P$ or $Q$ (or both). Given that $S = \{1,2,3,4,5\}$, compute $P(6) \cdot Q(6).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2400}
Given an obtuse triangle $ABC$ with an obtuse angle at $C$. Points $P$ and $Q$ are marked on its sides $AB$ and $BC$ respectively such that $\angle ACP = CPQ = 90^\circ$. Find the length of the segment $PQ$, given that $AC = 25$, $CP = 20$, and $\angle APC = \angle A + \angle B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Find the derivative.

\[ y = 2x - \ln \left(1 + \sqrt{1 - e^{4x}}\right) - e^{-2x} \cdot \arcsin \left(e^{2x}\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 e^{-2x} \cdot \arcsin(e^{2x})}
In relation to Smiths Falls, Ontario, the local time in St. John's, Newfoundland, is 90 minutes ahead, and the local time in Whitehorse, Yukon, is 3 hours behind. When the local time in St. John's is 5:36 p.m., what is the local time in Whitehorse? Express your answer in 12-hour format (HH:MM a.m./p.m.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:06 p.m.}
Find the cosine of the angle between adjacent lateral faces of a regular quadrilateral pyramid, where the lateral edge is equal to the side of the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{3}}
A certain company plans to build a container (neglecting thickness, with length unit in meters). The bottom of the container is cylindrical with a height of 1 and a bottom radius of r. The top is a hemisphere with a radius of r. According to the design requirements, the volume of the container is 28/3π cubic meters. Assuming that the construction cost of the container is only related to its surface area, it is known that the construction cost per square meter for the cylindrical part is 30,000 yuan and for the hemispherical part is 40,000 yuan. Calculate the value of the radius r that minimizes the construction cost of the container.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[3]{4}}
A total of 2015 tickets, numbered 1, 2, 3, 4, ..., 2014, 2015, are placed in an empty bag. Alfie removes ticket \( a \) from the bag. Bernice then removes ticket \( b \) from the bag. Finally, Charlie removes ticket \( c \) from the bag. They notice that \( a < b < c \) and \( a + b + c = 2018 \). In how many ways could this happen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{338352}
During the flower exhibition in Zhengzhou Green Expo Garden, 6 volunteers are arranged to provide services in 4 exhibition areas. It is required that areas A and B each have one person, and the remaining two areas each have two people. Among them, Little Li and Little Wang cannot be together. Determine the total number of different arrangements possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{156}
An $8 \times 8 \times 8$ cube is painted red on 3 faces and blue on 3 faces such that no corner is surrounded by three faces of the same color. The cube is then cut into 512 unit cubes. How many of these cubes contain both red and blue paint on at least one of their faces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
In the isosceles triangle \(ABC\) with equal sides \(AB\) and \(CB\), the sides \(AB\) and \(CB\) are extended beyond point \(B\) to points \(D\) and \(E\) respectively. The segments \(AE\), \(ED\), and \(DC\) are equal to each other, and \(\angle BED \neq \angle BDE\). Find the angle \(ABE\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken $-$ when $\underline{a}$ is pressed, the digit $b$ is registered in the system, and when $\underline{b}$ is pressed, the digit $a$ is registered in the system. Find the least number of attempts Andy needs to surely be able to open the lock.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Evaluate the limit:

$$\lim_{k \to \infty} \frac{\int_{0}^{1} x^{2k+1} e^x \, dx}{\int_{0}^{1} x^{2k} e^x \, dx}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999$? For example, both $121$ and $211$ have this property. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{226}