Context:
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{}.


Context:
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{}.


Context:
$已知函数f(x)=(\frac{1}{x}+a)ln(1+x).$
$若f(x)在(0，+\infty )存在极值点，求a的取值范围.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
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{}.


Context:
Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\frac 12,\frac 13,\frac 16\right).$  Find the area of $\mathcal{S}$ divided by the area of $\mathcal{T}.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
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{}.


Context:
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{}.


Context:
If circle $$C_{1}: x^{2}+y^{2}+ax=0$$ and circle $$C_{2}: x^{2}+y^{2}+2ax+ytanθ=0$$ are both symmetric about the line $2x-y-1=0$, then $sinθcosθ=$ \_\_\_\_\_\_ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
$已知函数f(x) = x^3 - 3x^2 + ax + 4, a \in  R.$
$若f(x)>0对x\in (1，2)恒成立，求a的取值范围.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Place as many points as possible on a plane such that any three points do not lie on a single straight line and form the vertices of an isosceles triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a cube \( ABCD A_1 B_1 C_1 D_1 \). A point is called "equidistant" if there are two vertices of the cube for which this point is the midpoint of the segment connecting them. How many "equidistant" points are there in the cube?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A triangle  $\vartriangle A_0A_1A_2$  in the plane has sidelengths  $A_0A_1 = 7$ , $A_1A_2 = 8$ , $A_2A_0 = 9$ . For  $i \ge 0$ , given  $\vartriangle A_iA_{i+1}A_{i+2}$ , let  $A_{i+3}$  be the midpoint of  $A_iA_{i+1}$  and let Gi be the centroid of  $\vartriangle A_iA_{i+1}A_{i+2}$ . Let point  $G$  be the limit of the sequence of points  $\{G_i\}^{\infty}_{i=0}$ . If the distance between  $G$  and  $G_0$  can be written as  $\frac{a\sqrt{b}}{c}$  , where  $a, b, c$  are positive integers such that  $a$  and  $c$  are relatively prime and  $b$  is not divisible by the square of any prime, find  $a^2 + b^2 + c^2$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Andrey, Boris, Vasily, Gennady, and Dmitry played table tennis in pairs so that each pair played against every other pair exactly once. There are no ties in tennis. It is known that Andrey lost exactly 12 times, and Boris lost exactly 6 times. How many times did Gennady win?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given circle $O$: $x^{2}+y^{2}=3$, line $l$: $x+3y-6=0$, and point $P(x_{0},y_{0})∈l$. If there exists a point $Q$ on circle $O$ such that $∠OPQ=60^{\circ}$, determine the range of $x_{0}$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{0 \leqslant x_0 \leqslant \frac{6}{5}}


Context:
In the triangle \(A B C\), angle \(C\) is a right angle, and \(AC: AB = 3: 5\). A circle with its center on the extension of leg \(AC\) beyond point \(C\) is tangent to the extension of hypotenuse \(AB\) beyond point \(B\) and intersects leg \(BC\) at point \(P\), with \(BP: PC = 1: 4\). Find the ratio of the radius of the circle to leg \(BC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $(X_k)$ be a sequence of random variables as following : $P(X_i=-1)=q$ and $P(X_i=1)=1-q$. I want to calculate $$E[\inf\{n:X_1+X_2+...+X_n>0\}]$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the shadow length $l$ is equal to the product of the table height $h$ and the tangent value of the solar zenith angle $\theta$, and $\tan(\alpha-\beta)=\frac{1}{3}$, if the shadow length in the first measurement is three times the table height, determine the shadow length in the second measurement as a multiple of the table height.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each square of a chessboard has a knight on it. What is the minimum number of knights that need to be removed so that there isn't a single knight left on the board that can attack exactly three other knights? (A knight attacks squares that are one square away horizontally and two squares vertically, or vice versa.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知勾函数$y=x+\frac{{{a}^{2}}}{x}(a>0)$在$(-\infty ,-a)$和$(a,+\infty )$内均为增函数，在$(-a,0)$和$(0,a)$ 内均为减函数。若勾函数$f(x)=x+\frac{t}{x}(t>0)$在整数集合$Z$内为增函数，则实数$t$的取值范围为___________。
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square board is divided by seven lines parallel to one side of the board and seven lines parallel to the other side, into 64 rectangular cells, which are painted in white and black in a checkerboard pattern. The distances between neighboring lines are not necessarily equal, so the cells can be of different sizes. It is known, however, that the ratio of the area of any white cell to the area of any black cell is no more than 2. Find the greatest possible ratio of the total area of the white cells to the total area of the black cells.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a triangular pyramid $P-ABC$ with its three side edges mutually perpendicular, and $AB=2, AC= \sqrt{7}, BC= \sqrt{13}$, then the volume of the circumscribed sphere of this pyramid is _______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A ranch has both horses and ponies. Some fraction of the ponies have horseshoes, and exactly 5/8 of the ponies with horseshoes are from Iceland. There are 4 more horses than ponies on the ranch. The minimum possible combined number of horses and ponies on the ranch is 164. What fraction of the ponies have horseshoes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Once, in a physics and mathematics school, September was declared the month of information technology. Therefore, every Monday in September there was 1 computer science lesson in each class, every Tuesday - 2 lessons, every Wednesday - 3 lessons, every Thursday - 4 lessons, and every Friday - 5 computer science lessons. Saturdays and Sundays were days off at this school. On September 1, there were ceremonial assemblies at the school, so there were no lessons. It is known that there were a total of 64 computer science lessons in each class throughout September. On which day of the week did September 1 fall that year? September has a total of 30 days.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Lisa bought a new iPhone for $1000. In addition to the iPhone, she bought a phone contract that costs $200/month, a case that costs 20% of the cost of the phone, headphones that cost half as much as the case, a portable charger that costs $60, and an extended warranty plan that costs $150 for the first two years. She also received a 10% discount on the case and headphones due to a promotion. If the phone contract remains the same price, how much will Lisa spend on her phone and related expenses after three years?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest three-digit number \(n\) such that if the three digits are \(a\), \(b\), and \(c\), then
\[ n = a + b + c + ab + bc + ac + abc. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
We have eight light bulbs, placed on the eight lattice points (points with integer coordinates) in space that are $\sqrt{3}$ units away from the origin. Each light bulb can either be turned on or off. These lightbulbs are unstable, however. If two light bulbs that are at most 2 units apart are both on simultaneously, they both explode. Given that no explosions take place, how many possible configurations of on/off light bulbs exist?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A and B began riding bicycles from point A to point C, passing through point B on the way. After a while, A asked B, "How many kilometers have we ridden?" B responded, "We have ridden a distance equivalent to one-third of the distance from here to point B." After riding another 10 kilometers, A asked again, "How many kilometers do we have left to ride to reach point C?" B answered, "We have a distance left to ride equivalent to one-third of the distance from here to point B." What is the distance between point A and point C? (Answer should be in fraction form.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( a_{1}, a_{2}, \ldots, a_{2000} \) be real numbers in the interval \([0,1]\). Find the maximum possible value of
\[
\sum_{1 \leq i < j \leq 2000}(j - i) \left| a_{j} - a_{i} \right|
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the plane, a rhombus \(ABCD\) and two circles are constructed, circumscribed around triangles \(BCD\) and \(ABD\). Ray \(BA\) intersects the first circle at point \(P\) (different from \(B\)), and ray \(PD\) intersects the second circle at point \(Q\). It is known that \(PD = 1\) and \(DQ = 2 + \sqrt{3}\). Find the area of the rhombus.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), and \(a, b\), and \(c\) are positive integers, then the number of ordered triples \((a, b, c)\), such that \(a+2b+c \leq 40\), is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
8 个女孩和 25 个男孩围成一圈,任意两个女孩之间至少站两个男孩, 那么共有多少种不同的排列方法 (只要把圆旋转一下就重合的排法认定是相同的).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $w_1$ and $w_2$ denote the circles $x^2+y^2+10x-24y-87=0$ and $x^2 +y^2-10x-24y+153=0,$ respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_2$ and internally tangent to $w_1.$ Given that $m^2=\frac{p}{q},$ where $p$ and $q$ are relatively prime integers, find $p+q.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $ABC$  be a triangle with  $\angle ABC=90^{\circ}$ . The square  $BDEF$  is inscribed in  $\triangle ABC$ , such that  $D,E,F$  are in the sides  $AB,CA,BC$  respectively. The inradius of  $\triangle EFC$  and  $\triangle EDA$  are  $c$  and  $b$ , respectively. Four circles  $\omega_1,\omega_2,\omega_3,\omega_4$  are drawn inside the square  $BDEF$ , such that the radius of  $\omega_1$  and  $\omega_3$  are both equal to  $b$  and the radius of  $\omega_2$  and  $\omega_4$  are both equal to  $a$ . The circle  $\omega_1$  is tangent to  $ED$ , the circle  $\omega_3$  is tangent to  $BF$ ,  $\omega_2$  is tangent to  $EF$  and  $\omega_4$  is tangent to  $BD$ , each one of these circles are tangent to the two closest circles and the circles  $\omega_1$  and  $\omega_3$  are tangents. Determine the ratio  $\frac{c}{a}$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
The interval in which the function $f(x) = \left(\frac{1}{2}\right)^{x^2 - 2x}$ is monotonically decreasing is $\_\_\_\_\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Kim uses five different electronic devices in her home – a TV, a refrigerator, an air conditioner, a washing machine, and a laptop. The TV uses 125 watts of electricity per hour and runs for 4 hours a day. The refrigerator uses 100 watts per hour and runs continuously throughout the day. The air conditioner consumes 2000 watts per hour and is used for 6 hours a day. The washing machine uses 500 watts per hour and is operated for 2 hours a day, but only three times a week. Lastly, the laptop uses 60 watts per hour and is in use for 5 hours a day.

The local electricity provider has implemented a tiered tariff system as follows:
- First 100 kWh/month: 14 cents per kWh
- Next 200 kWh/month: 18 cents per kWh
- Any usage above 300 kWh/month: 22 cents per kWh

Calculate the total cost of electricity for running all these devices for a week, considering the increasing electricity tariff.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two positive integers $m$ and $n$, we say that $m \mid\mid n$ if $m \mid n$ and $\gcd(m,\, n/m) = 1$. Compute the smallest integer greater than \[ \sum_{d \mid 2016} \sum_{m \mid\mid d} \frac{1}{m}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose two arithmetic sequences $\{a\_n\}$ and $\{b\_n\}$ have the sum of their first n terms as $S\_n$ and $T\_n$, respectively. If $a\_3 = 2b\_3$ and $\frac{S\_n}{T\_n} = \frac{3n - t}{2n + 2}$ for any $n \in \mathbb{N}^*$, find the value of $\frac{a\_9}{b\_6 + b\_8} + \frac{a\_5}{b\_3 + b\_11}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that an isosceles trapezoid is circumscribed around a circle, find the ratio of the area of the trapezoid to the area of the circle if the distance between the points where the circle touches the non-parallel sides of the trapezoid is related to the radius of the circle as $\sqrt{3}: 1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Equilateral triangles \( ABF \) and \( BCG \) are constructed outside regular pentagon \( ABCDE \). Compute \( \angle FEG \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence $\{a_n\}$ where $a_1 = -1$, $a_2 > a_1$, and $|a_{n+1}-a_n| = 2^{n}$ for $n \in \mathbb{N}^*$, if the subsequence $\{a_{2n-1}\}$ is monotonically decreasing and the subsequence $\{a_{2n}\}$ is monotonically increasing, then find the general term of the sequence $\{a_n\}$: $a_n = \_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $E, F, G$ 分别是正四面体 $A B C D$ 的棱 $A B, B C, C D$ 的中点，求二面角 $C-F G-E$ 的大小。如果答案为 $\pi-\operatorname{arccot} x$，其原始答案的形式为\frac{k\sqrt{n}}{m}，请计算k+m+n的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
$在\triangle ABC中，角A,B,C的对边分别为a,b,c，且满足b \sin B+c \sin C=a \sin A+\sqrt{3}b \sin C.$
求角A的值；
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that positive real numbers $a$ and $b$ satisfy $\frac{8}{{{{({b+1})}^3}}}+\frac{6}{{b+1}}≤{a^3}+3a$, find the minimum value of $2a+3b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the angle between $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is $90^{\circ}$, $|\overrightarrow{AB}|=2$, and $|\overrightarrow{AC}|=λ \overrightarrow{AB}+μ \overrightarrow{AC}$ $(λ,μ∈R)$, with $\overrightarrow{MF}\cdot \overrightarrow{BC}=0$, find the value of $\frac{\lambda }{\mu }$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given positive numbers $a$ and $b$ satisfying $\log _{6}(2a+3b)=\log _{3}b+\log _{6}9-1=\log _{2}a+\log _{6}9-\log _{2}3$, find $\lg \left(2a+3b\right)-\lg \left(10a\right)-\lg \left(10b\right)=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ \phi : L \to L $ be a linear operator with minimal polynomial $ m_\phi (x) = (x-1)x^8 $. What is the minimal polynomial of $ \phi^3 $?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Allie picked 100 wildflowers. 13 of the flowers were yellow and white, 17 of the flowers were red and yellow, 14 of the flowers were red and white, 16 of the flowers were blue and yellow, 9 of the flowers were blue and white, 8 of the flowers were red, blue, and yellow, and 6 of the flowers were red, white, and blue. How many more flowers contained the color red than contained the color white, considering the overlapping categories?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the fixed point (x, y) through which the graph of the quadratic function y=x^{2}-(2-m)x+m always passes, regardless of the value of the real number m. Express your answer as an ordered pair (x, y).
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are $n$ seats in a bus, and all tickets have been sold to $n$ passengers. The Absent-Minded Scientist enters first and sits in a randomly chosen seat without looking at their ticket. The remaining passengers enter one by one. If a passenger sees that their seat is vacant, they take their seat. If their seat is occupied, they sit in the first available seat. Find the probability that the last passenger to enter will sit in their assigned seat according to their ticket.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 12 items displayed on a shelf, arranged with 4 items on the upper layer and 8 items on the lower layer. Now, if 2 items are to be selected from the lower layer to be moved to the upper layer, and the relative order of the other items remains unchanged, calculate the number of different ways to adjust the items. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
John decides to go back to school to get his PhD. He first takes 1 year of courses to get acclimated back to school life before starting his PhD. After that, he spends 2 years learning the basics of his field. He then starts research and spends 75% more time on research than he did learning the basics. During his research phase, he takes a 6-month sabbatical for some personal development. After completing his research, he spends half as long on writing his dissertation as his acclimation period. However, he also attends a 3-month conference during his dissertation writing phase. How long did it take him to do everything, including the sabbatical and conference?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of quadrilateral ABCD given that $\angle A = \angle D = 120^{\circ}$, $AB = 5$, $BC = 7$, $CD = 3$, and $DA = 4$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that  $n$  is a natural number such that the leftmost digits in the decimal representations of  $2^n$  and  $3^n$  are the same, find all possible values of the leftmost digit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A permutation  $(a_1, a_2, a_3, \dots, a_{100})$  of  $(1, 2, 3, \dots, 100)$  is chosen at random. Denote by  $p$  the probability that  $a_{2i} > a_{2i - 1}$  for all  $i \in \{1, 2, 3, \dots, 50\}$ . Compute the number of ordered pairs of positive integers  $(a, b)$  satisfying  $\textstyle\frac{1}{a^b} = p$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Climbing a staircase with four steps, calculate the total number of possible ways to do so, considering different step sequences as distinct arrangements. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. [5 points] On a plane with a given rectangular Cartesian coordinate system, a square is drawn with vertices at points $(0 ; 0),(0 ; 69),(69 ; 69)$, and ( $69 ; 0)$. Find the number of ways to choose two grid nodes inside this square (not including its boundary) such that at least one of these nodes lies on one of the lines $y=x$ or $y=69-x$, but neither of the chosen nodes lies on any line parallel to any of the coordinate axes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square has eight points highlighted on its sides, dividing each side into three equal parts. How many right triangles can be drawn with their vertices at these points? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following limit:
$$\lim_{x\to\infty}\left(\sqrt{x^{6} +x^{4}\sqrt{x^{6} +1}}-\sqrt{2x^{6}}\right)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $O$ is the origin, and $P$ is a point on the hyperbola $\frac{x^{2}}{a^{2}}-y^{2}=1 (a > 0)$, two lines are drawn through $P$ parallel to the asymptotes, intersecting the asymptotes at points $A$ and $B$ respectively. The area of the parallelogram $OBPA$ is $1$. Find the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define $L(x) = x - \frac{x^2}{2}$ for every real number $x$.  If $n$ is a positive integer, define $a_n$ by
\[
  a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{17}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr),
\]where there are $n$ iterations of $L$.  For example,
\[
  a_4 = L \Bigl( L \Bigl( L \Bigl( L \Bigl( \frac{17}{4} \Bigr) \Bigr) \Bigr) \Bigr).
\]As $n$ approaches infinity, what value does $n a_n$ approach?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following sum: $$\frac{1}{2^{99}}\sum_{k=1}^{50} \sum_{r=0}^{k-1}\binom{50}{k}\binom{49}{r}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
If \begin{align}\prod_{k=1}^{n}\tan \left(\frac{k\pi}{2n+1}\right) = \prod_{k = 10}^{19}\cos \left(\frac{k\pi}{10}\right).\end{align} find the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that points A and B lie on the graph of y = \frac{1}{x} in the first quadrant, ∠OAB = 90°, and AO = AB, find the area of the isosceles right triangle ∆OAB.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ABC$, if $a= \sqrt {2}$, $b=2$, and $\sin B+\cos B= \sqrt {2}$, then $A=$ ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the domain of the function $f(x)$ is $\mathbb{R}$, and $f(0)=2$, if for any $\left\{x\in \mathbb{R},f(x)+{{f}^{{{'}}}}(x) < 1\right\}$, then the solution set of the inequality $${{e}^{x}}f(x) < {{e}^{x}}+1$$ is.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the angle between the generating line and the axis of a cone is $\frac{\pi}{3}$, and the length of the generating line is $3$, find the maximum value of the cross-sectional area through the vertex.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A simple graph $G$ on 2020 vertices has its edges colored red and green. It turns out that any monochromatic cycle has even length. Given this information, what is the maximum number of edges $G$ could have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
When Xiao Ma Hu was mentally calculating the product of a two-digit number and a whole ten number, he forgot to add a 0 at the end of the product. As a result, the product was less than the correct product by ____ times.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \mathcal{V} be the volume enclosed by the graph $x^{2016}+y^{2016}+z^{2}=2016$. Find \mathcal{V} rounded to the nearest multiple of ten.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the product
$$
\prod_{n=3}^{\infty} \frac{\left(n^{3}+3 n\right)^{2}}{n^{6}-64}
$$
The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Adam is playing Minesweeper on a $9\times9$ grid of squares, where exactly $\frac13$ (or 27) of the squares are mines (generated uniformly at random over all such boards). Every time he clicks on a square, it is either a mine, in which case he loses, or it shows a number saying how many of the (up to eight) adjacent squares are mines.

First, he clicks the square directly above the center square, which shows the number $4$. Next, he clicks the square directly below the center square, which shows the number $1$. What is the probability that the center square is a mine?

[i]Proposed by Adam Bertelli[/i]
Please reason step by step, and put your final answer within \boxed{}.


Context:
The integer $N$ is the smallest positive integer that is a multiple of 2024, has more than 100 positive divisors (including 1 and $N$), and has fewer than 110 positive divisors (including 1 and $N$). What is the sum of the digits of $N$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
"My phone number," said the trip leader to the kids, "is a five-digit number. The first digit is a prime number, and the last two digits are obtained from the previous pair (which represents a prime number) by rearrangement, forming a perfect square. The number formed by reversing this phone number is even." What is the trip leader's phone number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
6. In the Cartesian coordinate system $x O y$, the set of points
$$
K=\{(x, y) \mid x, y \in\{-1,0,1\}\} \text {. }
$$

If three points are randomly selected from the set $K$, the probability that there exist two points among these three with a distance of $\sqrt{5}$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mac loves the shape of quarters and is willing to trade money for them. He tells his brother that he will trade him some dimes for a quarter or some nickels. He trades for 20 quarters with dimes and 20 quarters with nickels. Mac lost $3. How many dimes did he trade for a quarter?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A polynomial of degree 10 has three distinct roots. What is the maximum number of zero coefficients it can have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a math competition, there are 8 fill-in-the-blank questions worth 4 points each if answered correctly, and 6 short-answer questions worth 7 points each if answered correctly. If 400 people participated in the competition, what is the minimum number of people who have the same total score?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a triangle, the lengths of the three sides are integers \( l, m, n \), with \( l > m > n \). It is known that \( \left\{ \frac{3^{l}}{10^{4}} \right\} = \left\{ \frac{3^{m}}{10^{4}} \right\} = \left\{ \frac{3^{n}}{10^{4}} \right\} \), where \( \{x\} \) denotes the fractional part of \( x \) and \( [x] \) denotes the greatest integer less than or equal to \( x \). Determine the smallest possible value of the perimeter of such a triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $O$ 是 $\triangle A B C$ 内一点, 满足 $S_{\triangle O A B}: S_{\triangle O B C}: S_{\triangle O C A}=4: 3: 2, \overrightarrow{A O}=\lambda \overrightarrow{A B}+$ $\mu \overrightarrow{A C}$, 则 $\lambda+\mu=$ $\qquad$.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{}.


Context:
The height of a triangle, equal to 2 cm, divides the angle of the triangle in the ratio 2:1 and the base of the triangle into parts, where the smaller part is 1 cm. Determine the area of the triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Kanul spent $3000 in buying raw materials, $2000 in buying machinery, 15% of the total amount he had as cash with him, and an additional $1000 for labor costs. What was the total amount he had initially?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 (a>0, b>0)$, where $F$ is the right focus of the hyperbola. A perpendicular line is drawn from $F$ to one of its asymptotes, with the foot of the perpendicular being $Q$. The line $FQ$ intersects the left and right branches of the hyperbola at points $M$ and $N$, respectively. If $|MQ|=3|QN|$, then the eccentricity of the hyperbola is ____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a given positive integer $k$, denote the square of the sum of its digits by $f_1(k)$. Define $f_{n+1}(k) = f_1(f_n(k))$. Determine the value of $f_{1991}(2^{1990})$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The expression $\sqrt{(\sqrt{56})(\sqrt{126})}$ can be simplified to $a\sqrt b$, where $a$ and $b$ are integers and $b$ is not divisible by any perfect square greater than 1. What is $a+b$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given vectors $\overrightarrow{a}=(-k,\;4)$ and $\overrightarrow{b}=(k,\;k+3)$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is acute. Determine the range of the real number $k$ (please write in interval form).
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a $3 \times 3$ chessboard, each square contains a knight with $\frac{1}{2}$ probability. What is the probability that there are two knights that can attack each other? (In chess, a knight can attack any piece which is two squares away from it in a particular direction and one square away in a perpendicular direction.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence $\{a_n\}$ with the first term $a_1=2$, and $a_{n+1} = \frac{2a_{n}}{a_{n+2}}$ (for $n=1,2,3,\ldots$), find the value of $a_{2012}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a parallelogram with vertices $A(0, 0)$, $B(0, 5)$, $C(x, 5)$, and $D(x, 0)$ where $x > 0$. The parallelogram is titled such that $AD$ and $BC$ make an angle of $30^\circ$ with the horizontal axis. If the area of the parallelogram is 35 square units, find the value of $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
30. Arrange the natural numbers 1 to 7 in a row, such that each number is either greater than all the numbers before it or less than all the numbers before it (for example: 4356271 satisfies the condition, starting from the second position, 3 is less than the first 4; 5 is greater than the first two 4, 3; 6 is greater than the first three 4, 3, 5; 2 is less than the first four 4, 3, 5, 6; 7 is greater than the first five 4, 3, 5, 6, 2; 1 is less than the first six 4, 3, 5, 6, 2, 7). Then, among the arrangements that satisfy the condition, the number of arrangements where 7 is not in the fourth position is  kinds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
136. How many times should a die be thrown so that the probability of the inequality

$$
|m / n-1 / 6| \leqslant 0.01
$$

is not less than the probability of the opposite inequality, where $m$ is the number of times one dot appears in $n$ throws of a die?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a positive integer \( n \), find the smallest real number \( \lambda \) such that there exist real numbers \( a_1, a_2, \ldots, a_n \) within the interval \([0, 1]\), for any real numbers \( x_1, x_2, \ldots, x_n \) satisfying \( 0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq 1 \), we have
$$
\min_{1 \leq i \leq n} \left|x_i - a_i\right| \leq \lambda
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A semicircle with radius 1 is centered at 0 and has AB as its diameter. A point C moves along the entire circumference of the circle. Extend AC to point D so that CD/CB = k (where k > 0 and k is a constant). Calculate the length of the curve traced by point D. Express your answer in terms of pi and the square root of 1 + k^2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \(ABC\), angle \(A\) is \(40^\circ\). The triangle is randomly thrown onto a table. Find the probability that vertex \(A\) ends up east of the other two vertices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the geometric sequence {a_n}, the product of the first three terms is 27, and the sum of the second and fourth terms is 30. Find the common ratio q. Express your answer as a numerical value, including the ± symbol if the answer has multiple solutions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an arithmetic sequence $\{a\_n\}$, there are two terms $a\_m=\frac{1}{k}$ and $a\_k=\frac{1}{m}$. Find the sum of the first $mk$ terms of this sequence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Once in a restaurant ***Dr. Strange*** found out that there were 12 types of food items from 1 to 12 on the menu. He decided to visit the restaurant 12 days in a row and try a different food everyday. 1st day, he tries one of the items from the first two. On the 2nd day, he eats either item 3 or the item he didn’t tried on the 1st day. Similarly, on the 3rd day, he eats either item 4 or the item he didn’t tried on the 2nd day. If someday he's not able to choose items that way, he eats the item that remained uneaten from the menu. In how many ways can he eat the items for 12 days?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ben's potato gun can launch a potato some number of football fields. If a football field is 200 yards long and Ben's dog can run 400 feet/minute, it takes his dog 9 minutes to fetch a potato he launches. How many football fields can Ben's potato gun launch a potato?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Problem 9. (4 points)

In all cells of a 4x8 table, except for the corner cells, non-negative numbers are arranged such that the sum of the numbers in each cross of five cells does not exceed 8. What is the maximum value that the sum of the numbers in the entire table can take?

#
Please reason step by step, and put your final answer within \boxed{}.


Context:
Circles \(\omega_{1}\) and \(\omega_{2}\) intersect at points \(A\) and \(B\). Segment \(PQ\) is tangent to \(\omega_{1}\) at \(P\) and to \(\omega_{2}\) at \(Q\), and \(A\) is closer to \(PQ\) than \(B\). Point \(X\) is on \(\omega_{1}\) such that \(PX \parallel QB\), and point \(Y\) is on \(\omega_{2}\) such that \(QY \parallel PB\). Given that \(\angle APQ=30^{\circ}\) and \(\angle PQA=15^{\circ}\), find the ratio \(AX / AY\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cube with edge length 2 cm has a dot marked at the center of the top face. The cube is on a flat table and rolls without slipping, making a full rotation back to its initial orientation, with the dot back on top. Calculate the length of the path followed by the dot in terms of $\pi. $
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points $G$ and $N$ are chosen on the interiors of sides $E D$ and $D O$ of unit square $D O M E$, so that pentagon GNOME has only two distinct side lengths. The sum of all possible areas of quadrilateral $N O M E$ can be expressed as $\frac{a-b \sqrt{c}}{d}$, where $a, b, c, d$ are positive integers such that $\operatorname{gcd}(a, b, d)=1$ and $c$ is square-free (i.e. no perfect square greater than 1 divides $c$ ). Compute $1000 a+100 b+10 c+d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the fractional equation in terms of $x$, $\frac{2}{x-2}-\frac{2x-m}{2-x}=3$, has a positive root, then the value of $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of functions $f:\mathbb{Z}\mapsto\mathbb{Z}$ for which $f(h+k)+f(hk)=f(h)f(k)+1$, for all integers $h$ and $k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
4. Inside a square, $n$ points are given. They are connected in pairs, as well as individual points with the vertices of the square, but in such a way that no two segments intersect at an interior point. How many segments can be constructed in this way?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Circles $P$, $Q$, and $R$ are externally tangent to one another. The external tangent of circles $P$ and $Q$ that does not intersect circle $R$ intersects circle $P$ at point $P_Q$ and circle $Q$ at point $Q_P$. Points $Q_R$, $R_Q$, $R_P$, and $P_R$ are defined similarly.

If the radius of circle $Q$ is $4$ and the line segment $\overline{Q_PP_Q}$ is parallel to $\overline{R_QQ_R}$, compute the length of $R_PP_R$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
One hundred number cards are laid out in a row in ascending order: \(00, 01, 02, 03, \ldots, 99\). Then, the cards are rearranged so that each subsequent card is obtained from the previous one by increasing or decreasing exactly one of the digits by 1 (for example, after 29 there can be 19, 39 or 28, but not 30 or 20). What is the maximum number of cards that could remain in their original positions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $F(x,\theta)$ be defined for $x>0$ and $\theta > 3$ by $$F(x,\theta) = 1 - {D\over(x-\theta+5)^2}\,.$$ If $F$ is a CDF in $x\in (0, \infty)$, then what must be the value of $D$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The value of the infinite series $$\sum_{n=2}^\infty \dfrac{n^4+n^3+n^2-n+1}{n^6-1}$$ can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive numbers. Compute $p+q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four numbers are in the ratio of 2 : 3 : 4 : 5 and their LCM is 3600. If the HCF of the two largest numbers in the ratio is 4, find the HCF of all four numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The charge for a standard single room at Hotel P is 50 percent less than the charge for a standard single room at Hotel R and 10 percent less than the charge for a standard single room at Hotel G. The charge for a deluxe single room at Hotel P is 30 percent less than the charge for a deluxe single room at Hotel R and 15 percent less than the charge for a deluxe single room at Hotel G. On weekends, Hotel P offers a 7 percent discount on all room types, while Hotel R and Hotel G offer 10 percent and 12 percent discounts respectively on standard single rooms. The charge for a standard single room on a weekday at Hotel R is what percent greater than the charge for a standard single room on a weekend at Hotel P?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two cylindrical cans have the same volume. The height of one can is triple the height of the other. If the radius of the narrower can is 12 units, how many units are in the length of the radius of the wider can? Express your answer in simplest radical form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 10 阶实方阵 A 满足 A^5=0，则 A 的秩的所有可能值之和为__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $0 < \alpha - \beta < \pi$, and $0 < \alpha + 2\beta < \pi$, find the range of values for $\alpha + \beta$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
John is a dedicated volunteer and is involved in many community service tasks at the local shelter. From January to March, he volunteers at the shelter twice a month for 3 hours at a time. Starting from April to June, he starts volunteering once a week, but only for 2 hours at a time, at the pet adoption events. However, in July and August, he volunteers every Saturday and Sunday for 4 hours each day for the summer community service program.
In September and October, John volunteers once a week, on Thursdays for 3 hours each time, at the shelter. Meanwhile, in November, due to his busy schedule, he can only volunteer for two Saturdays for a total of 6 hours. Finally, in December, due to holiday closures, he only volunteers for three Fridays for a total of 6 hours. However, John also volunteers on some bonus days throughout the year as follows: on the third Saturday of every month excluding May and June for 4 hours each, plus he joins the annual charity run event in June which requires him to volunteer for 8 hours. How many hours does he volunteer for that year?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $\displaystyle f(x) = \lim_{n\rightarrow \infty}\frac{x^2+2(x+1)^{2n}}{(x+1)^{2n+1}+x^2+1}\;,n\in \mathbb{N}$ and $\displaystyle g(x) = \tan \left(\frac{1}{2}\arcsin\left(\frac{2f(x)}{1+(f(x))^2}\right)\right)$ Then number of points where $g(x)$ is not differentiable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the slope of the line passing through $(-3,5)$ and $(2,-5)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which real numbers $c$ is there a straight line that intersects the curve
$$
y=x^{4}+9 x^{3}+c x^{2}+9 x+4
$$
in four distinct points? The original answer is in the format $c < \frac{m}{n}$. Please provide the value of $m + n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the minimum number of cells on a $6 \times 6$ board that need to be colored so that, regardless of the placement (including rotations and reflections) of an L-shaped figure consisting of 4 cells on the board, there is at least one colored cell in the figure?
Please reason step by step, and put your final answer within \boxed{}.


Context:
7.5. At a round table, 10 people are sitting, each of whom is either a knight, who always tells the truth, or a liar, who always lies. Each of them was given a token. Then each of them passed their token to one of their two neighbors. After that, 5 people said: "I have one token," while the other 5 said: "I have no tokens." What is the maximum number of knights that could have been among these 10 people?
Please reason step by step, and put your final answer within \boxed{}.


Context:
将方程 $(1+2+\cdots+k)-3(1+2+\cdots+t)=1$ 的全体正整数解 $(k, t)$ 按照从小到大的方式排列，得到序列 $\left\{\left(k_{n}, t_{n}\right)\right\}$ ，则 $k_{6}=$ $\qquad$。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value for \(a, b > 0\) of the expression

$$
\frac{|6a - 4b| + |3(a + b\sqrt{3}) + 2(a\sqrt{3} - b)|}{\sqrt{a^2 + b^2}}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an ellipse $\frac{x^{2}}{8} + \frac{y^{2}}{2} = 1$, and a point $A(2,1)$ on the ellipse. The slopes of the lines $AB$ and $AC$ connecting point $A$ to two moving points $B$ and $C$ on the ellipse are $k_{1}$ and $k_{2}$, respectively, with $k_{1} + k_{2} = 0$. Determine the slope $k$ of line $BC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A man started driving at a constant speed, from the site of a blast, the moment he heard the blast. He heard a second blast after some time. The second blast occurred exactly 30 mins after the first. He was 4950 meters from the site when he heard the second blast. (speed of sound = 330 m/s) How long did it take for him to hear the second blast?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Five men and five women stand in a circle in random order. The probability that every man stands next to at least one woman is $\tfrac m n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the work efficiency of each person is the same every day, $a$ people can make $c$ parts in $b$ days, then the number of days required for $bc$ people to make $a$ parts is.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On floor 0 of a weird-looking building, you enter an elevator that only has one button. You press the button twice and end up on floor 1. Thereafter, every time you press the button, you go up by one floor with probability \(\frac{X}{Y}\), where \(X\) is your current floor, and \(Y\) is the total number of times you have pressed the button thus far (not including the current one); otherwise, the elevator does nothing.

Between the third and the 100th press inclusive, what is the expected number of pairs of consecutive presses that both take you up a floor?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of pairs $(m,n)$ of positive integers with $1\le m<n\le 30$ such that there exists a real number $x$ satisfying
$$\sin(mx)+\sin(nx)=2.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mrs. Smith wanted to buy wears worth $500. She went to a boutique with the $500 but by the time she had picked out everything she liked, she realized that she would need two-fifths more money than she had. The shop owner gave her a certain discount. She still needed $95 more. What was the percentage of the discount given by the shop owner?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Bingley has 5 bracelets. Kelly gives Bingley a fourth of her 16 bracelets in sets of 3, meaning for every 3 bracelets Kelly gives, it counts as 1 bracelet. Then Bingley gives a third of his bracelets to his little sister. How many bracelets does Bingley have remaining?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sequence is given by: $a_1 = 1$, $a_{n+1} = \frac{a_n}{a_n + 2}$, (where $n \in \mathbb{N}^*$). If $b_{n+1} = (n - \lambda) \left( \frac{1}{a_n} + 1 \right)$, $b_1 = -\lambda$, and the sequence $\{b_n\}$ is monotonically increasing, then the range of values for the real number $\lambda$ is _______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ben "One Hunna Dolla" Franklin is flying a kite $KITE$ such that $IE$ is the perpendicular bisector of $KT$. Let $IE$ meet $KT$ at $R$. The midpoints of $KI, IT, TE, EK$ are $A, N, M, D,$ respectively. Given that $[MAKE] = 18$, $IT = 10$, $[RAIN] = 4$, find $[DIME]$.

Note: $[X]$ denotes the area of the figure $X$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an acute-angled triangle \( \triangle ABC \), find the solutions to the equation in \( x \):
\[ x^{2} \sin^{2} B + x \left( \cos^{2} C - \cos^{2} A + \sin^{2} B \right) + \sin^{2} A \cos^{2} C = 0 \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the triangle \( \triangle ABC \), the side lengths opposite to angles \( A \), \( B \), and \( C \) are \( a \), \( b \), and \( c \) respectively. If \( c - a \) equals the height \( h \) from angle \( A \) to side \( AC \), then find the value of \( \sin \frac{C - A}{2} + \cos \frac{C + A}{2} \). Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the heaviest weight that can be obtained using a combination of 2 lb and 4 lb, and 12 lb weights with a maximum of two weights used at a time.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct conjugacy classes of subgroups of $S_5$ that are isomorphic to the Klein four-group.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a_1, a_2, \ldots$ be an infinite sequence of integers such that $a_i$ divides $a_{i+1}$ for all $i \geq 1$. Let $b_i$ be the remainder when $a_i$ is divided by $210$. What is the maximal number of distinct terms in the sequence $b_1, b_2, \ldots$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given proposition P: ∃x₀ ∈ R, x₀² + 2x₀ + 3 ≤ 0, what is the negation of proposition P (∉P)? Express your answer as a logical statement in the format of a universal proposition.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a$ and $b$ be nonnegative real numbers such that
\[\sin (ax + b) = \sin 29x\]for all integers $x.$  Find the smallest possible value of $a.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equation \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), where \(a, b\), and \(c\) are positive integers, find the number of different ordered triples \((a, b, c)\) such that \(a+2b+c \leq 40\). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Vasya wrote a note on a piece of paper, folded it in quarters, wrote "MAME" on top, then unfolded the note, added something else, folded it again along the crease lines randomly (not necessarily as before), and left it on the table with a random side facing up. Find the probability that the "MAME" inscription is still on top.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An equilateral triangle is placed on top of a square with each side of the square equal to one side of the triangle, forming a pentagon. What percent of the area of the pentagon is the area of the equilateral triangle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
S is the set of all $(a, b, c, d, e, f)$ where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Find the largest $k$ which divides $abcdef$ for all members of $S$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a school there are  $1200$  students. Each student is part of exactly  $k$  clubs. For any  $23$  students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of  $k$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a certain company, there are full-time and part-time employees. Among the full-time employees, 60% are men and 40% are women. Among the part-time employees, 45% are men and 55% are women. During the annual company picnic, 20% of the full-time men, 30% of the full-time women, 25% of the part-time men, and 35% of the part-time women attended. If full-time employees make up 70% of the total workforce, what percentage of all employees attended the annual company picnic?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a circle of radius \( R \) centered at the origin on the Cartesian plane. Specify at least one value of \( R \) for which there are exactly 32 integer points (a point is called an integer point if both its x-coordinate and y-coordinate are integers) lying on this circle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each hotel room can accommodate no more than 3 people. The hotel manager knows that a group of 100 football fans, who support three different teams, will soon arrive. A room can only house either men or women; and fans of different teams cannot be housed together. How many rooms need to be booked to accommodate all the fans?
Please reason step by step, and put your final answer within \boxed{}.


Context:
袋内有 8 个白球和 2 个红球，每次从中随机取出一个球，然后放回一个白球，则第 4 次恰好取完所有红球的概率为 $\qquad$.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{}.


Context:
从正整数列1,2,3,4，···中划去3及4的倍数，但其中凡是5的倍数均保留（例如15,20,60,120均不划去），划完后，剩下的数依原有次序组成一个数列：$a _ { 1 } = 1,a _ { 2 } =2,a _ { 3 } = 5 ,a _ { 4 } = 7, \cdots$.试求$a _ { 2 0 0 3 } .$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the mapping function f: A → B defined as f(n) = 2n^2, where A and B are subsets of the natural numbers, find the original element n in set A that maps to 20 in set B. Express your answer as a single natural number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are \( N \) lockers, labeled from 1 to \( N \), placed in clockwise order around a circular hallway. Initially, all lockers are open. Ansoon starts at the first locker and always moves clockwise. When she is at locker \( n \) and there are more than \( n \) open lockers, she keeps locker \( n \) open and closes the next \( n \) open lockers, then repeats the process with the next open locker. If she is at locker \( n \) and there are at most \( n \) lockers still open, she keeps locker \( n \) open and closes all other lockers. She continues this process until only one locker is left open. What is the smallest integer \( N > 2021 \) such that the last open locker is locker 1?
Please reason step by step, and put your final answer within \boxed{}.


Context:
## Task 2 - 191242

Let $M$ be the set of all square areas $Q$ that lie in a given plane $\epsilon$, have a given point $Z$ of the plane $\epsilon$ as their center, and have a given segment length $a$ as their side length.

For arbitrary square areas $Q, Q^{\prime}$ from this set $M$, let $u\left(Q \cup Q^{\prime}\right)$ denote the perimeter of the polygonal area that results from the intersection of the square areas $Q$ and $Q^{\prime}$.

Investigate whether there are square areas $Q, Q^{\prime}$ in $M$ with the smallest possible $u\left(Q \cap Q^{\prime}\right)$.

If this is the case, determine (in dependence on $a$) this smallest possible value of $u\left(Q \cap Q^{\prime}\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Problem 40. Find the best positive real constant $k$ such that the following inequality holds for all positive real numbers $a, b$ and $c$
$$\frac{(a+b)(b+c)(c+a)}{a b c}+\frac{k(a b+b c+c a)}{a^{2}+b^{2}+c^{2}} \geq 8+k$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Xiaoqing is preparing a dish called "Xiang Qing Fried Eggs," which requires 7 steps with the following times:
- Wash scallions, chop scallions: 1 minute
- Beat eggs: half a minute
- Stir egg mixture and scallions: 1 minute
- Wash pan: half a minute
- Heat pan: half a minute
- Heat oil: half a minute
- Cook dish: 2 minutes

The minimum time to complete this dish is $\qquad$ minutes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the sequence: $$\lim_{n\to \infty} \sqrt[n]{\prod_{k=3}^{n+2}\left(1-\frac{3}{k}\right)^{k}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Rachel and Steven play games of chess. If either wins two consecutive games, they are declared the champion. The probability that Rachel will win any given game is 0.6, the probability that Steven will win any given game is 0.3, and the probability that any given game is drawn is 0.1. Find the value of \(1000P\), where \(P\) is the probability that neither is the champion after at most three games.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three different numbers are chosen such that when each of the numbers is added to the average of the remaining two, the numbers 65, 69, and 76 result. Find the average of the three original numbers. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the circle \( x^{2} + y^{2} = k^{2} \) at least covers one maximum point and one minimum point of the function \( f(x) = \sqrt{3} \sin \frac{\pi x}{k} \), find the range of the real number \( k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the numerator of \(\frac{1010\overbrace{11 \ldots 11}^{2011 \text{ ones}}0101}{1100\underbrace{11 \ldots 11}_{2011\text{ ones}}0011}\) when reduced.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $a_0 = 1$  and define the sequence  $\{a_n\}$  by \[a_{n+1} = \frac{\sqrt{3}a_n - 1}{a_n + \sqrt{3}}.\] If  $a_{2017}$  can be expressed in the form  $a+b\sqrt{c}$  in simplest radical form, compute  $a+b+c$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a square pyramid \( S-ABCD \) with a height of \( \frac{\sqrt{2}}{4} \), where the base is a square with side length 1 and points \( S, A, B, C, D \) all lie on the surface of a sphere with radius 1. Find the maximum length of the slant edge \( SA \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
求最大的整数 k\leq 2023，满足如下条件：当 A 将 \{1,2,\cdots,2023\} 中的任意 k 个数染红时，B 可将余下若干个数染蓝，使得红数之和等于蓝数之和.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function f(x) = sinx, whose graph intersects with the line kx - y - kπ = 0 (k > 0) at exactly three points. The x-coordinates of these points, in ascending order, are x₁, x₂, and x₃. Calculate the value of $$\frac{tan(x_{2} - x_{3})}{x_{1} - x_{2}}$$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
一张台球桌形状是正六边形 $A B C D E F$ ，一个球从 $A B$ 的中点 $P$ 击出，击中 $B C$ 边上的某点 $Q$ ，并且依次碰击 $C D 、 D E 、 E F 、 F A$ 各边，最后击中 $A B$ 边上的某一点. 设 $\angle B P Q=\theta$ ， $\theta$ 的范围为$\theta\in[arc\mathrm{tan}\frac{a\sqrt{3}}{b}, arc\mathrm{tan}\frac{c\sqrt{3}}{d}]$,求$a+b+c+d$。
提示：利用入射角等于反射角的原理。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two unit squares $S_{1}$ and $S_{2}$ have horizontal and vertical sides. Let $x$ be the minimum distance between a point in $S_{1}$ and a point in $S_{2}$, and let $y$ be the maximum distance between a point in $S_{1}$ and a point in $S_{2}$. Given that $x=5$, the difference between the maximum and minimum possible values for $y$ can be written as $a+b \sqrt{c}$, where $a, b$, and $c$ are integers and $c$ is positive and square-free. Find $100 a+10 b+c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
9. (10 points) The area of rectangle $A B C D$ is 2011 square centimeters. The vertex $F$ of trapezoid $A F G E$ is on $B C$, and $D$ is the midpoint of the non-parallel side $E G$. Find the area of trapezoid $A F G E$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the number of idempotent polynomials in the quotient ring $\Bbb R[x_1,x_2,\dots,x_n]/(x_1^2-x_1,x_2^2-x_2,\dots,x_n^2-x_n)$ that satisfy the relation $p^2=p$ modulo $x_i$ and $x_i-1$ for all $i$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
PQRST is a regular pentagon. The point U lies on ST such that ∠QPU is a right angle. Find the ratio of the interior angles in triangle PUT. Express your answer as a ratio of integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In base $R_1$, the expanded fraction $F_1$ becomes $0.373737\ldots$, and the expanded fraction $F_2$ becomes $0.737373\ldots$. In base $R_2$, fraction $F_1$, when expanded, becomes $0.252525\ldots$, while fraction $F_2$ becomes $0.525252\ldots$. Find the sum of $R_1$ and $R_2$, each written in base ten.
Please reason step by step, and put your final answer within \boxed{}.


Context:
David draws a $2 \times 2$ grid of squares in chalk on the sidewalk outside NIMO HQ. He then draws one arrow in each square, each pointing in one of the four cardinal directions (north, south, east, west) parallel to the sides of the grid. In how many ways can David draw his arrows such that no two of the arrows are pointing at each other?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $\frac{8}{19}$ of the product of the largest two elements of a positive integer set is not greater than the sum of the other elements, what is the minimum possible value of the largest number in the set?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The positional relationship between the circle $x^2+y^2=4$ and the circle $(x+3)^2+(y-4)^2=16$ is ____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, the median from vertex $A$ is perpendicular to the angle bisector of angle $B$, and the median from vertex $B$ is perpendicular to the angle bisector of angle $A$. It is known that side $AB = 1$. Find the perimeter of triangle $ABC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A traffic light runs repeatedly through the following cycle: green for \(30\) seconds, then yellow for \(3\) seconds, and then red for \(30\) seconds. Leah picks a random three-second time interval to watch the light. What is the probability, expressed as a fraction in simplest form \(\frac{k}{m}\), that the color changes while she is watching? Please provide the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many elements of order $p$ exist in the general linear group $GL_{2}(\Bbb Z_{p})$, where $p$ is a prime number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $H = H^1([0,1])$ and let $Tu = u(1)$. Let $\langle \cdot,\cdot \rangle_H$ denote the standard inner product in $H$. By the Riesz representation theorem, there exists a unique $v\in H$ such that $Tu = \langle u, v \rangle_H$ for all $u\in H$. Find $v$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an explicit formula for a function that is exactly equivalent to the power series $$\sum_{n=1}^\infty\frac{(-1)^n}{2n+1}x^{2n}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=\sin (2x+ \frac {π}{3})$, for any $x\_1$, $x\_2$, $x\_3$, and $0\leqslant x\_1 < x\_2 < x\_3\leqslant π$, the equation $|f(x\_1)-f(x\_2)|+|f(x\_2)-f(x\_3)|\leqslant m$ holds true. Find the minimum value of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The base of the pyramid SABCD is the parallelogram ABCD. Which figure is formed by the section of this pyramid with the plane ABM, where M is a point on the edge SC?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the sines of the internal angles of $\triangle ABC$ form an arithmetic sequence, what is the minimum value of $\cos C$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABC$ be a right triangle where $\measuredangle A = 90^\circ$ and $M\in (AB)$ such that $\frac{AM}{MB}=3\sqrt{3}-4$. It is known that the symmetric point of $M$with respect to the line $GI$ lies on $AC$. Find the measure of $\measuredangle B$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a mathematical test for grade 12 students in a certain city, it was found that the scores (X) approximately followed a normal distribution N(84, σ²) after a sample analysis. Also, P(78 < X ≤ 84) = 0.3. If there were 400 participants from a particular school, estimate the number of students from this school who scored no less than 90. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mac loves the shape of quarters and is willing to trade money for them, even if he loses value. He tells his brother that he will trade him 4 dimes and 2 pennies for a quarter with a 5% probability or 9 nickels and 1 penny for a quarter with a 10% probability, or he will trade 1 half-dollar coin and 3 cents for a quarter with an 85% probability. Each trade with the given probabilities is repeated 20 times for each option (60 trades in total). How many dollars did Mac lose in these transactions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Russom wants to give some friends envelopes with bus tickets and subway tickets in them. He has 18 bus tickets and some subway tickets to split equally among the envelopes, and wants no tickets left over. The greatest number of envelopes Russom can make is 6. How many subway tickets does he have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
All factors of a positive integer a are multiplied and the product obtained is a ^ 5 . If a is greater than 1, how many unique factors does the expression have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of prime numbers $p$ between $100$ and $200$ for which the congruence equation $x^{11} + y^{16} \equiv 2013 \pmod{p}$ has a solution in integers $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABC$ be an equilateral triangle of side length 6 inscribed in a circle $\omega$. Let $A_{1}, A_{2}$ be the points (distinct from $A$) where the lines through $A$ passing through the two trisection points of $BC$ meet $\omega$. Define $B_{1}, B_{2}, C_{1}, C_{2}$ similarly. Given that $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}$ appear on $\omega$ in that order, find the area of hexagon $A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many roots does the equation

$$
\overbrace{f(f(\ldots f}^{10 \text{ times }}(x) \ldots))+\frac{1}{2}=0
$$

where $f(x)=|x|-1$ have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of strictly increasing sequences of nonnegative integers with the following properties: - The first term is 0 and the last term is 12. In particular, the sequence has at least two terms. - Among any two consecutive terms, exactly one of them is even.
Please reason step by step, and put your final answer within \boxed{}.


Context:
在平面直角坐标系 $x O y$ 中, 设椭圆 $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}-9}=1(a>3)$.
若直线 $l: \frac{x}{a}+\frac{y}{a-3}=1$ 与椭圆 $C$ 交于 $A, B$ 两点, 其中对任意大于 3 的实数 $a$, 以 $A B$ 为直径的圆过定点, 求定点坐标.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the weights (in kilograms) of 4 athletes are all integers, and they weighed themselves in pairs for a total of 5 times, obtaining weights of 99, 113, 125, 130, 144 kilograms respectively, and there are two athletes who did not weigh together, determine the weight of the heavier one among these two athletes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest \( n \) such that the sum of the fourth powers of any \( n \) prime numbers greater than 10 is divisible by \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( A, B, C \) be positive integers such that the number \( 1212017ABC \) is divisible by 45. Find the difference between the largest and the smallest possible values of the two-digit number \( AB \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Milly chooses a positive integer  $n$  and then Uriel colors each integer between  $1$  and  $n$  inclusive red or blue. Then Milly chooses four numbers  $a, b, c, d$  of the same color (there may be repeated numbers). If  $a+b+c= d$  then Milly wins. Determine the smallest  $n$  Milly can choose to ensure victory, no matter how Uriel colors.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^3}$ is a real number and $|\alpha - \beta| = 6$. Find $|\alpha|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a function $f(x)$ that satisfies the following conditions:
- $f(x)$ is twice differentiable on $[0,1]$.
- $f(0) = 2$, $f'(0) = -2$, $f(1) = 1$.
- The differential equation $f''(x) + f(x) f'(x) = 0$ is satisfied for all $x \in (0,1)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( n \) be the largest integer for which \( 14n \) has exactly 100 digits. Counting from right to left, what is the 68th digit of \( n \)? Express your answer as a single digit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of triples $0 \leq k, m, n \leq 100$ of integers such that $$ 2^{m} n-2^{n} m=2^{k} $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Exercise 18. We say that a set $A$ of polynomials with real coefficients is beautiful if, whenever $P$ and $Q$ are two distinct elements of $A$, there exist positive integers $a_{1}>\cdots>a_{2020}$ such that

$$
P Q=\sum_{i=1}^{2020} i X^{a_{i}}
$$

What is the maximal cardinality of a beautiful set?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest positive integer $n$ ($n \ge 3$), such that there exists a convex $n$-gon where the tangent of each interior angle is an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 3 different pairs of shoes in a shoe cabinet. If one shoe is picked at random from the left shoe set of 6 shoes, and then another shoe is picked at random from the right shoe set of 6 shoes, calculate the probability that the two shoes form a pair.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$36 S$ is a subset of $\{1, 2, \cdots, 1989\}$, and any two numbers in $S$ cannot have a difference of 4 or 7. What is the maximum number of elements that $S$ can have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangular box has interior dimensions 6-inches by 5-inches by 10-inches. The box is filled with as many solid 3-inch cubes as possible, with all of the cubes entirely inside the rectangular box. What percent of the volume of the box is taken up by the cubes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=\frac{-4x+5}{x+1}$ and $g(x)=a\sin(\frac{\pi}{3}x)-2a+2(a>0)$, if there exist $x_{1}$ and $x_{2}\in \left[0,2\right]$ such that $f(x_{1})=g(x_{2})$, then the range of real number $a$ is ______
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose I have a connected graph with $n$ vertices and start at an arbitrary vertex $u$. I want to visit every vertex of the graph, and I don't care about returning to where I started. I can visit a node multiple times if needed. What is the worst possible length of the shortest path, as a function of $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the interquartile range (Q3 - Q1) of the following list of numbers: 42, 24, 30, 28, 26, 19, 33, 35, 47, 55, 61, 27, 39, 46, 52, 20, 22, 37, 48, 60, 50, 44, 31, 54, and 58.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A random integer $x \in \{1, 2, \cdots, 2016\}$ is equally likely to be generated. Determine the probability that the sum of the digits of $x$ in binary form does not exceed 8.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Xiaoming's father departs from home to go shopping at the supermarket. If he first rides a bicycle for 12 minutes and then walks for 20 minutes, he can reach the supermarket; if he first rides a bicycle for 8 minutes and then walks for 36 minutes, he can also reach the supermarket. How many minutes will it take to reach the supermarket if he first rides a bicycle for 2 minutes and then walks?
Please reason step by step, and put your final answer within \boxed{}.


Context:
求所有可用十进制表示$\overline {13xy45z}$，且能被792整除的正整数，其中，$x,y,z$为未知数.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the integers $1, 2, \ldots, n$ ($n \geq 2, n \in \mathbb{N^*}$), there are $n!$ different permutations $a_1, a_2, \ldots, a_n$. How many distinct permutations satisfy the condition that for all $k = 1, 2, \ldots, n$, $a_k \geq k-2$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Figure \(ABCD\) is a trapezoid with \(AB \parallel DC\), \(AB = 5\), \(BC = 3 \sqrt{2}\), \(\angle BCD = 45^\circ\), and \(\angle CDA = 60^\circ\). Find the length of \(DC\). The original answer is in the form \(m + \sqrt{n}\). Please give the value of \(m + n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many of the first 1000 positive integers can be expressed in the form $\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor$, where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of $50$ miles per hour, and Joe can run at a speed of $10$ miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. If the number of seconds he has to spare when he gets to the tunnel entrance is $x$, find the value of $10x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sells a bicycle to B and makes a profit of 20%. B sells the same bicycle to C at a profit of 25%. C then sells the bicycle to D at a profit of P%, where P% is an unknown profit percentage. If the final selling price of the bicycle after the transaction between C and D was Rs. 225, find out the cost price of the bicycle for A.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute
\[\frac{1}{2^3 - 2} + \frac{1}{3^3 - 3} + \frac{1}{4^3 - 4} + \dots + \frac{1}{100^3 - 100}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Car A and Car B start from points A and B, respectively, at the same time (Car A starts from point A), and travel towards each other. The speeds of Car A and Car B are 40 km/h and 50 km/h, respectively. The distance between points A and B is 900 km. When Car A reaches point B, it immediately turns around and heads back to point A at a speed of 50 km/h. When Car B reaches point A, it immediately turns around and heads back to point B at a speed of 40 km/h. When Car A reaches point A, it immediately turns around and heads back to point B, resuming a speed of 40 km/h. When Car B reaches point B, it immediately turns around and heads back to point A, resuming a speed of 50 km/h. This pattern continues, with both cars traveling back and forth between points A and B, and their speeds alternating between 40 km/h and 50 km/h.

How many kilometers has Car A traveled in total by the time the 2016th meeting occurs?
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. Given the number $5300 \ldots 0035$ (100 zeros). It is required to replace some two zeros with non-zero digits so that after the replacement, the resulting number is divisible by 495. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the largest real number $\theta$ less than $\pi$ (i.e. $\theta<\pi$ ) such that $\prod_{k=0}^{10} \cos \left(2^{k} \theta\right) \neq 0$ and $\prod_{k=0}^{10}\left(1+\frac{1}{\cos \left(2^{k} \theta\right)}\right)=1 ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation \(2 x^{3} + 24 x = 3 - 12 x^{2}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x) = x^2e^x + \ln t - a$, if for any $t \in [1,e]$, $f(x)$ always has a unique zero in the interval $[-1,1]$, then the range of the real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( S = \{1, 2, \ldots, 2005\} \). If any set of \( n \) pairwise co-prime numbers in \( S \) always contains at least one prime number, what is the minimum value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Sasha wrote down numbers from one to one hundred, and Misha erased some of them. Among the remaining numbers, 20 contain the digit one, 19 contain the digit two, and 30 contain neither one nor two. How many numbers did Misha erase?
Please reason step by step, and put your final answer within \boxed{}.