Context:
The sum of all real roots of the equation $|x^2-3x+2|+|x^2+2x-3|=11$ is .
Please reason step by step, and put your final answer within \boxed{}.


Context:
At a bus station, there are three buses departing to a school between 6:30 AM and 7:30 AM each day. The ticket prices for the buses are the same, but the comfort levels vary. Xiao Jie, a student, observes before boarding. When the first bus arrives, he does not get on but carefully observes its comfort level. If the comfort level of the second bus is better than the first, he will board the second bus; otherwise, he will take the third bus. Given that the comfort levels of the buses can be classified as high, medium, and low, the probability of Xiao Jie boarding a bus with a high comfort level is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
在$\triangle ABC$中，$\angle C=90^{\circ}$，$\angle B=30^{\circ}$，$AC=2$，$M$是$AB$的中点。将$\triangle ACM$沿$CM$折起，使$A、B$两点间的距离为$2\sqrt{2}$，此时三棱锥$A-BCM$的体积等于$\frac{m\sqrt{n}}{k}$的形式，请给出$m + n+k$的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Task 6. (30 points) A regular triangular prism $A B C A_{1} B_{1} C_{1} \mathrm{c}$ is inscribed in a sphere with base $A B C$ and lateral edges $A A_{1}, B B_{1}, C C_{1}$. Segment $C D$ is the diameter of this sphere, point $K$ and $L$ are the midpoints of edge $A A_{1}$ and $A B$ respectively. Find the volume of the prism if $D L=\sqrt{2}, D K=\sqrt{3}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define $a_k = (k^2 + 1)k!$ and $b_k = a_1 + a_2 + a_3 + \cdots + a_k$. Let \[\frac{a_{100}}{b_{100}} = \frac{m}{n}\] where $m$ and $n$ are relatively prime natural numbers. Find $n - m$.
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Context:
A dog is tied to a tree by a long nylon cord. There is a triangular-shaped obstacle nearby, with its vertices situated at points A, B, and C, where point A lies on the north side of the tree, point B is 10 feet directly east of point A, and point C is 20 feet diagonally southeast of point A. The dog starts at point A and runs around the triangular obstacle to finally reach point D, the due south side of the tree. During its run, the nylon cord is extended to its full length at all times without interfering with the obstacle. If the dog traveled approximately 60 feet, what was the approximate length of the nylon cord, L, in feet?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many values of $\theta$ in the interval $0<\theta\le 2\pi$ satisfy \[1-3\sin\theta+5\cos3\theta = 0?\]
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Context:
Given that $x_{n+2}=\dfrac{20x_{n+1}}{14x_n}$, $x_0=25$, $x_1=11$, it follows that $\sum_{n=0}^\infty\dfrac{x_{3n}}{2^n}=\dfrac{p}{q}$ for some positive integers $p$, $q$ with $\text{GCD}(p,q)=1$. Find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be a natural number divisible by $4$. Determine the number of bijections $f$ on the set $\{1,2,...,n\}$ such that $f (j )+f^{-1}(j ) = n+1$ for $j = 1,..., n.$
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Context:
Calculate the probability $P\left(X_{(2)}+X_{(3)}\geq1\right)$ for the order statistics $X_{(1)},X_{(2)},X_{(3)}$ of three independent random variables $X_1,X_2,X_3$ with uniform distribution in $(0,1)$.
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Context:
We colored the faces of a regular dodecahedron with red, blue, yellow, and green so that any two adjacent faces have different colors. How many edges of the dodecahedron have one face colored blue and the other green?
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Context:
Suppose we have a group $G$ of order $pq^3$ with a normal Sylow $q$-subgroup $H$. If $G$ has a subgroup $P$ of order $pq^2$ and a Sylow $p$-subgroup $R$ of $P$ with $|N_P(R)|=p$, can we find the order of $N_G(R)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Use Horner's method to find the value of V_2 for the polynomial function f(x) = 2x^5 - 3x^3 + 2x^2 - x + 5 when x = 2. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the angle $\alpha$ for which the vectors $(\rho^2 + i)^{16}$ and $(\rho^2 - i)^{16}$ on the complex plane are aligned with the imaginary axis.
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Context:
In trapezoid \( ABCD \), point \( X \) is taken on the base \( BC \) such that segments \( XA \) and \( XD \) divide the trapezoid into three similar but pairwise unequal, non-isosceles triangles. The side \( AB \) has a length of 5. Find \( XC \cdot BX \).
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Context:
Given circle $O$: $x^2+y^2=1$, and line $l$: $2x-y-4=0$. If point $A$ is on line $l$ such that there exists a point $B$ on circle $O$ and $\angle OAB=30^{\circ}$ (where $O$ is the origin), then the range of the $x$-coordinate of point $A$ is...
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Context:
How many such nine-digit positive integers can Ricardo make if he wants to arrange three 1s, three 2s, two 3s, and one 4 with the properties that there is at least one 1 before the first 2, at least one 2 before the first 3, and at least one 3 before the 4, and no digit 2 can be next to another 2?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of integer solutions to the equation $x^6 - 15x^5 - 10x^3 + 30x + 720 = 0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Divide the numbers 1 through 10 into three groups such that the difference between any two numbers within the same group does not appear in that group. If one of the groups is {2, 5, 9}, then the sum of all the numbers in the group that contains 10 is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangle with integer side-lengths is divided into four smaller rectangles. The perimeters of the largest and smallest of these smaller rectangles are 28 cm and 12 cm. What is a possible area of the original rectangle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
给定整数 $n(n \geq 2)$. 对于一个 $2 n$ 元有序数组 $$ T=\left(a_{1}, b_{1}, a_{2}, b_{2}, \cdots, a_{n}, b_{n}\right), $$ 若 $T$ 的每个分量均为 0 或 1 ，且对任意 $p, q(1 \leq p<q \leq n)$ ，均有 $\left(a_{p}, b_{p}, b_{q}\right) \neq(1,0,1)$ 且 $\left(a_{q}, b_{q}, a_{p}\right) \neq(1,0,0)$, 则称 $T$ 为 "有趣数组". 求有趣数组的个数。请提供当 n = 2 时的答案。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Create a fraction set $A_1$ with numerators being the integers in the interval $(0, m)$ ($m > 1$, $m \in \mathbb{N}$) and the denominator being $m$, with the sum of all elements denoted by $a_1$. Construct another fraction set $A_2$ with numerators being the integers in the interval $(0, m^2)$ ($m > 1$, $m \in \mathbb{N}$) and the denominator being $m^2$, such that $A_2$ contains no elements from set $A_1$, with the sum of all elements denoted by $a_2$. Continuing in this manner, create the fraction set $A_n$ using the integers in the interval $(0, m^n)$ as numerators ($m > 1$, $m \in \mathbb{N}$) and $m^n$ as the denominator, ensuring that $A_n$ includes no elements from $A_1$, $A_2$, ..., $A_{n-1}$, with the sum of all elements denoted by $a_n$. Find the sum $a_1 + a_2 + ... + a_n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In Wonderland, some cities are connected by roads so that one can reach any city from any other city, possibly by passing through other cities. No two cities are connected by more than one direct road.

A road is called "strange" if closing it would prevent travel between some pair of cities. It is known that there are 45 roads in total in the country, out of which 42 roads are strange. How many cities can there be in the country? If there are multiple correct answers, list them separated by commas or semicolons.
Please reason step by step, and put your final answer within \boxed{}.


Context:
You are playing a game called "Hovse." Initially, you have the number $0$ on a blackboard. At any moment, if the number $x$ is written on the board, you can perform one of the following operations:

- Replace $x$ with $3x + 1$
- Replace $x$ with $9x + 1$
- Replace $x$ with $27x + 3$
- Replace $x$ with $\left \lfloor \frac{x}{3} \right \rfloor$

You are not allowed to write a number greater than $2017$ on the board. How many positive numbers can you make with the game of "Hovse?"
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $\ln \left(1+x+x^{2}+x^{3}\right)$ 的考克劳林展开式为 $\sum_{n=0}^{\infty} a_{n} x^{n}$, 则 $\left[\left|a_{2024}\right|^{-1}\right]=$ $\qquad$
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Context:
Siva has the following expression, which is missing operations:  $$ \frac12 \,\, \_  \,\,\frac14 \,\, \_ \,\, \frac18 \,\, \_  \,\,\frac{1}{16} \,\, \_  \,\,\frac{1}{32}. $$   For each blank, he flips a fair coin: if it comes up heads, he fills it with a plus, and if it comes up tails, he fills it with a minus. Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coinflips and operations. If the probability that the positive difference between his computed values is greater than  $\frac12$  can be expressed as  $\frac{a}{b}$  for relatively prime positive integers  $a$ ,  $b$ , then find  $a + b$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
A wheel has a certain radius. The distance covered by the wheel in making 100 revolutions is 91.77 meters. What is the radius of the wheel in centimeters?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A bus departed from city $M$ to city $N$ at a speed of 40 km/h. After a quarter of an hour, it encountered a car traveling from city $N$. This car reached city $M$, waited for 15 minutes, and then returned to city $N$, overtaking the bus 20 km from city $N$. Find the distance between cities $M$ and $N$ if the car's speed is 50 km/h.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $I, T, E, S$  be distinct positive integers such that the product  $ITEST  =  2006$ .  What is the largest possible value of the sum  $I  +  T  +  E  +  S  +  T  +  2006$ ?
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Context:
What is the minimum percentage increase in the mean of set G { -7, -5, -4, -1, 0, 6, 9, 11 } if its four smallest elements are replaced with four different prime numbers?
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Context:
Find the number of positive integers $k \leq 2018$ for which there exist integers $m$ and $n$ such that $k = 2^m + 2^n$. For example, $64 = 2^5 + 2^5$, $65 = 2^0 + 2^6$, and $66 = 2^1 + 2^6$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+d e f$ 是奇数的概率为 $\qquad$.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
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Context:
Positive integers  $a_1, a_2, \ldots, a_{101}$  are such that  $a_i+1$  is divisible by  $a_{i+1}$  for all  $1 \le i \le 101$ , where  $a_{102} = a_1$ . What is the largest possible value of  $\max(a_1, a_2, \ldots, a_{101})$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sequence is defined by $A_{0}=0, A_{1}=1, A_{2}=2$, and, for integers $n \geq 3$, $$A_{n}=\frac{A_{n-1}+A_{n-2}+A_{n-3}}{3}+\frac{1}{n^{4}-n^{2}}$$ Compute $\lim _{N \rightarrow \infty} A_{N}$.
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Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. Given that $ac = \frac{1}{4}b^2$, $\sin A + \sin C = t \sin B$, and angle $B$ is acute, the range of the real number $t$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left( \frac{\sqrt{6}}{2}, \sqrt{2} \right)}


Context:
A cat broke into a wine store at night. It jumped onto a shelf where wine bottles were arranged in a long row - the first third of bottles were priced at 160 CZK each, the next third were priced at 130 CZK each, and the last third at 100 CZK each. First, the cat knocked over a bottle priced at 160 CZK, which was at the very beginning of the row, and then continued knocking over one bottle after another without skipping any. It knocked down a total of 25 bottles before it stopped. In the morning, the owner regretted that the cat did not start its rampage from the other end of the shelf. If it had broken the same number of bottles from the other end, the damage would have been 660 CZK less. How many bottles were originally on the shelf?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Regular pentagon $ABCDE$ and regular hexagon $AEFGHI$ are drawn on opposite sides of line segment $AE$ such that they are coplanar. What is the degree measure of exterior angle $DEF$? [asy]
draw((0,2.5)--(0,7.5)--(4,10)--(8,7.5)--(8,2.5)--(4,0)--cycle,linewidth(1));
draw((8,2.5)--(11.5,-1)--(9,-5)--(5,-4.5)--(4,0),linewidth(1));
dot((0,2.5)); dot((0,7.5)); dot ((4,10)); dot((8,7.5)); dot((8,2.5)); dot((4,0));

label("I",(0,2.5),W); label("H",(0,7.5),W); label("G",(4,10),N);

label("F",(8,7.5),E); label("E",(8,2.5),NW); label("A",(4,0),SW);
dot((11.5,-1)); dot((9,-5)); dot((5,-4.5));
label("D",(11.5,-1),E); label("C",(9,-5),SE); label("B",(5,-4.5),SW);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
John decides to buy some birds. He received $50 each from his 4 grandparents. In the pet store, parrots cost $30, pigeons cost $20, and canaries come at $15 each. Assuming all birds have 2 wings, and he spent all his money, how many wings did all the birds he bought have, if he bought at least one of each type of bird?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangle \( ABC \) has \( AB=24 \), \( AC=26 \), and \( BC=22 \). Points \( D \) and \( E \) are located on \( \overline{AB} \) and \( \overline{AC} \), respectively, so that \( \overline{DE} \) is parallel to \( \overline{BC} \) and contains the center of the inscribed circle of triangle \( ABC \). Calculate \( DE \) and express it in the simplest form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A volunteer organization is recruiting new members throughout the year. In the fall, they manage to increase their number by 5%. In the winter, membership grows by 14%. By the spring, however, membership falls by 19%. Finally, in the summer, membership decreases further by 7%. What is the overall percentage change in membership from the beginning of fall to the end of summer?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a cycling competition with  $14$  stages, one each day, and  $100$  participants, a competitor was characterized by finishing  $93^{\text{rd}}$  each day.What is the best place he could have finished in the overall standings? (Overall standings take into account the total cycling time over all stages.)
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Context:
Given the function $f(x) = |2^{x} - 2|, (x \in (-1, 2))$, find the range of the function $y = f(x - 1)$.
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Context:
A cyclist rides a bicycle 7 km at an average speed of 10 km/hr, then travels 10 km at an average speed of 7 km/hr. After that, the cyclist covers another 5 km at an average speed of 12 km/hr, and finally rides 8 km at an average speed of 6 km/hr. What is the average speed for the entire trip?
Please reason step by step, and put your final answer within \boxed{}.


Context:
25. Arrange the numbers $0,0,1,1,2,2,4,4$ in a row (0 can be placed first), so that no four consecutive numbers form 2014. There are $\qquad$ ways.
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Context:
What is the sum of $x + y$ if the sequence $3, ~8, ~13, \ldots, ~x, ~y, ~33$ forms an arithmetic sequence?
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Context:
In the context of the Joint Entrance Examination (JEE, a national level entrance test), determine the largest non-negative integer \( a \) for which the following limit holds:

\[
\lim_{x \to 1} \left( \dfrac{-ax + \sin(x-1) + a} { x + \sin(x-1) -1 } \right)^{\dfrac{1-x}{1-\sqrt x} } = \dfrac 1 4
\]
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Context:
Consider two fields with rectangular shapes rather than squares. The area of the first rectangular field is 1 hectare, and the second field has 1% greater width than the first field with a length that is 5% less than the length of the first field. Determine the difference in area between these two fields, given that the width of the first field is proportional to the square root of its length.
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Context:
Determine the number of pairs \((a, b)\) of natural numbers such that \(b\) is a 3-digit number, \(a+1\) divides \(b-1\), and \(b\) divides \(a^2 + a + 2\).
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Context:
The Crackhams were supposed to make their first stop in Baglmintster and spend the night at a family friend's house. This family friend in turn was to leave home at the same time as them and stop in London at the Crackhams' house. Both the Crackhams and the friend were traveling on the same road, looking out for each other, and they met 40 km from Baglmintster. That same evening, George came up with the following small puzzle:

- I realized that if, upon arrival at our destinations, each of our cars immediately turned back and set off in the opposite direction, we would meet 48 km from London.

If George is correct, what is the distance from London to Baglmintster?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of the discretization step $h$ for which the Euler explicit method is stable when applied to the differential equation $y'(x) + \frac{1}{4}y(x) = x$ with the initial condition $y(1) = 1$. Provide your answer as a specific value of $h$.
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Context:
Given a tetrahedron with vertices at coordinates \((3,1,5), (-7,0,13), (27,5,-13), (-17,4,21)\), find the minimum distance required to visit each vertex exactly once without crossing through the interior of the tetrahedron. Round your answer to the nearest hundredth.
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Context:
Six boys stood equally spaced on a circle of radius 40 feet. Each boy walked to all of the other non-adjacent persons on the circle, shook their hands and then returned to his original spot on the circle before the next boy started his trip to shake hands with all of the other non-adjacent boys on the circle. After all six boys had done this, what is the least distance in feet that could have been traveled? Express your answer in simplest radical form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. Natural numbers divisible by 3 were painted in two colors: red and blue, such that the sum of a blue and a red number is red, and the product of a blue and a red number is blue. In how many ways can the numbers be colored so that the number 546 is blue?
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Context:
Calculate the minimal polynomial of the matrix $E(i,j) = I_n + \alpha I_{(i,j)}$, where $I_n$ is the identity matrix of order $n$, $I_{(i,j)}$ is the matrix with 1 in the $(i,j)$ position and 0 elsewhere ($i \neq j$), and $\alpha \in \mathbb{K}$ with $\mathbb{K}$ a field, $\alpha \neq 0$. Provide your answer in terms of $n$, $i$, $j$, and $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the rectangular coordinate plane $xOy$, the angle $\angle MON$ is formed by the ray $y = x$ $(x \geq 0)$ and the positive x-axis. Points $A(6,5)$ and $B(10,2)$ are fixed points within $\angle MON$. Points $P$ and $Q$ are movable points on the two sides of $\angle MON$ respectively. Determine the minimum value of the perimeter $l$ of the quadrilateral $ABQP$. \( \boxed{\text{Minimum Perimeter}} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
On $2020 \times 2021$ squares, cover the whole area with several L-Trominos and S-Tetrominos (Z-Tetrominos) along the square so that they do not overlap. The tiles can be rotated or flipped over. Find the minimum possible number of L-Trominos to be used.
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Context:
Find the coordinates of the foci of the hyperbola $\frac{x^2}{8} - \frac{y^2}{17} = 1$. Express your answer as coordinates (x, y), including the ± symbol if necessary.
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Context:
In a certain company, there are 100 shareholders, and any 66 of them own no less than 50% of the company's shares. What is the maximum percentage of all shares that one shareholder can own?
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Context:
What is the largest integer $n$ with no repeated digits that is relatively prime to $6$? Note that two numbers are considered relatively prime if they share no common factors besides $1$.
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Context:
Determine the number of distinct 7-permutations of the multiset \( \mathcal{M} = \{0^1,\ldots, 0^7,1^1,\ldots, 1^7, \cdots,9^1,\ldots, 9^7  \} \), where each digit from 0 to 9 appears 7 times. Two permutations are considered indistinguishable if they differ only by the exchange of identical digits. Calculate this number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the second homotopy group of the space obtained by removing a line and a point (where the line does not intersect the point) from \( \mathbb{R}^3 \).
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Context:
A binary message is emitted by a source, where the probability of a '1' appearing is 0.6 and the probability of a '0' appearing is 0.4. The interpreter of the message incorrectly interprets 10% of the symbols. If the interpreter receives the message "101", what is the probability that this is the same message that was sent by the source? Provide your answer to three decimal places.
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Context:
Evaluate $\int^4_1\frac{x-2}{(x^2+4)\sqrt x}dx$
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Context:
Determine the dimension of the quotient space $c_0/M$, where $c_0 = \{ (x_n) : x_n \in \mathbb{R}, x_n \to 0 \}$ and $M = \{(x_n) \in c_0 : x_1 + x_2 + \cdots + x_{10} = 0\}$. What is $\dim(c_0/M)$?
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Context:
Calculate the flux of the vector field \( \mathbf{F} = (3x^3y^2 + 3)\mathbf{i} + \left(\frac{y^2 + 2x}{3}\right)\mathbf{j} + (3yz^3 + 3)\mathbf{k} \) through the box with opposite corners at \((0, 1, 0)\) and \((2, 2, 1)\), where the faces are parallel to the coordinate planes. Use the divergence theorem to find the correct flux value.
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Context:
In a hockey tournament with $n$ teams, each team plays every other team exactly once. A win awards 2 points, a draw gives 1 point to each team, and a loss gives 0 points. Determine the maximum possible point difference between two consecutively ranked teams.
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Context:
Find the value of \( a \) such that \( l = m \), where \( l = \lim_{x \rightarrow a}\frac{x^x-a^x}{x-a} \) and \( m = \lim_{x \rightarrow a}\frac{a^x-x^a}{x-a} \), given that \( a > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the outward flux of the vector field \( V = x(x^2+y^2+z^2)^{-3/2}\mathbf{i} + y(x^2+y^2+z^2)^{-3/2}\mathbf{j} + z(x^2+y^2+z^2)^{-3/2}\mathbf{k} \) through the boundary of the rectangular box \( R \), where \( R \) consists of all points \((-1 \leq x \leq 1, -1 \leq y \leq 1, -1 \leq z \leq 1)\). Note that the vector field is not defined at the origin, so the Divergence Theorem cannot be directly applied to the box and its interior.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the differential equation $y'(t) = \frac{t}{4} - \frac{1}{y^2 + 1}$ with the initial condition $y(0) = 1$, and the function $x(t) = 1 - \frac{t}{2}$, approximate $|x(t) - y(t)|$ for $|t| < \frac{1}{2}$ using a corollary of Gronwall's theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^a\!\!\!\int_0^a\frac{dx\,dy}{(x^2+y^2+a^2)^{\frac32}} \) using polar coordinates.
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Context:
Determine the number of roots of the equation: \[ 10^{\lg \cos x} = \frac{1}{7} x - 1 \]
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Context:
Find the nine-digit number of the form \( \overline{a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} a_{1} a_{2} a_{3}} \) that is equal to the square of the product of five different prime numbers, where \( \overline{b_{1} b_{2} b_{3}} = 2 \cdot \overline{a_{1} a_{2} a_{3}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient \( a_{2^n} \) in the power series expansion of \( \left(\frac{x}{1-x^2}+\frac{3x^3}{1-x^6}+\frac{5x^5}{1-x^{10}}+\frac{7x^7}{1-x^{14}}+\cdots\right)^2 = \sum_{i=0}^{\infty}a_{i}x^i \).
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Context:
Compute the integral \( \int_R (x^2-y^2)\,dx\,dy \) over the region \( R \) in the first quadrant defined by the inequalities \( 3 \geq y-x \geq 0 \) and \( 5 \geq xy \geq 2 \). Use the change of variables \( u = y-x \) and \( v = xy \) to evaluate the integral.
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Context:
In the $xy$-coordinate plane, the $x$-axis and the line $y=x$ act as mirrors. A laser beam is shot from the point $(126, 21)$ towards a point on the positive $x$-axis. There are three points on the $x$-axis where the beam can be aimed such that it will bounce off the mirrors and return to $(126, 21)$. These points are $(126, 0)$, $(105, 0)$, and a third point $(d, 0)$. Find the value of $d$. (Note: When light bounces off a mirror, the angle of incidence equals the angle of reflection.)
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Context:
Determine the values of \( \alpha \) for which the series \( \sum_{n=0}^{\infty} \left( \frac{\alpha n}{n+1} \right)^n \) converges.
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Context:
Consider the double integral \( \int_0^{\infty} \int_0^{\infty} f(xy) g(x,y) \, dx \, dy \). Perform the substitution \( y=\frac{t}{x} \) with \( dy=\frac{dt}{x} \). Determine the correct bounds for the variable \( t \) in the transformed integral.
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Context:
In triangle $ABC$, a midline $MN$ that connects the sides $AB$ and $BC$ is drawn. A circle passing through points $M$, $N$, and $C$ touches the side $AB$, and its radius is equal to $\sqrt{2}$. The length of side $AC$ is 2. Find the sine of angle $ACB$.
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Context:
Given that \(a_{1}, a_{2}, a_{3}, a_{4}\) are positive real numbers, satisfying:
$$
\sum_{i=1}^{4} i a_{i} \leqslant 10, \quad a_{i} \geqslant \frac{1}{2^{4-i}} \quad (i=1, 2, 3, 4).
$$
Define \(f=\sum_{i=1}^{4} \frac{1}{1+a_{i}^{i}}\). Find the minimum value of \(f\).
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Context:
In triangle $ABC$, the lateral sides $AB$ and $BC$ are equal. A line parallel to the base $AC$ intersects side $AB$ at point $D$ and side $BC$ at point $E$, and each of the segments $AD$, $EC$, and $DE$ is equal to 2. Point $F$ is the midpoint of segment $AC$, and point $G$ is the midpoint of segment $EC$, connected by a straight line segment. It is known that the measure of angle $GFC$ is $\beta$. Find the area of triangle $ABC$.
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Context:
3. If real numbers $a, b, x$ and $y$ satisfy the system of equations
$$
\left\{\begin{array}{l}
a x+b y=3, \\
a x^{2}+b y^{2}=7, \\
a x^{3}+b y^{3}=6, \\
a x^{4}+b y^{4}=42 .
\end{array}\right.
$$

Find the value of $a x^{5}+b y^{5}$.
(8th American Invitational Mathematics Examination)
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Context:
For the power function $y = (m^2 - m - 1)x^{2m+1}$, if it is a decreasing function for $x \in (0, +\infty)$, then the value of the real number $m$ is ______.
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Context:
Today, Ivan the Confessor prefers continuous functions $f:[0,1] \rightarrow \mathbb{R}$ satisfying $f(x)+f(y) \geq|x-y|$ for all pairs $x, y \in[0,1]$. Find the minimum of $\int_{0}^{1} f$ over all preferred functions. The original answer is in \frac{m}{n} format, please give the value of m + n.
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Context:
Professors Alpha, Beta, Gamma, and Delta choose their chairs so that each professor will be between two students. Given that there are 13 chairs in total, determine the number of ways these four professors can occupy their chairs.
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Context:
25. $[\mathbf{1 4}]$ Let $n$ be an integer greater than 3 . Let $R$ be the set of lattice points $(x, y)$ such that $0 \leq x, y \leq n$ and $|x-y| \leq 3$. Let $A_{n}$ be the number of paths from $(0,0)$ to $(n, n)$ that consist only of steps of the form $(x, y) \rightarrow(x, y+1)$ and $(x, y) \rightarrow(x+1, y)$ and are contained entirely within $R$. Find the smallest positive real number that is greater than $\frac{A_{n+1}}{A_{n}}$ for all $n$.
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Context:
Problem 3. Points $A$ and $B$, located on a circular avenue, are connected by a straight highway segment 4 km long, which is the diameter of the circular avenue. A pedestrian left his house and started walking along the avenue from point $A$. After 1 hour, he realized he had forgotten his keys and asked a neighbor-cyclist to bring them as quickly as possible. What is the minimum time it will take for the pedestrian to receive the keys if the cyclist's speed on the highway is 15 km/h, on the avenue - 20 km/h, and the pedestrian's speed is 7 km/h? The pedestrian can walk towards the cyclist.
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Context:
Let $M$ and $N$ be any two points on the parabola $C$: $y^{2}=2px(p > 0)$. The coordinates of point $E$ are $(-\lambda,0)$ where $\lambda > 0$. If the minimum value of $\overrightarrow{EM}\cdot \overrightarrow{EN}$ is $0$, then $\lambda=$ ______.
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Context:
How many  $5$ -digit numbers  $N$  (in base  $10$ ) contain no digits greater than  $3$  and satisfy the equality  $\gcd(N,15)=\gcd(N,20)=1$ ? (The leading digit of  $N$  cannot be zero.)

*Based on a proposal by Yannick Yao*
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Context:
There are $n$ parking spaces along a road, and $n$ drivers each driving a car. Each driver parks their car in their preferred parking space. If their preferred space is already occupied, they park in the nearest available space along the road. If both their preferred and subsequent spaces are occupied, they drive away and do not park. How many sequences $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ exist such that every parking space is occupied? Here, $a_{i}$ represents the preferred parking space of the $i$-th driver, and the values of $a_{1}, a_{2}, \cdots, a_{n}$ do not need to be distinct.
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Context:
A chemical factory needs to sequentially add two types of chemical raw materials during production. It is known that there are 5 types of raw materials available, but materials A and B cannot be used at the same time, and when adding materials in sequence, if material A is used, it must be added first. Find the total number of different adding schemes available. Express your answer as a whole number.
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Context:
Given a right quadrilateral pyramid \( V-ABCD \) with the height equal to half the length of \( AB \), \( M \) is the midpoint of the lateral edge \( VB \), and \( N \) is a point on the lateral edge \( VD \) such that \( DN=2VN \). Find the cosine of the angle between the skew lines \( AM \) and \( BN \).
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Context:
The flag-raising time at Dongfanghong Elementary School in 2012 varies depending on the date as follows: From January 1 to January 10, it is fixed at 7:13 AM. From January 11 to June 6, it gradually advances from 7:13 AM to 4:46 AM, advancing by 1 minute every day. From June 7 to June 21, it is fixed at 4:46 AM. What is Dongfanghong Elementary School's flag-raising time today (March 11)?
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Context:
5. We define a "table" of type $\left(n_{1}, n_{2}, \ldots, n_{m}\right), n_{1} \geq n_{2} \geq \ldots \geq n_{m}$ as follows: We place $n_{1}$ squares horizontally next to each other, then we place $n_{2}$ squares horizontally next to each other below the $n_{1}$ squares starting from the first square on the left. We continue this process until we get a "table" with $n_{1}$ squares in the first row, $n_{2}$ squares in the second row, $\ldots, n_{m}$ squares in the $m$-th row, with a total of $n=n_{1}+n_{2}+\ldots+n_{m}$ squares. The rows of placed squares always start from the square below the leftmost square of the first row (see example). We fill the obtained "table" with numbers from 1 to n such that in each row and each column, the numbers from left to right and from top to bottom, respectively, are in increasing order. Below is an example of a table of type $(5,4,2,1)$ and one of its fillings.

| 1 | 3 | 4 | 7 | 11 |
| :---: | :---: | :---: | :---: | :---: |
| 2 | 5 | 10 | 12 |  |
| 6 | 9 |  |  |  |
| 8 |  |  |  |  |
|  |  |  |  |  |

Determine the number of fillings of a "table" of type $(4,3,2)$.
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Context:
Dan owns an ice cream shop and every sixth customer gets a free ice cream cone. Cones cost $2 each. If he sold $100 worth of cones, how many free ones did he give away?
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Context:
Let $C_1$ and $C_2$ be circles defined by:
\[
(x - 10)^2 + y^2 = 36
\]
and
\[
(x + 15)^2 + y^2 = 81
\]
respectively. Find the length of the shortest line segment $\overline{PQ}$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$.
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Context:
It takes 40 minutes to freeze ice cubes. Once they're frozen, it takes 3 minutes per smoothie to turn them into smoothies. Additionally, you have to chop fruits for each smoothie: 2 minutes for apples, 3 minutes for bananas, 4 minutes for strawberries, 5 minutes for mangoes, and 6 minutes for pineapples. Assuming you need 1 of each fruit for 5 different smoothies, how long does it take to make the 5 smoothies?
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Context:
Evaluate the limit:
$$\lim_{n \to \infty} n\int_0^1 \frac{x^n}{4x + 5} \, dx$$
Given that the sequence $(I_n)_{n \geq 1}$ is defined by $I_n = \int_0^1 \frac{x^n}{4x + 5} \, dx$ and is known to be decreasing and convergent to $0$, determine the limit's value.
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Context:
Given \( A, B, C, D \in\{1, 2, \cdots, 6\} \), and each is distinct from the others. If the curves \( y = Ax^{2} + B \) and \( y = Cx^{2} + D \) intersect, then there are \(\quad\) different ways to choose \( A, B, C, D \) (the intersection of the curves is independent of the order of \( A, B, C, D \); for example, \( A=3, B=2, C=4, D=1 \) and \( A=4, B=1, C=3, D=2 \) are considered the same).
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Context:
Let's say the function q is defined by the formula q = (5 w / (4 h (z ^ 2))) + kx, where k is a constant and x is an additional variable. By what factor will q be multiplied if w is quadrupled, h is doubled, z is tripled, and x is halved, while keeping k constant?
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Context:
Find the smallest positive integer \( n \) such that if \( n \) squares of a \( 1000 \times 1000 \) chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.
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Context:
Rahim went on a book shopping spree and visited 4 different shops. At the first shop, he bought 42 books for Rs. 520 and received a 10% discount on the total purchase. At the second shop, he bought 22 books for Rs. 248 and paid an additional 5% in taxes. Then, at the third shop, he bought 35 books for Rs. 740 and received a 15% discount on the total purchase. Finally, at the fourth shop, he bought 18 books for Rs. 360 and paid an additional 8% in taxes. What is the average price Rahim paid per book after considering all the discounts and taxes?
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Context:
If $y<0$, find the range of all possible values of $y$ such that $\lceil{y}\rceil\cdot\lfloor{y}\rfloor=110$.  Express your answer using interval notation.
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Context:
Jason is trying to figure out whether he can afford a new car. The car he wants costs $32,000, and he can make a down payment of $8,000. He'll have to get a loan for the rest of the amount and make 48 equal monthly payments on it. Each month, he also has to pay interest equal to 5% of that month's payment. How much will Jason's monthly payments be?
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Context:
Given a geometric sequence $\{a_n\}$ with positive terms, the sum of the first $n$ terms is $S_n$. If $-3$, $S_5$, and $S_{10}$ form an arithmetic sequence, calculate the minimum value of $S_{15} - S_{10}$.
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Context:
The Chinese Academy of Social Sciences needs to conduct a stratified sampling survey of the economic conditions in the counties within the Poyang Lake Eco-Economic Zone in Jiangxi Province. The survey will cover 20 counties within 9 prefecture-level cities (excluding urban districts and county-level cities). If exactly 2 samples were taken from the 8 counties in the Jiujiang area that are part of the economic zone, then the total number of samples taken is __________.
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Context:
Let \( A \in M_{5\times 6}(\mathbb{R}) \) be a matrix with rank 4. Consider the block matrix \( D = \begin{pmatrix} I_5 & A \\ A^T & 0 \end{pmatrix} \). Determine the rank of \( D \).
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Context:
Given the function y = sin(2x + π/4), translate its graph left by π/4 units and then up by 2 units. Calculate one of the symmetry centers of the resulting graph. Express your answer as an ordered pair (x, y).
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Context:
已知函数 $f: \mathbf{R} \rightarrow \mathbf{R}$, 若对任意实数 $x 、 y 、 z$ 都有
$$\frac{1}{3} f(x y)+\frac{1}{3} f(x z)-f(x) f(y z) \geqslant \frac{1}{9},$$
求 $\sum\limits_{i=1}^{100}[i f(i)]$. 其中, $[x]$ 表示不超过实数 $x$ 的最大整数.
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Context:
设a,b,c,a+b-c,a+c-b,b+c-a,a+b+c是7个两两不同的素数，且a，b，c中有两数之和是800．设d是这7个素数中最大数与最小数之差，求d的最大可能值．
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Context:
In a checkered square with a side length of 2018, some cells are painted white and the rest are black. It is known that from this square, one can cut out a 10x10 square where all the cells are white, and a 10x10 square where all the cells are black. What is the smallest value for which it is guaranteed that one can cut out a 10x10 square in which the number of black and white cells differ by no more than?
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Context:
If the domain of the function $f(x)=x^{2}$ is $D$, and its range is ${0,1,2,3,4,5}$, then there are \_\_\_\_\_\_ such functions $f(x)$ (answer with a number).
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Context:
The manager of a discount clothing store received a shipment of Fall clothing. The shipment contained 10 identical sweaters and 20 identical jackets. The second shipment at the same prices, contained 5 of the same sweaters and 15 of the same jackets. The second shipment was 550 dollars. What was the cost of the first shipment?
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Context:
Given $p$: $|1-\frac{x-1}{3}|\leqslant 2$, $q$: $x^2-2x+1-m^2\leqslant 0 (m > 0)$, if $\lnot p$ is a sufficient but not necessary condition for $\lnot q$, find the range of values for the real number $m$.
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Context:
Compute the limit: $$L=\lim_{x\to 0}\frac{\sin(\ln(x+1))-\ln(\sin(x)+1)}{\sin^4\left(\frac{x}{2}\right)}$$
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Context:
Given that $\sqrt{51.11}\approx 7.149$ and $\sqrt{511.1}\approx 22.608$, determine the value of $\sqrt{511100}$.
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Context:
Binhao has a fair coin. He writes the number $+1$ on a blackboard. Then he flips the coin. If it comes up heads (H), he writes $+\frac12$ , and otherwise, if he flips tails (T), he writes $-\frac12$ . Then he flips the coin again. If it comes up heads, he writes $+\frac14$ , and otherwise he writes $-\frac14$ . Binhao continues to flip the coin, and on the nth flip, if he flips heads, he writes $+ \frac{1}{2n}$ , and otherwise he writes $- \frac{1}{2n}$ . For example, if Binhao flips HHTHTHT, he writes $1 + \frac12 + \frac14 - \frac18 + \frac{1}{16} -\frac{1}{32} + \frac{1}{64} -\frac{1}{128}$ . The probability that Binhao will generate a series whose sum is greater than $\frac17$ is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + 10q$.
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Context:
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13, 19, 20, 25$, and $31$, although this is not necessarily their order around the pentagon. What is the area of the pentagon?
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Context:
Two circles with radii 2 and 3 touch each other externally at point $A$. Their common tangent passing through point $A$ intersects their other two common tangents at points $B$ and $C$. Find $BC$.
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Context:
Find the Galois group of $x^6+1$ over $F=\mathbb{F}_2=\{0,1\}$.
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Context:
How many of the smallest 2401 positive integers written in base 7 include the digits 4, 5, or 6?
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Context:
Given a tetrahedron $ABCD$, each edge is colored red or blue with equal probability. What is the probability that point $A$ can reach point $B$ through red edges within the tetrahedron $ABCD$?
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Context:
In the right-angled triangle $ABC$, the foot of the altitude from point $A$ to the hypotenuse $AB$ is point $D$. The angle bisectors of $\angle ACD$ and $\angle BCD$ intersect the hypotenuse $AB$ at points $E$ and $F$ respectively. Determine the ratio of the inradius of triangle $ABC$ to the circumradius of triangle $CEF$.
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Context:
从一个正方体的八个顶点中取出三个. 则以这三个点为顶点构成直角三角形的概率是 $\qquad$。
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
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Context:
In a triangle $XYZ$, angle $\angle ZXY$ is $72^\circ$. It is given that side $XY$ is twice as long as side $YZ$. What is the measure of angle $\angle XYZ$?
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Context:
Given the equation:

$$\int_{0}^ {\pi/2}(\sin x+a\cos x)^3dx-\frac{4a}{\pi-2}\int_{0}^{\pi/2}x\cos x dx=2$$

find the coefficient of the linear term in the cubic polynomial expression inside the first integral.
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Context:
Usually, I go up the escalator in the subway. I have calculated that when I walk up the moving escalator, I ascend 20 steps, and it takes me exactly 60 seconds. My wife walks up the stairs more slowly and only ascends 16 steps; therefore, her total time to ascend the escalator is longer - it is 72 seconds.

How many steps would I have to climb if the escalator suddenly breaks down?
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Context:
57 Given a regular $(2n+1)$-gon. Consider all triangles whose vertices are the vertices of this $(2 n+1)$-gon. Find the number of triangles that contain the center of the regular $(2 n+1)$-gon.
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Context:
Given that the lengths of the sides of an acute triangle are $3$, $4$, and $a$, the range of values for $a$ is \_\_\_\_\_\_\_.
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Context:
A school club designed an unconventional rectangular checkerboard that measures 10 by 6. Calculate the probability that a randomly chosen unit square does not touch any edge of the board or is not adjacent to any edge.
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Context:
Each cell of a $2 \times 5$ grid of unit squares is to be colored white or black. Compute the number of such colorings for which no $2 \times 2$ square is a single color.
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Context:
Distinct points $A$, $B$, $C$, and $D$ lie on a line, with $AB=BC=CD=1$. Points $E$ and $F$ lie on a second line, parallel to the first, with $EF=1$. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?
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Context:
In a cube with 7 visible corner cubies, determine the number of different permutations and orientations possible for the hidden edges, given that the visible corners and edges are correctly positioned and oriented.
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Context:
Find all integers $n$ such that $\frac{5^n - 1}{3}$ is a prime or a perfect square of an integer.
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Context:
Given a parallelepiped \( ABCD A_1 B_1 C_1 D_1 \), point \( X \) is chosen on edge \( A_1 D_1 \) and point \( Y \) is chosen on edge \( BC \). It is known that \( A_1 X = 5 \), \( B Y = 3 \), and \( B_1 C_1 = 14 \). The plane \( C_1 X Y \) intersects the ray \( DA \) at point \( Z \). Find \( DZ \).
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Context:
Consider a unit cube in a coordinate system with vertices $A(0,0,0)$, $A'(1,1,1)$, and other vertices placed accordingly. A regular octahedron has vertices placed at fractions $\frac{1}{3}$ and $\frac{2}{3}$ along the segments connecting $A$ with $A'$'s adjacent vertices and vice versa. Determine the side length of this octahedron.
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Context:
Determine the minimum distance $d(C)$ of a linear code $C$ with a parity-check matrix $H$ over $\Bbb F_2$. The first seven rows of $H$ are the identity matrix $I_7$, and the remaining rows consist of all weight-$3$ vectors in $\{0,1\}^7$. The rank of $H$ is 7. What is the minimum distance $d(C)$?
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Context:
Let $f(x)=\cos(\omega x+\varphi)$, where $\omega\gt 0$ and $0\lt \varphi\lt \pi$. The smallest positive period of the function is $T$. If $f(T)=\frac{\sqrt{3}}{2}$ and $x=\frac{π}{9}$ is a zero of $f(x)$, then the minimum value of $\omega$ is ____.
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Context:
Compute the flux of water through the parabolic cylinder $S: y = x^2$, for $0 \le x \le 2$ and $0 \le z \le 3$, given the velocity vector field $F(x,y,z) = 3z^2\mathbf{i} + 6\mathbf{j} + 6xz\mathbf{k}$. What is the flow rate of the fluid through this surface?
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Context:
Based on a $city^{'}s$ rules, the buildings of a street may not have more than $9$ stories. Moreover, if the number of stories of two buildings is the same, no matter how far they are from each other, there must be a building with a higher number of stories between them. What is the maximum number of buildings that can be built on one side of a street in this city?
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Context:
Given the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \; (a, b>0)$ with left and right foci at $F_{1}(-c, 0)$ and $F_{2}(c, 0)$, respectively, and an eccentricity of 2. If the line $y=2(x-c)$ intersects the hyperbola at points $A$ and $B$, and the incenters of $\triangle A F_{1} F_{2}$ and $\triangle B F_{1} F_{2}$ are $I_{1}$ and $I_{2}$ respectively, and $\left|I_{1} I_{2}\right|=2 \sqrt{5}$, find the value of $a$.
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Context:
Suppose that $f'(z)$ exists on $\mathbb{C}$ and $\Im(f'(z))=6x(2y-1)$, with $f(0)=3-2i$ and $f(1)=5-5i$. Find $f(1+i)$. \(z = x + yi\) is a complex number.
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Context:
Let $D$ and $E$ be points on respectively $[AB]$ and $[AC]$ of $\triangle ABC$ where $|AB|=|AC|$, $m(\widehat{BAC})=40^{\circ}$. Let $F$ be a point on $BC$ such that $C$ is between $B$ and $F$. If $|BE|=|CF|$, $|AD|=|AE|$ and $m(\widehat{BEC})=60^{\circ}$, then what is $m(\widehat{DFB})$?
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Context:
Calculate the limit of the following integral as \( n \to \infty \):

\[ \lim_{n \to \infty } \frac{1}{n^2} \int_0^n \frac{ \sqrt{n^2-x^2} }{2+x^{-x}} \, dx \]
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Context:
Find an upper bound for the smallest positive integer $b$ such that the inequality \(2^{3^{4^{\cdots^a}}} \leq b^{(b-1)^{\cdots^{3^{2}}}}\) holds for a given positive integer \(a \geq 2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{\infty} \frac{d x}{\left(x^{3}+\frac{1}{x^{3}}\right)^{2}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
On circle in clockwise order are written positive integers from $1$ to $2010$. Let us cross out number $1$, then number $10$, then number $19$, and so on every $9$th number in that direction.  How many numbers at that moment are not crossed out?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Anya, Vanya, Danya, Sanya, and Tanya were collecting apples. It turned out that each of them collected a whole percentage of the total number of apples, and all these numbers were different and greater than zero. Then Tanya, who collected the most apples, ate her apples. After that, it turned out that each of the remaining children still had a whole percentage, but now from the number of apples left. What is the minimum number of apples that could have been collected?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a relation $R$, determine the smallest set of pairs of elements that must be added to $R$ to make the resulting relation $R^*$ reflexive, symmetric, and transitive.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the product of an empty family of similar algebras, \( \prod\langle \mathbf{A}_i \mid i \in I \rangle \), where \( I = \emptyset \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area enclosed by the inverse function of \( f(x) = x^3 + 3x + 1 \), between \( x = -3 \) and \( x = 5 \), and the x-axis. Use techniques of finding area by definite integrals.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $A$ and $B$ are $N \times N$ matrices with $\text{rank}(A) = N$ and $\text{rank}(B) = 1$. Determine the lower bound of $\text{rank}(A + B)$. Provide your answer as a single number or expression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the limit: \[ \lim_{n\to +\infty}\frac{1}{n^2}\sum_{i=1}^{n}\log\binom{n}{i}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\rightarrow \infty } \int_0^1 \sqrt{ \frac 1 x + n^2 x^{2n} } \ \mathrm dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the domain of \( z \) such that the Möbius transformation \( w = T(z) = \frac{1+z}{1-z} \) maps to the region \( \{ \Re(w) > 0 \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest value of $a > 1$ such that the inequality \( \frac{a + \sin x}{a + \sin y} \leq e^{(y-x)} \) holds for all $x \leq y$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given 2006 points on a plane, determine the maximum number of points among them that can form an acute-angled triangle with any other two points.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( \int_{S^2} \frac{1}{\sqrt{(x-a)^2+(y-b)^2+(z-c)^2}} \, dS \) over the unit sphere \( S^2 \) in \( \mathbb{R}^3 \), given that \( a^2+b^2+c^2 < 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
p13. Isosceles trapezoid \( {ABCD} \) has \( {AB} = {CD} \) and \( {AD} = {6BC} \) . Point \( X \) is the intersection of the diagonals \( {AC} \) and
 \( {BD} \) . There exist a positive real number \( k \) and a point \( P \) inside \( {ABCD} \) which satisfy
 \( \left\lbrack  {PBC}\right\rbrack   : \left\lbrack  {PCD}\right\rbrack   : \left\lbrack  {PDA}\right\rbrack   = 1 : k : 3, \)
 where \( \left\lbrack  {XYZ}\right\rbrack \) denotes the area of triangle \( {XYZ} \) . If \( {PX} \parallel  {AB} \) , find the value of \( k \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
On an island, there are 100 knights and 100 liars. Every resident has at least one friend. Knights always tell the truth, while liars always lie. One morning, each resident said either the phrase "All my friends are knights" or the phrase "All my friends are liars," with exactly 100 people saying each phrase. Find the minimum possible number of pairs of friends where one is a knight and the other is a liar.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(OA\) be a fixed radius of a circle with center \(O\); \(B\) be the midpoint of the radius \(OA\); and \(M\) be an arbitrary point on the circumference of the circle. Find the maximum value of the angle \(\angle OMB\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the average score of six students is 92.5, the highest score is 99, and the lowest score is 76, and that all scores are unique integers, calculate the minimum possible score of the student who ranks 3rd from the highest.
Please reason step by step, and put your final answer within \boxed{}.


Context:
给定 $A(-2,2)$, 已知 $B$ 是椭圆 $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$ 上的动点, $F$ 是左焦点, 当 $|A B|+\frac{5}{3}|B F|$ 取最小值时, $B$ 的坐标为$\left(-\frac{a \sqrt{b}}{c}, b\right)$.求$a+b+c+d$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $r$ in the congruence $$\sum_{0\leq{k}<50}\binom{100}{2k+1}5^k \equiv r\pmod{ 2^{99}}$$ where $0 \leq{r} < 2^{99}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $p: -3 \leqslant x \leqslant 3$, $q: x < a$ ($a$ is a real number). If $q$ is a sufficient and necessary condition for $p$, then the range of real number $a$ is ______.
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Context:
若直线 \(x=\frac{\pi}{4}\) 被曲线 \(C:(x-\arcsin a)(x-\arccos a)+(y-\arcsin a)(y+\arccos a)=0\) 所截得的弦长为 \(d\)，当 \(a\) 变化时，求 \(d\) 的最小值。答案的形式为 \(\frac{k\pi}{m}\)，请给出 \(k + m\) 的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Angela, Bill, and Charles each independently and randomly choose a subset of $\{ 1, 2, 3, 4, 5, 6, 7, 8 \}$ that consists of consecutive integers (two people can select the same subset). The expected number of elements in the intersection of the three chosen sets is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the sequence of Fibonacci numbers $F_0, F_1, F_2, ... $, given by $F_0 = F_1= 1$ and $F_{n+1} =F_n + F_{n-1}$ for $n\ge 1$. Define the sequence $x_0, x_1, x_2, ....$ by $x_0 = 1$ and $x_{k+1} = x^2_k + F^2_{2^k}$ for $k \ge 0$. Define the sequence $y_0, y_1, y_2, ...$ by $y_0 = 1$ and $y_{k+1} = 2x_ky_k - y^2_k$ for $k \ge 0$. If $$\sum^{\infty}_{k=0} \frac{1}{y_k}= \frac{a -\sqrt{b}}{c}$$
for positive integers a$, b, c$ with $gcd (a, c) = 1$, find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The eight vertices of a cube are randomly labelled with the integers from 1 to 8 inclusive. Judith looks at the labels of the four vertices of one of the faces of the cube. She lists these four labels in increasing order. After doing this for all six faces, she gets the following six lists: $(1,2,5,8),(3,4,6,7),(2,4,5,7),(1,3,6,8),(2,3,7,8)$, and $(1,4,5,6)$. What is the label of the vertex of the cube that is farthest away from the vertex labelled 2? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the interval of real numbers $a$ for which the function $f(x) = \lg(1-ax)$ is monotonically decreasing on the interval $(0,1)$. Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $\frac{\pi}{4}$ is a zero point of the function $f(x) = \sin 2x + a \cos^2 x$ ($a \in \mathbb{R}$, and it's a constant), then the maximum value of $f(x)$ is $\_\_\_\_\_\_\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A set of sample data $11$, $13$, $15$, $a$, $19$ has an average of $15$. Calculate the standard deviation of this data set.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the lines ${l}_{1}:x-y+5=0$ and ${l}_{2}:x+4=0$, and the parabola $C:{y}^{2}=16x$, let $P$ be a moving point on $C$. The minimum value of the sum of the distances from $P$ to ${l}_{1}$ and ${l}_{2}$ is _______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangle $PQR$ has $\angle Q = 45^{\circ}$ and $QR = 6$. Point $S$ is the midpoint of $QR$. If the altitude from point $P$ to $QR$ is 3 units, what is the largest possible value of $\tan{\angle PSQ}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a certain company, a third of the workers do not have a retirement plan. Some percentage of the workers who do not have a retirement plan are women, and 40% of the workers who do have a retirement plan are men. If 120 of the workers of that company are men, and there are 120 women workers, what is the percentage of women workers without a retirement plan?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Greg and Sarah are going to split some apples evenly. Susan has twice as many apples as Greg will end up with. Mark has 5 fewer apples than Susan. Their mom needs 40 apples to make an apple pie and will have 9 apples left over. How many apples do Greg and Sarah have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two trains of length 100 m and 220 m are running towards each other on parallel lines at a certain speed and 30 kmph respectively. They will be clear of each other in 15.99872010239181 seconds from the moment they meet. What is the speed of the first train in kmph?
Please reason step by step, and put your final answer within \boxed{}.


Context:
 $ m$  and  $ n$  are positive integers. In a  $ 8 \times 8$  chessboard,  $ (m,n)$  denotes the number of grids a Horse can jump in a chessboard ( $ m$  horizontal  $ n$  vertical or  $ n$  horizontal  $ m$  vertical ). If a  $ (m,n) \textbf{Horse}$  starts from one grid, passes every grid once and only once, then we call this kind of Horse jump route a  $ \textbf{H Route}$ . For example, the  $ (1,2) \textbf{Horse}$  has its  $ \textbf{H Route}$ . Find the smallest positive integer  $ t$ , such that from any grid of the chessboard, the  $ (t,t\plus{}1) \textbf{Horse}$  does not has any  $ \textbf{H Route}$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a cone with a height and base radius of 7, four identical spheres are placed. Each sphere touches two others (externally), the base, and the lateral surface of the cone. A fifth sphere touches the lateral surface of the cone and all four identical spheres (externally). Find the radius of the fifth sphere.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The differentiable function \( F: \mathbb{R} \rightarrow \mathbb{R} \) satisfies \( F(0) = -1 \) and
$$
\frac{d}{d x} F(x) = \sin (\sin (\sin (\sin (x)))) \cdot \cos (\sin (\sin (x))) \cdot \cos (\sin (x)) \cdot \cos (x).
$$

Find \( F(x) \) as a function of \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the arithmetic sequence $\{a\_n\}$, $4a_{12}=-3a_{23} > 0$. Let $b\_n= \frac {a\_n a_{n+1}}{a_{n+2}}$, $S\_n$ is the sum of the first $n$ terms of $\{b\_n\}$, and $S\_{n_0}$ is the maximum term of the sequence $\{S\_n\}$. Find $n\_0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each face of a hexahedron and each face of a regular octahedron are squares with side length $a$. The ratio of the radii of the inscribed spheres of these two polyhedra is an irreducible fraction $\frac{m}{n}$. What is the product $m \cdot n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\mathrm{Rt} \triangle A B C$, $\angle A=90^{\circ}$, the inscribed square $P Q R S$ has one side $R S$ on the hypotenuse $B C$, $P$ is on $A B$, and $Q$ is on $A C$. Given $|A B|=a(a>0, a$ is a constant), $\angle B=\theta$, when $\theta$ varies, the minimum value of $S_{\triangle A B C}$ : $S_{P Q R S}$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
【Question 29】
8 different positive integers, among which there are at least: 6 multiples of 2, 5 multiples of 3, 3 multiples of 5, 1 multiple of 7. To make the maximum number among the 8 numbers as small as possible, then this maximum number is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the determinant of the matrix \( D_n = \begin{vmatrix} n & -1 & -3 & 0 & 0 & \cdots & 0 & 0 & 0 \\ n & 1 & 2 & -3 & 0 & \cdots & 0 & 0 & 0 \\ n & 0 & 1 & 2 & -3 & \cdots & 0 & 0 & 0 \\ n & 0 & 0 & 1 & 2 & \cdots & 0 & 0 & 0 \\ n & 0 & 0 & 0 & 1 & \cdots & 0 & 0 & 0 \\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots \\ n & 0 & 0 & 0 & 0 & \cdots & 1 & 2 & -3 \\ n & 0 & 0 & 0 & 0 & \cdots & 0 & 1 & 2 \\ n & 0 & 0 & 0 & 0 & \cdots & 0 & 0 & 1 \\ \end{vmatrix} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
 $22$  football players took part in the football training. They were divided into teams of equal size for each game ( $11:11$ ). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of sequences consisting of 100 R's and 2011 S's that satisfy the property that among the first \( k \) letters, the number of S's is strictly more than 20 times the number of R's for all \( 1 \leq k \leq 2111 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
By selling 20 pencils for a rupee, a man loses 30%. He wants to adjust the number of pencils he sells for a rupee to make a certain gain. He finds out that he should sell approximately 10.77 pencils for a rupee. What is the percentage of the gain he wants to make?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Rahim visited three different bookstores to purchase books for his collection. At Store A, he bought 25 books for a total of Rs. 1,500, and received a 15% discount on the entire purchase. At Store B, he bought 35 books for a total of Rs. 3,000, and got a "Buy 3, get 1 free" deal on the books, effectively getting 4 less books to pay for. At Store C, he bought 40 books for Rs. 3,500, and got a 10% discount on every 10th book he bought there. Calculate the average price Rahim paid per book considering the discounts and deals he got at each store.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sum \(\sum_{k=3}^{\infty} \frac{1}{k(k^4-5k^2+4)^2}\) is equal to \(\frac{m^2}{2n^2}\), 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:
Find the largest five-digit positive integer such that it is not a multiple of 11, and any number obtained by deleting some of its digits is also not divisible by 11.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Rural School USA has 105 students enrolled. There are 60 boys and 45 girls. If $\frac{1}{10}$ of the boys and $\frac{1}{3}$ of the girls are absent on one day, what percent of the total student population is absent?
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Context:
Five people participate in a relay game, each randomly choosing between two options: left path and right path. If more than 2 people choose the same path, each of them gets 1 point; if fewer than 3 people choose the same path, each of them gets 0 points. Let $\xi$ denote the score of a player named Xiaoqiang. Calculate the expected value of $\xi$, $E\xi$. Express your answer as a fraction in the format $\boxed{[answer]}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A moth starts at vertex $A$ of a certain cube and is trying to get to vertex $B$, which is opposite $A$, in five or fewer "steps," where a step consists in traveling along an edge from one vertex to another. The moth will stop as soon as it reaches $B$. How many ways can the moth achieve its objective?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A right circular cone and a right cylinder with same height $20$ does not have same circular base but the circles are coplanar and their centers are same. If the cone and the cylinder are at the same side of the plane and their base radii are $20$ and $10$, respectively, what is the ratio of the volume of the part of the cone inside the cylinder over the volume of the part of the cone outside the cylinder?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the set \(\{1, 2, 3, \cdots, 99, 100\}\), how many numbers \(n\) satisfy the condition that the tens digit of \(n^2\) is odd?
(45th American High School Mathematics Examination, 1994)
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Context:
Given the polynomial \( P(x) = x^{n+1} - 2x^n + 1 \) for all \( n \geq 2 \), and knowing that \( P(x) \) has a unique root \( a_n \in (1,2] \), find the limit \( L = \lim_{n\to\infty} 2^n(2-a_n) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the genus of the curve $y^{2}+y=\frac{\alpha}{x^{2}+x}$ over a finite field with even characteristic?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an isosceles right triangle \(AOB\), points \(P, Q, R,\) and \(S\) are chosen on the sides \(OB, OA,\) and \(AB\) respectively, such that \(PQRS\) forms a square. Given that the lengths of \(OP\) and \(OQ\) are \(a\) and \(b\) respectively, and the area of the square \(PQRS\) is \(\frac{2}{5}\) of the area of the triangle \(AOB\), find the value of \(\frac{a}{b}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function
\[ f(x) = \log_{2}(4^x + 1) + mx \quad (m > 0) \]
the equation regarding \( x \)
\[ f\left(8\left(\log_{4} x\right)^{2} + 2 \log_{2} \frac{1}{x} + \frac{4}{m} - 4\right) = 1 \]
has exactly two different real solutions in the interval \([1, 2\sqrt{2}]\). Then the set of all possible values of \( m \) is
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCDE$ be be a cyclic pentagon. Given that $\angle CEB = 17^o$, find $\angle CDE  + \angle EAB$, in degrees.
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Context:
Determine the order of the quotient ring \( \mathbb{Z}_7[i]/\langle 2 \rangle \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine for which values of $n$ the improper integral $$\int_{0}^{2} \left(\log\left(\frac{2}{2-x}\right)\right)^n\,\mathrm dx$$ converges.
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Context:
3. (CZS) A tetrahedron $A B C D$ is given. The lengths of the edges $A B$ and $C D$ are $a$ and $b$, respectively, the distance between the lines $A B$ and $C D$ is $d$, and the angle between them is equal to $w$. The tetrahedron is divided into two parts by the plane $\pi$ parallel to the lines $A B$ and $C D$. Calculate the ratio of the volumes of the parts if the ratio between the distances of the plane $\pi$ from $A B$ and $C D$ is equal to $k$.
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Context:
Given that in a class of 36 students, more than half purchased notebooks from a store where each notebook had the same price in cents greater than the number of notebooks bought by each student, and the total cost for the notebooks was 990 cents, calculate the price of each notebook in cents.
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Context:
Given the function \( f(x) = 1 + \sum_{k=0}^{100}\frac{(-1)^{k+1}}{(k+1)!}x(x-1)(x-2)\cdots (x-k) \), find the value of \( S(f(x)) - S(f'(x)) \), where \( S \) denotes the sum of the real roots of the respective function.
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Context:
集合 $U=\{1,2, \cdots, 366\}$ ，则 $U$ 的互不相交各元素之和为 $17$ 的倍数的二元子集最多有 $\qquad$个.
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Context:
On January 1st, the day of the New Year celebration, an Elderly Sage was reflecting on his life. He noticed that over the past 5 years, all the days of the week occurred equally often, and 10 years ago, the New Year celebration fell on a Friday. What day of the week did the Sage's somber reflections fall on? It is known that the Sage's reflections occurred in the 21st century.
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Context:
Determine which ordinal is larger: \(\tau_1\), the supremum of ordinals writable by iterated Infinite Time Turing Machines, or \(\tau_2\), the smallest \(\Sigma_2^1\)-reflecting ordinal.
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Context:
Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes $4''$ or $10''$ or $19''$ to the total height of the tower. How many different tower heights can be achieved using all ninety-four of the bricks?
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Context:
10. Let real numbers $x, y$ be such that $x-y, x^{2}-y^{2}, x^{3}-y^{3}$ are all prime numbers, then the value of $x-y$ is
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Context:
Find the maximum positive integer \( n \) such that
\[
n^{2} \leq 160 \times 170 \times 180 \times 190
\]
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Context:
Let exp ( m , n ) = m to the power n . If exp ( 10 , some number ) = 25 exp ( 22 ) where m and n are integers. What is that number?
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Context:
Let $a_{1}, a_{2}, \ldots, a_{n}$ be a sequence of distinct positive integers such that $a_{1}+a_{2}+\cdots+a_{n}=2021$ and $a_{1} a_{2} \cdots a_{n}$ is maximized. If $M=a_{1} a_{2} \cdots a_{n}$, compute the largest positive integer $k$ such that $2^{k} \mid M$.
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Context:
Light of a blue laser (wavelength  $\lambda=475 \, \text{nm}$ ) goes through a narrow slit which has width  $d$ . After the light emerges from the slit, it is visible on a screen that is  $ \text {2.013 m} $  away from the slit. The distance between the center of the screen and the first minimum band is  $ \text {765 mm} $ . Find the width of the slit  $d$ , in nanometers.

*(Proposed by Ahaan Rungta)*
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Context:
A train of a certain length is running at 60 kmph. It takes 21.598272138228943 seconds to pass a platform 240 m long. How long is the train?
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Context:
A town has a current population of 14,000 people. Over the next three years, the annual population growth rates and additional contributing factors change as follows:

- In the first year, the population increases by 12% due to a high birth rate, and an additional 150 new residents move to the town.
- In the second year, the population growth rate decreases to 8%, and only 100 new residents move to the town due to increased emigration.
- In the third year, the growth rate further declines to 6% due to economic challenges, and at the same time, 200 employees and their families move to the town for a new business, increasing the population by an additional 500 people.

What will the population be at the end of these three years?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A Vandal and a Moderator are editing a Wikipedia article.  The article originally is error-free.  Each day, the Vandal introduces one new error into the Wikipedia article.  At the end of the day, the moderator checks the article and has a  $ 2/3$  chance of catching each individual error still in the article.  After  $ 3$  days, what is the probability that the article is error-free?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Place all terms of the arithmetic sequence \(2, 6, 10, 14, \cdots, 2006\) closely together to form a large number \(A = 261014 \cdots 2006\). Find the remainder when \(A\) is divided by 9.
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Context:
The graph of the power function y=f(x) passes through the point (-2, -1/8). Find the value of x that satisfies f(x)=27. Express your answer as a single numerical value.
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Context:
A store owner ordered different types of soda bottles in April, May, and June. The detailed order was as follows:

April:
- 100 cases of Soda A, with 24 bottles per case, priced at $1.50 per bottle.
- 50 cases of Soda B, with 30 bottles per case, priced at $2.00 per bottle.
April's orders had a discount of 15% applicable only to Soda A.

May:
- 80 cases of Soda A, with 24 bottles per case, priced at $1.50 per bottle.
- 40 cases of Soda B, with 30 bottles per case, priced at $2.00 per bottle.
May's orders had a 10% sales tax applicable on the total price.

June:
- 120 cases of Soda A, with 24 bottles per case, priced at $1.50 per bottle.
- 60 cases of Soda B, with 30 bottles per case, priced at $2.00 per bottle.
June's orders had a special offer: Buy one case of Soda B and get 10% off on one case of Soda A.

Find the net total expense of purchasing soda bottles for these three months, considering the varying discounts and taxes.
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Context:
3 设X={0,1,,2,⋯,9},$F = \{ A _ { 1 } ,$,$A _ { 2 } , \cdots , A _ { k } \}$中的每个元素$A _ { i }$都是X 的非空子集，且对任何1≤i<j≤k，有$\vert A _ { i } \cap A _ { j } \vert \leq 2$，求k的最大值.（第26届IMO备选题）
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Context:
If $P = \{1, 2\}$, $Q = \{1, a^2\}$, and $P = Q$, then find the value of $a$. Express your answer as a mathematical expression, including the ± symbol if necessary.
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Context:
Let \( ABC \) be a triangle with circumcenter \( O \) such that \( AC = 7 \). Suppose that the circumcircle of \( AOC \) is tangent to \( BC \) at \( C \) and intersects the line \( AB \) at \( A \) and \( F \). Let \( FO \) intersect \( BC \) at \( E \). Compute \( BE \).
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Context:
There are two identical cups, A and B. Cup A is half-filled with pure water, and cup B is fully filled with a 50% alcohol solution. First, half of the alcohol solution from cup B is poured into cup A and mixed thoroughly. Then, half of the alcohol solution in cup A is poured back into cup B. How much of the solution in cup B is alcohol at this point?
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Context:
For every integer n ≥ 3, the function g(n) is defined as the product of all the odd integers from 1 to n, inclusive. The difference between g(101) and g(some number) is 100. What is that some number?
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Context:
Determine the number of terms required in the Taylor series expansion of cos(x) to approximate cos(0.5) with a relative approximate error of less than 0.2%.
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Context:
Given a sequence $\{a_n\}$ that satisfies: for any $n\in\mathbb{N}^*$, we have $a_{n+1}=ka_n+3k-3$, where $k$ is a constant not equal to $0$ or $1$. If $a_i \in \{-678, -78, -3, 22, 222, 2222\}$ for $i=2, 3, 4, 5$, then the sum of all possible values of $a_1$ that meet the conditions is ______.
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Context:
Given the function $f(x)=a^{x}+x^{2}-x\ln a-b$ ($a > 1,b\in\mathbb{R}$), where $e$ is the base of the natural logarithm. If there exist $x_{1}, x_{2}\in[-1,1]$ such that $|f(x_{1})-f(x_{2})|\geqslant e-1$, then the range of values for the real number $a$ is ______. (Reference formula: $(a^{x})'=a^{x}\ln a$)
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Context:
In a global company with a diverse workforce, 60% are men, and 40% are women. Among the male employees, 40% speak only French, 20% speak only Spanish, 15% speak only German, 10% speak both French and Spanish, 5% speak both French and German, 5% speak both Spanish and German, and 5% speak all three languages. Among the female employees, 30% speak only French, 25% speak only Spanish, 20% speak only German, 10% speak both French and Spanish, 5% speak both French and German, 5% speak both Spanish and German, and 5% speak all three languages. What is the percentage of women in the company who do not speak French but speak one or more of the other two languages (Spanish and/or German)?
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Context:
Fnargs are either red or blue and have 2, 3, or 4 heads. A group of six Fnargs consisting of one of each possible form (one red and one blue for each number of heads) is made to line up such that no immediate neighbors are the same color nor have the same number of heads. How many ways are there of lining them up from left to right?
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Context:
In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points.
A natural number $m$ is called [i]good[/i] if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on.
Find the minimum value of a [i]good[/i] number $m$.
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Context:
Twenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered 1 through 10 in some order. Each block in layers 2,3 and 4 is assigned the number which is the sum of numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block.
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Context:
32at8 ** Let $a, b, c \in \mathbf{R}^{+}, a+b+c=1$, find the minimum value of $\frac{a+3 c}{a+2 b+c}+\frac{4 b}{a+b+2 c}-$ $\frac{8 c}{a+b+3 c}$.
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Context:
Two concentric circles have radii of $12$ meters and $24$ meters respectively. A rabbit starts at point X and runs along a sixth of the circumference of the larger circle, followed by a straight line to the smaller circle, then along a third of the circumference of the smaller circle, and finally across the diameter of the smaller circle back to the closest point on the larger circle's circumference. Calculate the total distance the rabbit runs.
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Context:
$x_{1}, x_{2}, x_{3}, x_{4}$ 为互不相等的正实数, $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 为 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意顺序排列, $X=\max \left\{\min \left\{x_{i 1}, x_{i 2}\right\}, \min \left\{x_{i 3}, x_{i 4}\right\}, Y=\min \left\{\max \left\{x_{i 1}, x_{i 2}\right\}, \max \left\{x_{i 3}, x_{i 4}\right\}\right.\right.$, 求 x 大于 y 的概率.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
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Context:
Using squares of side 1, a stair-like figure is formed in stages following a specific pattern. For example, the first stage uses 1 square, the second stage uses 5 squares, and so on. Determine the last stage for which the corresponding figure uses less than 2014 squares.
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Context:
A line whose $y$-intercept is $(0,8)$ intersects the ellipse $4x^2 + 25y^2 = 100.$ Find all possible slopes of this line.
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\boxed{(-\infty, -\sqrt{2.4}] \cup [\sqrt{2.4}, \infty)}


Context:
Grandma tells Xiao Ming, "There are 53 Saturdays in 2016." The clever Xiao Ming immediately tells Grandma: The New Year's Day of 2017 must be on a \_\_\_\_\_\_.
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Context:
A square contains \( K \) points (\( K > 2 \)). What is the minimum number of triangles needed to divide the square so that each triangle contains at most one point?
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Context:
Let $S$ be the set of all nondegenerate triangles formed from the vertices of a regular octagon with side length 1. Find the ratio of the largest area of any triangle in $S$ to the smallest area of any triangle in $S$.
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Context:
Find all possible three-digit numbers that can be obtained by removing three digits from the number 112277. Sum them and write the result as the answer.
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Context:
10,11

During an interview, ten people were offered a test consisting of several questions. It is known that any five people together answered all the questions (that is, at least one of the five gave the correct answer to each question), but any four did not. What is the minimum number of questions for which this could have been the case?

#
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Context:
$n$ integers are arranged along a circle such that each number is equal to the absolute value of the difference of the two numbers following it in the clockwise direction. If the sum of all numbers is $278$, find how many different values $n$ can take.
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Context:
Triangle $ABC$ has side lengths $AB = 20$, $AC = 14$, and $BC = 22$. The median from $B$ intersects $AC$ at $M$, and the angle bisector from $C$ intersects $AB$ at $N$ and the median from $B$ at $P$. Let $\frac{p}{q} = \frac{[AMPN]}{[ABC]}$ for positive integers $p$ and $q$ that are coprime. Note that $[ABC]$ denotes the area of triangle $ABC$. Find $p + q$.
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Context:
In triangle $ABC$, $\angle C = \frac{\pi}{3}$, and let $\angle BAC = \theta$. If there exists a point $M$ on line segment $BC$ (distinct from points $B$ and $C$) such that $\triangle BAM$ is folded over line $AM$ to obtain $\triangle B'A'M$, and the condition $AB' \perp CM$ is satisfied, then determine the range of values for $\theta$.
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\boxed{\left( \frac{\pi}{6}, \frac{2\pi}{3} \right)}


Context:
The smallest number which, when diminished by 16, is divisible by some numbers, including 4, 8, and 10, and the result is 136. What is the second divisor?
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Context:
Let \( a_{1} = 3 \), and for \( n \geq 1 \), let \( a_{n+1} = (n+1) a_{n} - n \). Find the smallest \( m \geq 2005 \) such that \( a_{m+1} - 1 \mid a_{m}^{2} - 1 \).
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Context:
What is the least positive integer such that the product of its digits is $9!$?
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Context:
Given a quadratic function $f(x) = x^2 + ax + 5$, it holds that $f(t) = f(-4-t)$ for any $t$ and the function has a maximum value of 5 and a minimum value of 1 on the closed interval $[m, 0]$. Determine the range of values for $m$.
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Context:
Let $n \ge 2$ be an integer. Find the largest integer $k \ge 1$ with the following property: for any $k$ matrices: $A_1,A_2,...,A_k \in \mathcal{M}_n(\mathbb{C})$, if $I_n-A_1A_2...A_k$ is invertible, then so is $I_n-A_{\pi(1)}A_{\pi(2)}...A_{\pi(k)}$ for every permutation $\pi \in S_k$.
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Context:
Anna joined coaching on the first day of the year. Anna took coaching every day without fail till Anna discontinued on September 4. How many days Anna took coaching, if the year is not a leap year?
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Context:
Given \(x\) and \(y\) are real numbers, and:
\[
z_1 = x + (y + 2)i,
\]
\[
z_2 = (x - 2) + yi,
\]
with the condition \(\left|z_1\right| + \left|z_2\right| = 4\),

Find the maximum value of \(|x + y|\).
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Context:
Evaluate the limit $$\lim\limits_{n\to\infty}(f(n+1)-f(n))$$ where $f(n)=|B_{2n}|^{1/2n}$ and $B_{2n}$ is the $(2n)$-th Bernoulli's number.
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Context:
Given a natural number \( n \), consider the function
\[ f_n(x, y) = x^n + x^{n-1}y + x^{n-2}y^2 + \dots + xy^{n-1} + y^n \]
of two real variables. Determine the minimum number \( k \) for which there exist functions \( g_1, \dots, g_k \) and \( h_1, \dots, h_k \) of one real variable such that
\[ f_n(x, y) = \sum_{i=1}^k g_i(x)h_i(y). \]
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Context:
11.1. In each of the four volleyball teams, there are six players, including a captain and a setter, and these are different people. In how many ways can a team of six players be formed from these four teams, such that there is at least one player from each team and there must be a pair of captain and setter from at least one team?
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Context:
An isosceles trapezoid \(ABCD\) with the larger base \(AB\) is circumscribed around a circle that touches side \(BC\) at point \(M\). Segment \(AM\) intersects the circle at point \(N\). Find the ratio \(\frac{AB}{CD}\) if \(\frac{MN}{AN}=k\).
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Context:
The area of triangle $ABC$ is $2007$. One of its sides has length $18$, and the tangent of the angle opposite that side is $\frac{2007}{24832}$. When the altitude is dropped to the side of length $18$, it cuts that side into two segments. Find the sum of the squares of those two segments.
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Context:
In the election for the class president, Petya and Vasya competed. Over three hours, 27 students voted for one of the two candidates. During the first two hours, Petya received 9 more votes than Vasya. In the last two hours, Vasya received 9 more votes than Petya. In the end, Petya won. By the largest possible margin, how many more votes could Petya have received than Vasya?
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Context:
The hyperbola $C: x^{2}-y^{2}=2$ has its right focus at $F$. Let $P$ be any point on the left branch of the hyperbola, and point $A$ has coordinates $(-1,1)$. Find the minimum perimeter of $\triangle A P F$.
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Context:
9. How many 9 -term sequences $a_{1}, \ldots, a_{9}$ of nonnegative integers are there such that
- $0 \leq a_{i}<i$ for all $i=1, \ldots, 9$; and
- there are no ordered triples $(i, j, k)$ with $1 \leq i<j<k \leq 9$, such that $a_{i}, a_{j}, a_{k}$ are all distinct?
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Context:
A sequence of complex numbers $z_{0}, z_{1}, z_{2}, ...$ is defined by the rule \[z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},\] where $\overline {z_{n}}$ is the complex conjugate of $z_{n}$ and $i^{2}=-1$. Suppose that $|z_{0}|=1$ and $z_{2005}=1$. How many possible values are there for $z_{0}$?
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Context:
How many different patterns can be made by shading exactly two of the nine squares?  Patterns that can be matched by flips and/or turns are not considered different.  For example, the patterns shown below are not considered different.
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Context:
Given the proposition $p$: "The equation $x^{2}+mx+1=0$ has two real roots"; and the proposition $q$: "$\forall x \in \mathbb{R}$, $4x^{2}+4(m-2)x+1 \neq 0$", if $p \wedge q$ is false, and $\neg q$ is false, find the range of values for the real number $m$.
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Context:
在椭圆 $\Gamma$ 中, $F$ 为一个焦点, $A, B$ 为两个顶点. 若 $|F A|=3,|F B|=2$,求 $|A B|$ 的所有可能值。请提供所有可能值平方和的值。
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Context:
$a$ , $b$ , $c$  are real. What is the highest value of  $a+b+c$  if  $a^2+4b^2+9c^2-2a-12b+6c+2=0$
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Context:
Given propositions $p$ and $q$:

- $p$: The function $f(x) = \frac{1}{3}x^3 - x^2 + (5 - a^2)x + a$ is increasing on $\mathbb{R}$.
- $q$: The function $g(x) = \frac{e^x}{x}$ is strictly increasing on $[a, +\infty)$.

If the proposition "$p \vee (\neg q)$" is true, then "$(\neg p) \vee q$" is also true. Find the range of values for $a$.
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Context:
Given a triangle \(ABC\) with \(\angle C = 120^{\circ}\). Point \(D\) is the foot of the perpendicular dropped from point \(C\) to side \(AB\); points \(E\) and \(F\) are the feet of the perpendiculars dropped from point \(D\) to sides \(AC\) and \(BC\) respectively. Find the perimeter of triangle \(ABC\) given that triangle \(EFC\) is isosceles and its area is \(\sqrt{3}\).
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Context:
Jeffrey rolls three fair six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as \( \frac{m}{n} \) for relatively prime positive integers \( m \) and \( n \). Compute \( m+n \).
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Context:
Matt did his homework for 150 minutes. He spent 20% of that time on math, 25% on science, 10% on history, and 15% on English. He had to spend at least 30 minutes on each remaining subject, and he could only spend the remaining time on other subjects. How much time did Matt spend on homework in other subjects?
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Context:
Archie received his medical receipt from his doctor, and he needs to take two different types of antibiotics for a week. The first antibiotic, Antibiotic A, costs $3 per dose and he needs to take it twice a day on Monday, Wednesday, and Friday. The second antibiotic, Antibiotic B, costs $4.50 per dose and he needs to take it once a day on Tuesday, Thursday, Saturday, and Sunday. Additionally, there is a promotion available: if Archie buys 10 doses of Antibiotic A, he will get a 20% discount on those doses. How much money does Archie need to spend on the antibiotics for the week, considering the discount?
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Context:
The largest divisor of $2,014,000,000$ is itself. What is its fifth-largest divisor?
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Context:
[The Pigeonhole Principle (continued).]

In a photo studio, 20 birds flew in - 8 sparrows, 7 wagtails, and 5 woodpeckers. Each time the photographer clicks the camera shutter, one of the birds flies away (permanently). How many shots can the photographer take to be sure: he will have at least four birds of one species left, and at least three of another?

#
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Context:
"a > 0" is a \_\_\_\_\_\_\_\_\_\_\_ condition for "a^2 + a ≥ 0".
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Context:
Given that $\sin(\alpha - 70^\circ) = \alpha$, find the value of $\cos(\alpha + 20^\circ)$.
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Context:
Evaluate the following limit:
$$\lim_{x \to 0^{+}} \left[\frac{\exp(\sin x \log x) - \exp(x \log \sin x)}{x^3} + \frac{\log x}{6}\right]$$
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Context:
There are $n\geq 3$ cities in a country and between any two cities $A$ and $B$, there is either a one way road from $A$ to $B$, or a one way road from $B$ to $A$ (but never both). Assume the roads are built such that it is possible to get from any city to any other city through these roads, and define $d(A,B)$ to be the minimum number of roads you must go through to go from city $A$ to $B$. Consider all possible ways to build the roads. Find the minimum possible average value of $d(A,B)$ over all possible ordered pairs of distinct cities in the country.
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Context:
4. The numbers $x$ and $y$ are such that the equations $\sin y + \sin x + \cos 3x = 0$ and $\sin 2y - \sin 2x = \cos 4x + \cos 2x$ are satisfied. What is the greatest value that the sum $\cos y + \cos x$ can take?
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Context:
Point $P_{1}(1, y_{1})$ and point $P_{2}(-3, y_{2})$ are two points on the graph of the linear function $y=-4x+3$. The relationship between $y_{1}$ and $y_{2}$ is ______.
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