inclusionAI/Ring-lite-rl-data huggingface dataset:
On a plane, there is an invisible rabbit hiding on a lattice point. We want to place $n$ hunters on some lattice points to catch the rabbit. In each turn, the following steps occur in order:

1. The rabbit jumps to an adjacent lattice point in the top, bottom, left, or right direction.
2. Each hunter moves to an adjacent lattice point in the top, bottom, left, or right direction (each hunter can move in a different direction). Then, they shoot a bullet which travels $\frac{334111214}{334111213}i$ units in the direction they chose.

If a bullet hits the rabbit, then it is caught. Find the smallest number $n$ such that the rabbit would definitely be caught in a finite number of turns.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
As a special treat, Georgia makes muffins and brings them to her students on the first day of every month. Her muffin recipe only makes 6 muffins. Georgia makes 36 batches of muffins in 9 months. How many students does Georgia have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $p$ be the probability that the object reaches $(2,2)$ in six or fewer steps. Given that $p$ can be written in the form $m/n,$ 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{2141}
Given the parabola $y^{2}=4x$ with focus $F$, let $A$ and $B$ be the points of intersection between the directrix line of $F$ and the parabola, and let $P$ be the point of intersection between the parabola and the line $x=-1$. If $\overrightarrow{PA}=λ \overrightarrow{AF}$ and $\overrightarrow{PB}=μ \overrightarrow{BF}$ for some $λ,μ∈\mathbb{R}$, then find the value of $λ+μ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
$已知函数f(x)=(\frac{1}{x}+a)ln(1+x).$
$若f(x)在(0，+\infty )存在极值点，求a的取值范围.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(0,\frac {1}{2}\right)}
What is the maximum number of checkers that can be placed on a $6 \times 6$ board so that no three checkers (more precisely, the centers of the cells they occupy) are collinear (in any direction)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Twelve mayoral candidates each made a statement about how many times lies had been told before their turn. The first candidate said, "Before me, one lie was told." The second candidate said, "Now, two lies have been told." The third candidate said, "Now, three lies have been told," and so on, until the twelfth candidate who said, "Now, twelve lies have been told." It was later revealed that at least one candidate correctly counted how many times lies had been told up to their turn. How many lies in total were told by the candidates?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Suppose that \( x \) and \( y \) are chosen randomly and uniformly from \( (0,1) \). What is the probability that \( \left\lfloor \sqrt{\frac{x}{y}} \right\rfloor \) is even? 

Hint: \( \sum_{n=1}^{\infty} \frac{1}{n^{2}} = \frac{\pi^{2}}{6} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \frac{\pi^2}{24}}
A farmer has a certain amount of land, of which 80% is owned, and 20% is rented. Of the owned land, 90% was cleared for planting. From the cleared land, 20% was planted with potatoes, 30% was planted with tomatoes, 25% was planted with onions, and 10% was planted with carrots. The remaining cleared land, which was 5% of the owned land, was planted with corn. Additionally, 30% of the rented land was left unutilized. If the total unutilized land, including owned and rented, was 720 acres, how many acres of land did the farmer own?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5760}
A sequence is defined recursively as follows: \( t_{1} = 1 \), and for \( n > 1 \):

- If \( n \) is even, \( t_{n} = 1 + t_{\frac{n}{2}} \).
- If \( n \) is odd, \( t_{n} = \frac{1}{t_{n-1}} \).

Given that \( t_{n} = \frac{19}{87} \), find the sum of the digits of \( n \). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Quadrilateral $ABCD$ is neither a kite nor a rectangle. It is known that its side lengths are integers, $AB = 6$, $BC = 7$, and $\angle B = \angle D = 90^\circ$. Find the area of $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given that the circumference of a sector is $20\,cm$ and its area is $9\,cm^2$, find the radian measure of the central angle of the sector.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{9}}
Triangle $DEF$ has side lengths $DE = 15$, $EF = 39$, and $FD = 36$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \omega$, the area of $WXYZ$ can be expressed as a quadratic polynomial \[Area(WXYZ) = \gamma \omega - \delta \omega^2.\] Then the coefficient $\delta = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{229}
In the Cartesian coordinate plane $(xOy)$, a point $A(2,0)$, a moving point $B$ on the curve $y= \sqrt {1-x^{2}}$, and a point $C$ in the first quadrant form an isosceles right triangle $ABC$ with $\angle A=90^{\circ}$. The maximum length of the line segment $OC$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1+2 \sqrt {2}}
Dima calculated the factorials of all natural numbers from 80 to 99, found the reciprocals of them, and printed the resulting decimal fractions on 20 endless ribbons (for example, the last ribbon had the number \(\frac{1}{99!}=0. \underbrace{00\ldots00}_{155 \text{ zeros}} 10715 \ldots \) printed on it). Sasha wants to cut a piece from one ribbon that contains \(N\) consecutive digits without any decimal points. For what largest \(N\) can he do this so that Dima cannot determine from this piece which ribbon Sasha spoiled?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{155}
Consider pairs of functions $(f, g)$ from the set of nonnegative integers to itself such that:
- $f(0) + f(1) + f(2) + \cdots + f(42) \le 2022$;
- For any integers $a \ge b \ge 0$, we have $g(a+b) \le f(a) + f(b)$.

Determine the maximum possible value of $g(0) + g(1) + g(2) + \cdots + g(84)$ over all such pairs of functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7993}
19. $A B C$ and $G B D$ are straight lines. $E$ is a point on $C D$ produced, and $A D$ meets $E G$ at $F$. If $\angle C A D=\angle E G D, E F=F G, A B: B C=1: 2$ and $C D: D E=3: 2$, find $B D: D F$.
(3 marks)
$A B C$ 和 $G B D$ 均為直線, $E$ 是 $C D$ 延線上的一點, 且 $A D$ 交 $E G$ 於 $F$ 。若 $\angle C A D=$ $\angle E G D, E F=F G, A B: B C=1: 2$, 且 $C D: D E=3: 2$, 求 $B D: D F$ 。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}:2}
In the expansion of $$(3 \sqrt {x}-2 3x )^{11}$$, randomly select one term. Suppose the probability that the selected term is a rational term is $$\alpha$$. Then, $$\int_{ 0 }^{ 1 }x^{\alpha} \, dx= \_\_\_\_\_\_ ．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {6}{7}}
A semicircular paper with a radius of 4cm is rolled into a cone and placed on a table. After a gust of wind blows it over, calculate the distance from the highest point of the cone to the table. Express your answer in centimeters (cm).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}}
求最小的正整数 $m$, 使得对于任意大于 3 的质数 $p$ ，都有 $1051\left(9^{p^{2}}-29^{p}+m\right)$ 。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given that the line $ax+y-4=0$ is parallel to the line $x+(a+ \dfrac{3}{2})y+2=0$, find the real number $a=\boxed{\phantom{answer}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
This is a problem from Makarov’s Selected Problems in Real Analysis. Put $S(n) = \sum_{k=0}^{2n}{\frac{k}{k+n^2}}$. Find the limit as n tends to infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the equation $x^2 + y^2 + 4x - 2y - 4 = 0$, find the maximum value of $x^2 + y^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14 + 6\sqrt{5}}
In a regular truncated quadrilateral pyramid with lateral edges $A A 1, B B 1, C C 1, D D 1$, the side of the upper base $A 1 B 1 C 1 D 1$ is 1, and the side of the lower base is 7. A plane passing through the edge $B 1 C 1$ perpendicular to the plane $A D 1 C$ divides the pyramid into two parts of equal volume. Find the volume of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{38}{\sqrt{5}}}
A five-digit number is called irreducible if it cannot be expressed as the product of two three-digit numbers.

What is the maximum number of consecutive irreducible five-digit numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
A regular tetrahedron is formed by joining the centers of four non-adjacent faces of a cube. Determine the ratio of the volume of the tetrahedron to the volume of the cube.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Given \(a > b > 0\) and \(m > 0\), determine the relationship between the fractions \(\frac{b}{a}\) and \(\frac{b+m}{a+m}\). Express your answer using the symbols <, >, or =.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{<}
Given an isosceles right triangle with a leg length of 10, an infinite number of equilateral triangles are inscribed within it as shown in the figure: the vertices lie on the hypotenuse, and the bases are sequentially placed on one of the legs starting from the vertex of the right angle. Find the sum of the areas of the equilateral triangles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars would increase sales. If the diameter of the jars is increased by $25\%$ without altering the volume, by what percent must the height be decreased? If the answer is $y\%$, what is the value of $y$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Maria had 1450 bottles of water in her fridge. If she drank 38.5% of them and then bought 725 more, how many bottles would she have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1617}
In the 1448th exercise, let the circles centered at $A$ and $B$ be denoted by $k_{1}$ and $k_{2}$, their radii by $r_{1}$ and $r_{2}$, respectively, and the radius of the circle $k$ that touches $k$ from the inside and $k_{1}$ and $k_{2}$ from the outside by $r_{3}$. What does the ratio $r_{3} / r_{1}$ tend to, if - with fixed $A, B$ - $C$ tends to $A$, that is, $r_{1}$ tends to 0?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
How many ordered pairs of real numbers $(x, y)$ are there such that $x^2 + y^2 = 200$ and \[ \sqrt{(x-5)^2 + (y-5)^2} + \sqrt{(x+5)^2 + (y+5)^2} \] is an integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given the sets \( A = \{2, 4, a^3 - 2a^2 - a + 7\} \) and \( B = \{-4, a + 3, a^2 - 2a + 2, a^3 + a^2 + 3a + 7\} \), and \( A \cap B = \{2, 5\} \), find the possible values of \( a \) that form a set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{-1, 2\}}
从 $1,2,3, \cdots, 100$ 这 100 个正整数中任取 $n$ 个数, 在这 $n$ 个数中总能得到 4个数, 它们两两都互素,求 $n$ 的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
A one-dollar coin should have a mass of 7.0 g. Each individual coin may be lighter or heavier by as much as 2.14%. Joshua has a number of these coins and determines that they have a total mass of 1 kg. What is the difference between the greatest possible number and the least possible number of these coins that he could have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
15. Let  $a_{n}$  denote the number of ternary strings of length  $n$  so that there does not exist a  $k<n$  such that the first  $k$  digits of the string equals the last  $k$  digits. What is the largest integer  $m$  such that  $3^{m} \mid a_{2023}$  ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2022}
9. Let $A B C$ be a triangle, and let $B C D E, C A F G, A B H I$ be squares that do not overlap the triangle with centers $X, Y, Z$ respectively. Given that $A X=6, B Y=7$, and $C Z=8$, find the area of triangle $X Y Z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{21\sqrt{15}}{4}}
已知 f:\bZ\rightarrow\bZ 为偶函数，且 f(1)=1, f(2017)\neq1. 若对任意整数 x, y，均有2f(x+y)-f(x)-f(y)\leq |f(x)-f(y)|,求 f(2018) 的所有可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given a Brownian motion with drift parameter $\mu = 3$ and variance parameter $\sigma^2 = 9$, denoted as $\{X(t), t \geq 0\}$, where $X(0) = 10$, find the probability $P(X(0.5) > 10)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.7602}
A regular octagon has a shaded trapezium within it. Find the ratio of the area of the shaded trapezium to the area of the whole octagon. Express your answer as a ratio in the format a:b, where a and b are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:4}
John decides to get gym memberships so he can get in shape. The gym close to his house is close but doesn't have everything he wants, so he gets two different gym memberships. The cheap one costs $10 a month and has a sign-up fee of $50. The second gym is 3 times more expensive and it has a sign-up fee of 4 months membership. John refers one friend to each gym, receiving a 10% discount on the sign-up fees. In addition, John also books personal training sessions; he pays $25 per session at the first gym and $45 per session at the second gym. He attends one personal training session per week at each gym for an entire year. How much total did he pay in the first year for gym membership and personal training sessions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$4273}
Eunice spent $7500 on a used car, which had a series of three discounts applied to its original price. The first discount was 15%, followed by the second discount of 20%, and finally a third discount of 25%. In addition, there was a 10% sales tax applied to the final discounted price and a $200 documentation fee. What was the original price of the car before any discounts, taxes, or additional charges were applied?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$13,012.48}
在 $\triangle A B C$ 中, $\sin ^{2} A+\sin ^{2} C=2018 \sin ^{2} B$, 则 $\frac{(\tan A+\tan C) \tan ^{2} B}{\tan A+\tan B+\tan C}=\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}
Two right triangles $\triangle ABC$ and $\triangle ABD$ share a side $AB$ in such way that $AB=8$, $BC=12$, and $BD=10$. Let $E$ be a point on $BC$ such that $BE=9$. Determine the area of $\triangle ABE$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.8}
The cost price is a certain percentage of the selling price, and the profit percentage is 12.359550561797752%. What is the cost price as a percentage of the selling price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89.00\%}
There are $2n$ complex numbers that satisfy both $z^{28} - z^{8} - 1 = 0$ and $\mid z \mid = 1$. These numbers have the form $z_{m} = \cos\theta_{m} + i\sin\theta_{m}$, where $0\leq\theta_{1} < \theta_{2} < \ldots < \theta_{2n} < 360$ and angles are measured in degrees. Find the value of $\theta_{2} + \theta_{4} + \ldots + \theta_{2n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$840$}
Given that each airplane engine has a failure rate of $1-P$ during flight, and whether each engine fails is independent, determine the range of values for $P$ that makes a 4-engine airplane safer than a 2-engine airplane. Assume that for an airplane to fly successfully, at least 50% of its engines must function normally. Express your answer as an interval or range of values for $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{2}{3}, 1\right)}
What is the sum of those eight-digit numbers that consist only of the digits 1, 2, 3, 4, 5, 6, 7 and contain each of them at least once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8!14\cdot\frac{10^{8}-1}{9}}
Triangle $ABC$ has an obtuse angle at $\angle A$. Points $D$ and $E$ are placed on $\overline{BC}$ in the order $B$, $D$, $E$, $C$ such that $\angle BAD=\angle BCA$ and $\angle CAE=\angle CBA$. If $AB=10$, $AC=11$, and $DE=4$, determine $BC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Exactly at noon, a truck left a village heading towards a city, and at the same time, a car left the city heading towards the village. If the truck had left 45 minutes earlier, they would have met 18 kilometers closer to the city. If the car had left 20 minutes earlier, they would have met $k$ kilometers closer to the village. Find $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
There are 8 lottery tickets, among which there are first, second, and third prizes with one ticket each, and the remaining 5 tickets are without prizes. These 8 tickets are to be distributed among 4 people, with each person receiving 2 tickets. How many different winning scenarios are there? (Answer with a number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Two types of anti-inflammatory drugs must be selected from $X_{1}$, $X_{2}$, $X_{3}$, $X_{4}$, $X_{5}$, with the restriction that $X_{1}$ and $X_{2}$ must be used together, and one type of antipyretic drug must be selected from $T_{1}$, $T_{2}$, $T_{3}$, $T_{4}$, with the further restriction that $X_{3}$ and $T_{4}$ cannot be used at the same time. Calculate the number of different test schemes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let \( S \) be a subset of the set of numbers \(\{1,2,3, \ldots, 1989\}\) such that the difference between any two numbers in \( S \) is not equal to 4 or 7. What is the maximum number of elements that \( S \) can contain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{905}
What is the value of $n$ so that the Diophantine equation $12x + 13y = n$ has 11 non-negative integer solutions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1560}
Let $P_0 = (3,1)$ and define $P_{n+1} = (x_n, y_n)$ for $n \ge 0$ by \[ x_{n+1} = - \frac{3x_n - y_n}{2}, \quad y_{n+1} = - \frac{x_n + y_n}{2} \] Find the area of the quadrilateral formed by the points $P_{96}, P_{97}, P_{98}, P_{99}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let \( a_k = 0.\overbrace{0 \ldots 0}^{k - 1 \: 0's} 1 \overbrace{0 \ldots 0}^{k - 1 \: 0's} 1 \).

The value of \( \sum_{k = 1}^\infty a_k \) can be expressed as a rational number \( \frac{p}{q} \) in simplest form. Find \( p + q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Suppose  $f$  and  $g$  are differentiable functions such that \[xg(f(x))f^\prime(g(x))g^\prime(x)=f(g(x))g^\prime(f(x))f^\prime(x)\] for all real  $x$ .  Moreover,  $f$  is nonnegative and  $g$  is positive.  Furthermore, \[\int_0^a f(g(x))dx=1-\dfrac{e^{-2a}}{2}\] for all reals  $a$ .  Given that  $g(f(0))=1$ , compute the value of  $g(f(4))$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-16}}
将一枚骰子连续投掷五次，则事件 "五次出现的点数既不全相同也不两两互异，且从第二次起每一次的点数都不小于前一次的点数"的概率为 $\qquad$。  原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{167}
The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y| \leq 1000$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$. Points $P$ and $Q$ are placed on segments $\overline{CD}$ and $\overline{DA}$ respectively such that $AP \perp CD$ and $BQ \perp DA$, and point $X$ is the intersection of these two altitudes. Suppose that $BX = 3$ and $XQ = 1$. Compute the largest possible area of $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Let  $T_1$  and  $T_2$  be the points of tangency of the excircles of a triangle  $ABC$  with its sides  $BC$  and  $AC$  respectively. It is known that the reflection of the incenter of  $ABC$  across the midpoint of  $AB$  lies on the circumcircle of triangle  $CT_1T_2$ . Find  $\angle BCA$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Given that \( x \) and \( y \) are integers and satisfy the equation
$$
\left(\frac{1}{x} + \frac{1}{y}\right) \left(\frac{1}{x^2} + \frac{1}{y^2}\right) = -\frac{2}{3} \left(\frac{1}{x^4} - \frac{1}{y^4}\right),
$$
find the number of possible values of \( x + y \). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that a product originally priced at an unknown value is raised by 20% twice consecutively, and another product originally priced at an unknown value is reduced by 20% twice consecutively, ultimately selling both at 23.04 yuan each, determine the profit or loss situation when one piece of each product is sold.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.92}
Problem 4. Square $A B C D$ is inscribed in circle $\omega$. On the smaller arc $C D$ of circle $\omega$, an arbitrary point $M$ is chosen. Inside the square, points $K$ and $L$ are marked such that $K L M D$ is a square. Find $\angle A K D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135}
设集合 $S$ 是 $\{0,1, \cdots, 98\}$ 的 $m(m \geqslant 3)$元子集，满足对任意的 $x, y \in S$ ，均存在 $z \in$ $S$, 使得 $x+y \equiv 2 z(\bmod 99)$. 求 $m$ 的所有可能值。请给出所有可能值的和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{155}
A giant panda is transporting bamboo from location A to location B. Each time, the panda can carry 50 bamboo sticks, but it eats 5 bamboo sticks both on the way from A to B and on the way back from B to A. There are currently 200 bamboo sticks at location A. What is the maximum number of bamboo sticks that the panda can transport to location B?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}